KMS Meeting - Program and Abstracts
– Click title to show/hide contents.
– Code preceding talk title indicates date-slot-time; for example, "25th-A-09:00 - 09:20" means the talk will be given on 25th, at slot A on time 09:00 - 09:20.
- Plenary Lectures
- ⋅ 21st-O-10:10 − 11:00 Overcoming the curse of dimensionality for Hamilton-Jacobi equations with applications to control and differential games (Stanley Osher, Jerome Darbon, Yat-Tin Chow, Wotao Yin )
- Stanley Osher*, UCLA, Jerome Darbon, Brown University, Yat-Tin Chow, UCLA, Wotao Yin, UCLA
It is well known that certain Hamilton-Jacobi partial differential equations (HJ PDE's) play an important role in analyzing control theory and differential games. The cost of standard numerical algorithms for HJ PDE's is exponential in the space dimension and time, with huge memory requirements. Here we propose and test methods for solving a large class of these problems without the use of grids or significant numerical approximation. We begin with the classical Hopf and Hopf-Lax formulas which enable us to solve state independent problems via variational methods originating in compressive sensing with remarkable results.
We can evaluate the solution in $10^{-4}$ to $10^{-8}$ seconds per evaluation on a laptop. The method is Embarrassingly parallel and has low memory requirements.
Recently, with a slightly more complicated, but still embarrassingly parallel, we have extended this in great generality to state dependent HJ equations, apparently, with the help of parallel computers, overcoming the curse of dimensionality for these problems.
The term, ``curse of dimensionality” was coined by Richard Bell man in 1957 when he did his classic work on dynamic optimization.
2010 Mathematics Subject Classification: 35F21, 49J04, 65K15
Key Words and Phrases: Hamilton-Jacobi, Hopf-Lax formulas, control, game, variational, parallel, convex optimization
- ⋅ 21st-O-11:20 − 12:10 Rational curves on algebraic varieties-minimal models and extremal rays (Shigefumi Mori)
- Shigefumi Mori, KUIAS/RIMS, Kyoto University
In my talk I will present my personal views on the area around my research; I have been studying algebraic varieties through rational curves on them. I was first interested in a special problem called the Hartshorne Conjecture, and when I solved it I encountered a notion called an extremal ray as a biproduct, through which I got attracted to the biregular classification and the minimal model program, and furthermore to a general theory of higher dimensional birational classification. Reviewing them, I will also touch the study of 3-dimensional extremal contractions which I have been interested in.
2010 Mathematics Subject Classification: 14E30
Key Words and Phrases: extremal ray, minimal model program, terminal singularity, rational curve
- ⋅ 21st-O-14:00 − 14:50 Crossroads of symplectic rigidity and flexibility (Yakov Eliashberg)
- Yakov Eliashberg, Stanford University
In Mathematics there always exist two directions of explorations.
On the rigid side, mathematicians are concerned with finding constraints and restrictions, while on the flexible side they are developing techniques for new constructions revealing new properties which are often unexpected and counter-intuitive. In symplectic topology flexible and rigid methods shaped the development of the subject from its inception. In the talk, I will discuss some recent progress towards finding the border between rigid and flexible symplectic worlds.
2010 Mathematics Subject Classification: 57R17
Key Words and Phrases: symplectic topology
- Public Lecture
- ⋅ 20th-O-17:00 − 17:50 Mathematics and Physics: Companion forever (Dong Pyo Chi)
- 지동표(서울대 \& UNIST)
Dong Pyo Chi, Seoul National University \& UNIST
We discuss companionship between mathematics and physics old and present with several examples.
2010 Mathematics Subject Classification: 57R17
Key Words and Phrases: companionship between mathematics and physics
- Number Theory
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk] Kohnen plus space for Hilbert modular forms (Tamotsu Ikeda)
- Tamotsu Ikeda, Kyoto University
The classical Kohnen plus subspace is a subspace of the space $S_{k+1/2}(\Gamma_0(4))$ of modular forms of half-integral weight $k+1/2$.
A modular form of weight $k+1/2$ with $n$-th Fourier coefficient $c(n)$ belongs to the Kohnen plus space if and only if $c(n)=0$ unless $(-1)^k n$ is congruent to a square modulo $4$.
It is known that the Kohnen plus subspace is isomorphic to the space $S_{2k}(\Gamma(1))$ of modular forms of weight $2k$ by the Shimura correspondence.
In this talk, we give a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight.
2010 Mathematics Subject Classification: 11F37, 11F41
Key Words and Phrases: Hilbert modular forms, modular forms of half-integral weight, Kohnen plus space
- ⋅ 22nd-A-09:50 − 10:10 Mod $p$ local-global compatibility for ordinary Galois representations (Chol Park, Zicheng Qian)
- 박철*(고등과학원), Zicheng Qian(University Paris-Sud)
Chol Park*, KIAS, Zicheng Qian, University Paris-Sud
Let $F/\mathbb{Q}$ be a CM field in which $p$ splits completely and $\overline{r}:\mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow\mathrm{GL}_4(\overline{\mathbb{F}}_p)$ a continuous modular Galois representation. We assume that $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ is an ordinary Galois representation at a place $w$ above $p$. In this talk, we discuss a problem about local-global compatibility in the mod $p$ Langlands program for $\mathrm{GL}_4(\mathbb{Q}_p)$. It is expected that if $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ is tamely ramified, then it is determined by the set of modular Serre weights and the Hecke action on its constituents. However, this is not true if $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ is wildly ramified, and the question of determining $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ from a space of mod $p$ automorphic forms lies deeper than the weight part of Serre's conjecture. We define a local invariant associated to $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ in terms of Fontaine-Laffaille theory, and discuss a way to prove that the local invariant associated to $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_w)}$ can be obtained in terms of a refined Hecke action on a space of mod $p$ algebraic automorphic forms on a compact unitary group.
The talk is based on the ongoing joint work with Zicheng Qian.
2010 Mathematics Subject Classification: 11F80
Key Words and Phrases: mod $p$ Langlands programs, Serre weights, Galois representations, automorphic forms
- ⋅ 22nd-A-10:10 − 10:30 Arithmetic Siegel-Weil formula on $X_{0}(N)$ (Tuoping Du, Tonghai Yang)
- Tuoping Du*, PMI, POSTECH, Tonghai Yang, Wisconsin University
Kudla, Rapport and Tonghai give some certain quantities from the arithmetic of Shimura curves to the coefficients of Siegel Eisenstein series (or the derivatives of the Eisenstein series).
They constructed an interesting generating function for the arithmetic divisors on the arithmetic surface associated to the Shimura curve. Moreover, they found that the degree of this function equals to the Eisenstein series and the height of this function equals to the derivative, which all at the value $s=1$.
Tonghai and I give an formula on $X_{0}(N)$, we will show that the coefficients of one Eisenstein series are deree of Heegner diviors. Also the coefficients of derivaitve of this Eisenstein series are the heights of Heegner divisors.
2010 Mathematics Subject Classification: 11G15, 11F41, 14K22
Key Words and Phrases: modular curve, intersection number, Eisenstein series
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Period of automorphic forms-global and local aspects (Erez Lapid)
- Erez Lapid, Weizmann Institute of Science
I will review some recent progress about questions concerning periods of automorphic forms
and related local problems.
2010 Mathematics Subject Classification: 11F70
Key Words and Phrases: automorphic forms
- ⋅ 22nd-B-11:35 − 11:55 Two versions (weak and strong) of the generic Arthur packet conjecture for $GSpin$ groups (Yeansu Kim)
- 김연수(전남대)
Yeansu Kim, Chonnam National University
Langlands program is a set of conjectures that construct the bridge between two different areas: Number Theory (Galois representations) and Representation Theory (Automorphic forms). Recently, Heiermann and I have constructed the generic local Langlands correspondence for $GSpin$ groups (one main conjecture in the Langlands program). More precisely, we construct the local Langlands parameter that corresponds to an irreducible admissible generic representation of $GSpin$ groups. I further study the structure of the $L$-packet which contains a generic representation. As an application, I prove the strong version of the generic Arthur packet conjecture in the case of $GSpin$ groups. The strong version of the generic Arthur packet conjecture states that if the $L$-packet attached to an Arthur parameter has a generic member, then it is a tempered $L$-packet. If time permits, I will explain the case of classical groups. This case is in progress.
2010 Mathematics Subject Classification: 11F70
Key Words and Phrases: the generic Arthur packet conjecture, generic local Langlands correspondence
- ⋅ 22nd-B-11:55 − 12:15 The minimal number of Frobenius elements of $G_{K,S}$ whose conjugacy classes generate the whole group $G_{K,S}$ (Kwang-Seob Kim)
- 김광섭(고등과학원)
Kwang-Seob Kim, KIAS
Assume that $K$ is a number field and $S$ is a finite set of primes of $K$. Let $G_S(K)$ be the Galois group ${\rm Gal}(K_S/K)$, where $K_S$ is the maximal extension of $K$, which is unramified outside the primes in $S$. Suppose that the rank of $G^{ab_{K,S}}$ is $r$. In this article, we use topological methods to show that $G_{K,S}$ can be generated by $r$ (or 1 if $r = 0$) Frobenius classes and $r$ is minimal.
2010 Mathematics Subject Classification: 11R21
Key Words and Phrases: Frobenius elements, generators of $G_{K,S}$
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] The Gross-Keating invariant and its applications (Sungmun Cho, Takuya Yamauchi)
- 조성문*(Kyoto University), Takuya Yamauchi(Tohoku University)
Sungmun Cho*, Kyoto University, Takuya Yamauchi, Tohoku University
The Gross-Keating invariant (GK invariant) of a quadratic form over $\mathbb{Z}_p$ plays an important role to make a direct connection between arithmetic intersection numbers for some moduli stack and the Fourier coefficients of Siegel Eisenstein series for $\mathrm{Sp}_{6}$. Recently Ikeda and Katsurada described the Fourier coefficients in terms of GK invariant.
In this talk, after reviewing the definition and a formula of GK invariant, I will explain two major applications. Firstly, this would be used to prove an equality between some arithmetic intersection numbers and the Fourier coefficients of Siegel Eisenstein series for $\mathrm{Sp}_{8}$. Secondly, the local density (or the associated smooth integral group scheme) of a quadratic form over $\mathbb{Z}_p$ would be formulated in terms of GK invariant.
This is a joint work with Takuya Yamauchi.
2010 Mathematics Subject Classification: 11E08, 11F41, 14L15, 20G25
Key Words and Phrases: Gross-Keating invariant, arithmetic intersection number, Siegel Eisenstein series, local density, smooth integral model
- ⋅ 22nd-C-14:20 − 14:40 Arithmetic Chern-Simons theory (Hwajong Yoo, Minhyong Kim, Dohyeong Kim, Jeehoon Park, Hee-Joong Chung)
- 유화종*(기초과학연구원), 김민형(Univ. of Oxford \& KIAS CMC), 김도형(Univ. of, Michigan, Ann Arbor), 박지훈(포항공대), 정희중(고등과학원)
Hwajong Yoo*, IBS-CGP, Minhyong Kim, University of Oxford/KIAS CMC, Dohyeong Kim, University of Michigan, Ann Arbor, Jeehoon Park, POSTECH, Hee-Joong Chung, KIAS
We present Arithmetic Chern-Simons theory and its arithmetic application. By using computation formula for Chern-Simons invariants, we prove non-solvability of certain embedding problems in inverse Galois problem.
2010 Mathematics Subject Classification: 11R04, 11R23, 11R29
Key Words and Phrases: Arithmetic Chern-Simons theory, Iwasawa theory, inverse Galois problem
- ⋅ 22nd-C-14:40 − 15:00 On the calculation of the number of Galois orbits (Hyunsuk Moon)
- 문현숙(경북대)
Hyunsuk Moon, Kyungpook National University
Let $A$ be an abelian variety over a global field $K$. We know that Chen-Kuan's invariant $M(A[n])$, that is the average value of $n$-torsion points of $A$ over various residue field of $K$, is equal to the number of orbits of the absolute Galois group. In this talk, we explain that in many cases, $M(A[n])$ has the minimal possible value and we compute several defect cases by counting the number of Galois orbits.
2010 Mathematics Subject Classification: 11F80, 11G05
Key Words and Phrases: Galois representations, Galois orbits, Galois images
- ⋅ 22nd-D-15:15 − 15:35 Simple zeros of automorphic L-functions (Jaehyun Cho(Peter), Myoungil Kim, Andrew Booker)
- 조재현*(울산과학기술원), 김명일(울산과학기술원), Andrew Booker(University of Bristol)
Jaehyun Cho(Peter)*, UNIST, Myoungil Kim, UNIST, Andrew Booker, University of Bristol
It is believed that all autmorphic L-functions not only satisfy the Generalized Riemann Hypothesis and their zeros are simple if there is no geometric reason. It is known that more than 40$\%$ of zeros of the Riemann zeta function are located on the line $re(s)=\frac{1}{2}$ and they are simple. For the case of degee 2 L-functions, not much about it is known. Recently, A. Booker showed that L-functions attached to Hecke modular forms have infinitely many simple zeros. We (Booker, Cho, and Kim) extend this result to the Maass forms. The case of weight 0 is done and the case of weight one is in process.
2010 Mathematics Subject Classification: 11M26
Key Words and Phrases: automorphic L-functions, simple zeros
- ⋅ 22nd-D-15:35 − 15:55 Generation of ray class fields modulo 2 or 3 by using the Weber function (Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin)
- 정호윤*(성균관대 응용대수 및 최적화 연구센터), 구자경(한국과학기술원), 신동화(한국외대)
Ho Yun Jung*, Sungkyunkwan University, AORC , Ja Kyung Koo, KAIST, Dong Hwa Shin, Hankuk University of Foreign Studies
Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_K$, and let $h_E$ be the Weber function on $E$. Let $N\in\{2,~3\}$. We show that the Weber function alone when evaluated at a certain $N$-torsion point on $E$ generates the ray class field of $K$ modulo $N\mathcal{O}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.
2010 Mathematics Subject Classification: 11R37, 11G15, 11G16
Key Words and Phrases: class field theory, complex multiplication, Weber function
- ⋅ 22nd-D-16:05 − 16:25 On the structure of certain valued fields (Junguk Lee, Wan Lee)
- 이정욱*(연세대), 이완(연세대)
Junguk Lee*, Yonsei University, Wan Lee, Yonsei University
For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if each pair of $n$-th residue rings is isomorphic for each $n\ge1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. We can find a lower bound for $n_2$ depending only on $K_2$. Moreover, we get a functor from a category of certain principal Artinian local rings of length $n$ to a category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. The result improves Basarab's generalization of the AKE-principle for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields. This is joint work with Wan Lee.
2010 Mathematics Subject Classification: 11U09, 03C20, 13L05
Key Words and Phrases: finitely ramified valued fields, functorial property of the ring of Witt vectors, Krasner's lemma, lifting number, Ax-Kochen-Ershov prinicple
- ⋅ 22nd-D-16:25 − 16:45 Genus-correspondences respecting spinor genus (Jangwon Ju, Byeong-Kweon Oh)
- 주장원*(서울대), 오병권(서울대)
Jangwon Ju*, Seoul National University, Byeong-Kweon Oh, Seoul National University
For two positive definite integral ternary quadratic forms $f$ and $g$ and a positive integer $n$, if $n\cdot g$ is represented by $f$ and $n\cdot dg=df$, then the pair $(f,g)$ is called a representable pair by scaling $n$. The set of all representable pairs in $\text{gen}(f)\times \text{gen}(g)$ is called a genus-correspondence. Jagy conjectured that if $n$ is square free and the number of spinor genera in the genus of $f$ equals to the number of spinor genera in the genus of $g$, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs $(f,g), (f',g')$ by scaling $n$, $f' \in \text{spn}(f)$ if and only if $g' \in \text{spn}(g)$. In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.
2010 Mathematics Subject Classification: 11E12, 11E20
Key Words and Phrases: genus-correspondence, spinor genus
- ⋅ 22nd-E-17:00 − 17:20 Bounds for ranks of rational points of abelian varieties and elliptic curves over $F(\mu_{p^{\infty}})$ when $p$ is ramified over $F/\mathbb Q$, and the reduction type is non-ordinary/super\-singular (Byoung Du Kim)
- 김병두(Victoria University of Wellington)
Byoung Du Kim, Victoria University of Wellington
Efforts to study elliptic curves and the Galois representations attached to modular forms are often hampered by reduction types. Especially for Iwasawa Theory, good ordinary reduction and split multiplicative reduction are comparatively easier to study, and good supersingular/non-ordinary reduction and additive/potentially semistable reduction are considered harder.
The model that everyone wants to emulate is Barry Mazur's ``Rational Points of Abelian Varieties
with Values in Towers of Number Fields'', Inventiones Math. 18, 183--266 (1972). In it, assuming $p$ is good ordinary or multiplicative, he showed that one can precisely express the meanings of the characteristic ideals of the $p$-adic Selmer groups of elliptic curves, their $p$-adic $L$-functions, and the relation between the two (i.e., a \textit{Main Conjecture}). In particular, if the $p$-adic $L$-functions are not 0, then the rational points over $\mathbb Q(\mu_{p^{\infty}})$ have a finite rank.
In this presentation, I will show my work on the ranks of the rational points of elliptic curves and abelian varieties over $\mathbb Z_p$-extensions of number fields $F$ when the reduction type is good \textit{supersingular/non-ordinary}. The biggest difference (apart from the methods) between my work and the existing work is that I do not assume that $p$ is unramified over the number fields $F$, therefore my result applies to a larger class of elliptic curves and abelian varieties.
First, I show a weak bound for the ranks of rational points over $F(\mu_{p^n})$ of abelian varieties over $F$ over which $p$ is ramified as $n\to \infty$ by building up an Iwasawa Theory for non-ordinary primes, and second, I establish a stronger Iwasawa Theory for elliptic curves over $F$ over which $p$ is ramified under certain conditions, and show a finite bound for the ranks of the rational points over $F(\mu_{p^{\infty}})$.
2010 Mathematics Subject Classification: 11G, 11R
Key Words and Phrases: ranks of rational points, Iwasawa theory for ramified primes, non-ordinary reduction, supersingular reduction
- ⋅ 22nd-E-17:20 − 17:40 The class numbers of ternary quadratic forms (Toshiya Kawakubo, Byeong-Kweon Oh)
- Toshiya Kawakubo*(서울대), 오병권(서울대)
Toshiya Kawakubo*, Seoul National University, Byeong-Kweon Oh, Seoul National University
Let $K$ be a positive definite ternary $\mathbb{Z}$-lattice and let
$L$ be a $\mathbb{Z}$-lattice such that $\lambda_2(L)\cong K$, where $\lambda_2$ is a Watson transformation. In this talk, we count the number of isometry classes in the genus of $L$ that are transformed to $K$ by $\lambda_2$-transformation. We provide an inductive method on computing class number of an arbitrary ternary lattice. As a simple, we give a complete closed formula for the class number of a Bell ternary form. This is a joint work with B. K. Oh.
2010 Mathematics Subject Classification: 11E12, 11E20, 11E41
Key Words and Phrases: class numbers, quadratic forms
- ⋅ 22nd-E-17:50 − 18:10 The number of representations of squares by integral ternary quadratic forms (II) (Kyoungmin Kim, Byeong-Kweon Oh)
- 김경민*(서울대), 오병권(서울대)
Kyoungmin Kim*, Seoul National University, Byeong-Kweon Oh, Seoul National University
Let $f$ be a positive definite ternary quadratic form. We assume that $f$ is non-classic integral, that is, the norm ideal of $f$ is $\mathbb{Z}$. We say $f$ is {\it strongly $s$-regular } if the number of representations of squares of integers by $f$ satisfies the condition in Cooper and Lam's conjecture. In this article, we prove that there are only finitely many strongly $s$-regular ternary forms up to isometry if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly $207$ non-classic integral strongly $s$-regular ternary forms that represent one. This result might be considered as a complete answer to a natural extension of Cooper and Lam's conjecture.
2010 Mathematics Subject Classification: 11E12, 11E20
Key Words and Phrases: representations of ternary quadratic forms, squares
- ⋅ 22nd-E-18:10 − 18:30 Sturmian words and cantor sets arising from unique expansions over ternary alphabets (DoYong Kwon)
- 권도용(전남대)
DoYong Kwon, Chonnam National University
Over a finite alphabet $A$ of real numbers, unique expansions in base $\beta$ are considered. A real number $G_A$ called the generalized golden ratio is a border of situation of unique expansions. If $\beta<G_A$ then there are only trivial unique expansions in base $\beta$, while we have non-trivial unique expansions in base $\beta$ whenever $\beta>G_A$. Komornik, Lai, and Pedicini (2011) investigated the case where $A$ consists of three real numbers, and demonstrated that Sturmian words curiously emerge out of the generalized golden ratio. The present paper focuses on Sturmian words under this context. For a given alphabet $A=\{a_1,a_2,a_3\}$ with $a_1<a_2<a_3$, we give a complete characterization of the corresponding Sturmian words effectively and algorithmically, which reveals interesting structures behind the generalized golden ratios.
2010 Mathematics Subject Classification: 11A63
Key Words and Phrases: Sturmian word, $\beta$-expansion, univoque sequence, ternary alphabet
- Representation Theory
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Heller lattices and AR quivers (Susumu Ariki)
- Susumu Ariki, Osaka University
In modern mathematics, answer to classification problem of indecomposable matrix representations of an algebra is given by the Auslander-Reiten quiver. If the algebra is over a field, there is a wealth of concrete examples. Further, general theory was already developed by Auslander himself, for algebras whose base ring is not a field. However, few examples may be found in the literature even in the case when the base ring is a complete discrete valuation ring. In this talk, we propose a research direction to find concrete examples, that is, to determine the shape of components of AR quivers of lattices over Brauer graph algebras over a discrete valuation ring that contain Heller lattices. Then we give two examples. The first one was obtained in joint work with R. Kase and K. Miyamoto, and the second one was given by K. Miyamoto.
2010 Mathematics Subject Classification: 16G20, 16G30, 16G70
Key Words and Phrases: Heller lattice, AR quiver, Brauer graph algebra
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Jucys-Murphy elements of the walled Brauer algebra (Ji Hye Jung, Myungho Kim)
- 정지혜, 김명호*(경희대)
Ji Hye Jung,, Myungho Kim*, Kyung Hee University
In this talk, I will introduce a family of commuting elements of the walled Brauer algebra, called the Jucys-Murphy elements. This family is a variant of the ones introduced by Brundan-Stroppel and Sartori-Stroppel. As similar in the case of symmetric groups, the supersymmetric polynomials in the Jucys-Murphy elements belong to the center of the walled Brauer algebra. We show that if the walled Brauer algebra is semisimple, then these supersymmetric polynomials generate the center. This fact enables us to mimic the approach of Okounkov-Vershik on the representation theory of symmetric groups. Furthermore, we have an analogue of the Jucys-Murphy elements for the quantized walled Brauer algebra and a similar connection between the supersymmetric polynomials in the Jucys-Murphy elements and the center. Interestingly enough, this connection was already observed by H. Morton in 2001 in terms of HOMFLYPT skein on the annulus, and our result gives a proof of a conjecture of Morton. This is a joint work with Ji Hye Jung and was posted on arXiv:1508.06469.
2010 Mathematics Subject Classification: 16U70, 16T30, 05E05
Key Words and Phrases: Jucys-Murphy elements, walled Brauer algebras, supersymmetric polynomials
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Pieri-type tensor products for classical groups (Sangjib Kim)
- 김상집(고려대)
Sangjib Kim, Korea University
The Pieri rule in representation theory describes the decomposition of tensor product of finite dimensional irreducible representation of the general linear group $GL_n(\mathbb{C})$ and the symmetric power $Sym^k(\mathbb{C}^n)$ of $\mathbb{C}^n$. I will construct a family of algebras encoding the decomposition of Pieri-type tensor products for the symplectic and orthogonal groups (within some stable range) and then investigate their algebraic and combinatorial structures.
2010 Mathematics Subject Classification: 2G05, 13A50
Key Words and Phrases: Pieri rule, representation, classical group
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Covariant differential operators and Heckman-Opdam hypergeometric systems (Hiroyuki Ochiai)
- Hiroyuki Ochiai, Kyushu University
This is a joint work with Tomoyoshi Ibukiyama and Takako Kuzumaki. We consider holomorphic linear differential operators with constant coefficients acting on Siegel modular forms, which preserve the automorphy when restricted to a subdomain. We give a characterization of the symbols of such differential operators, and mention an explicit form in terms of hypergeometric functions with respect to root systems introduced by G. Heckman and E. Opdam.
2010 Mathematics Subject Classification: 11F60
Key Words and Phrases: covariant differential operator, Siegel modular form, hypergeometric function
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Quantization and semiclassical limit (Sei-Qwon Oh)
- 오세권(충남대)
Sei-Qwon Oh, Chungnam National University
Weyl introduced a quantization to explain a mechanical problem appearing in classical mechanics and many people have developed this method called a quantization. In general, algebraic structures appearing in classical mechanics are Poisson algebras $A$ and their quantizations are noncommutative algebras obtained by deforming the Poisson bracket in $A$. Here quantizations and their reversing processes, semiclassical limits, are explained. Moreover a natural map between quantized spaces and their semiclassical limits and recent results are introduced.
2010 Mathematics Subject Classification: 17B63
Key Words and Phrases: Poisson algebra, quantization, semiclassical limit
- ⋅ 22nd-D-16:05 − 16:45 [Invited Talk] Quantum Teichm\"uller space from the quantum plane (Igor Frenkel, Hyun Kyu Kim)
- Igor Frenkel(Yale University), 김현규*(고등과학원)
Igor Frenkel, Yale University, Hyun Kyu Kim*, KIAS
The quantum plane is one of the most basic non-commutative non-cocommutative Hopf algebra. It has a unique irreducible `integrable' representation, say $H$. The tensor product of $H$ with itself is a representation of the quantum plane via the coproduct, and it decomposes into the direct integral of $H$. This decomposition map yields a unitary map $T$ between the two possible decompositions of the tensor cube of $H$. For the tensor fourth power, there are five ways of decomposing, leading to the quantum pentagon relation of $T$. On the other hand, the maps between different decompositions of tensor powers of $H$ can be graphically encoded using triangulations of polygons. We show that the operator $T$ coincides with the operator associated to the elementary change of triangulations in the quantum Teichmüller theory.
2010 Mathematics Subject Classification: 16T05, 17B37, 20G42
Key Words and Phrases: quantum Teichm\"uller theory, quantum plane, representation theory, quantum groups
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Explicit bases of some highest weight modules of the general linear algebra (Soo Teck Lee)
- Soo Teck Lee, National University of Singapore
Let $n,p,q$ be positive integers, and let $V_{n,p,q}=(\mathbb{C}^n)^p\oplus
(\mathbb{C}^{n\ast})^q$ be the $\mathrm{GL}_n$ module formed by taking the direct sum of $p$ copies of the standard module $\mathbb{C}^n$ of $q$ copies of the dual $\mathbb{C}^{n\ast}$. Then the algebra $ \mathcal{P}(V_{n,p,q})$ of polynomial functions on $V_{n,p,q} $ is a module for $\mathrm{GL}_n\times \mathfrak{gl}_{p+q}$, and it admits the decomposition
\[ \mathcal{P}(V_{n,p,q})\cong \bigoplus_\lambda \rho^\lambda\otimes \sigma^\lambda\]
where for each $\lambda$, $\rho^\lambda$ is an irreducible finite dimensional representation of $\mathrm{GL}_n$ and $\sigma^\lambda$ is an irreducible highest weight module of $\mathfrak{gl}_{p+q}$. In this talk, we shall construct a basis for each of the highest weight module $\sigma^\lambda$.
2010 Mathematics Subject Classification: 20G05
Key Words and Phrases: general linear algebra, highest weight module
- ⋅ 23rd-F-09:50 − 10:30 [Invited Talk] Nahm's conjecture and representation theory of classical and quantum affine algebras (Chul-hee Lee)
- 이철희(The Univ. of Queensland)
Chul-hee Lee, The University of Queensland
Finding a criterion when a $q$-hypergeometric series can be a modular function is a widely open problem in number theory. Nahm's conjecture is an attempt to give a partial answer to this. Nahm considered a question of when a certain $r$-fold $q$-hypergeometric series associated to a positive definite symmetric matrix $A$ is modular and made a conjecture relating this question to an equation associated to $A$ and some properties of its solutions in terms of algebraic K-theory, motivated by mathematical physics.
In this talk, I will focus on a certain family of $q$-hypergeometric series, which fits into the framework of Nahm's conjecture and explain how it is related to integrable representations of affine Kac-Moody algebras and finite-dimensional representations of quantum affine algebras.
2010 Mathematics Subject Classification: 11P84, 20G42, 33D80
Key Words and Phrases: Nahm's conjecture, Rogers-Ramanujan identities, Kirillov-Reshetikhin modules, Q-systems, string functions
- ⋅ 23rd-G-10:45 − 11:25 [Invited Talk] W-algebras and related problems (Uhi Rinn Suh)
- 서의린(한국과학기술원)
Uhi Rinn Suh, KAIST
W-algebras are studied in many fields of mathematics. For example, classical (affine) W-algebras are underlying algebraic structures of some Hamiltonian integrable systems and quantum (affine and finite) W-algebras are interesting objects in representation theories. Algebraic structures of classical W-algebras, such as free generators and Poisson brackets, are completely discovered recently. On the other hand, there are still many open problems associated to algebraic structures of quantum W-algebra. In this talk, I will introduce classical and quantum W-algebras and relations between them. Also, I will mention interesting results and open problems.
2010 Mathematics Subject Classification: 17B69, 17B08, 17B63, 17B35, 17B80
Key Words and Phrases: W-algebra, vertex algebra
- ⋅ 23rd-G-11:35 − 12:15 [Invited Talk] Twisted Coxeter elements, the longest elements of Weyl group and denominator formulas (Se-Jin Oh, Uhi Rinn Suh)
- 오세진*(이화여대), 서의린(한국과학기술원)
Se-Jin Oh*, Ewha Womans University, Uhi Rinn Suh, KAIST
We study the reduced expressions of the longest element of simply laced finite Weyl groups arising from
twisted Coxeter elements. As an application, As applications of the study, we prove that folded AR-quivers encode crucial information on the representation theory of quantum affine algebra of non-simply laced type such as Dorey's rule and denominator formulas.
2010 Mathematics Subject Classification: 81R50, 05E10, 16T30, 17B37
Key Words and Phrases: twisted Coxeter elements, Dorey's rule, denominator formulas
- Algebraic Geometry
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk] Components rigid in moduli and the irreducibility of the Hilbert scheme of smooth projective curves (Changho Keem)
- 김창호(서울대)
Changho Keem, Seoul National University
Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves,
which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in ${\mathbb P}^r$.
A component of $\mathcal{H}_{d,g,r}$ is rigid in moduli if its image under the natural map $\pi:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ is a one point set. In this talk, we discuss about the fact that $\mathcal{H}_{d,g,r}$ has no components rigid in moduli for $g > 0$ and $r=3$. In case $r \geq 4$, we also discuss the non-existence of a component of $\mathcal{H}_{d,g,r}$ rigid in moduli in a certain restricted range of $d$, $g>0$ and $r$. These results are partly by products of the irreducibility of $\mathcal{H}_{d,g,3}$ beyond the range which has been known before.
2010 Mathematics Subject Classification: Primary 14H10; Secondary 14C05
Key Words and Phrases: Hilbert scheme, algebraic curves, linear series, component rigid in moduli
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Higher dimensional projective manifolds with primitive automorph\-isms of positive entropy (Keiji Oguiso)
- Keiji Oguiso, University of Tokyo \& KIAS
We explain how one can apply basic tools in birational geometry and complex dynamics to study primitive automorphisms of projective manifolds, a kind of irreducible automorphisms. In particular, we show that, in any dimension greater than one, there are abelian varieties with primitive automorphisms of positive topological entropy, and smooth complex projective Calabi-Yau manifolds and smooth rational manifolds, of any even dimension, with primitive biregular automrphisms of positive topological entropy.
2010 Mathematics Subject Classification: 14E07, 14J50, 37A35
Key Words and Phrases: Primitive automorphisms, topological entropy, Calabi-Yau manifolds, rational manifolds
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Automorphisms of K3 surfaces (JongHae Keum)
- 금종해(고등과학원)
JongHae Keum, KIAS
K3 surfaces are 2-dimensional Calabi-Yau manifolds.
I will start with standard examples of K3 surfaces, then review basic result and recent progress on automorphisms of K3 surfaces, e.g.,
how to determine the finiteness of the full automorphism group of a given K3 surface,
how to compute the automorphism group for some nice classes of K3 surfaces, what are the possible orders of automorphisms, etc.
I will also include the case of K3 surfaces defined over fields of positive characteristic.
2010 Mathematics Subject Classification: 14J28
Key Words and Phrases: K3 surfaces, automorphisms
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Globally F-regular type of moduli spaces of parabolic sheaves (Xiaotao Sun, Mingshuo Zhou)
- Xiaotao Sun*, Chinese Academy of Sciences, Mingshuo Zhou, Hangzhou Dianzi University
A variety over a perfect field of characteristic $p>0$ is called globally F-regular if it is stably
Frobenius split along every effective divisor. A variety over a field of characteristic zero is called
globally F-regular type if its modulo $p$ reduction is globally F-regular for almost $p$. In this talk, I will
report a joint work with Mingshuo Zhou that moduli spaces of semi-stable parabolic bundles and
generalized parabolic sheaves on curves are globally F-regular type.
2010 Mathematics Subject Classification: 14J45, 14B05
Key Words and Phrases: moduli spaces, globally F-regular type, parabolic sheaves
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Infinitesimal automorphisms of cubic hypersurfaces (Jun-Muk Hwang)
- 황준묵(고등과학원)
Jun-Muk Hwang, KIAS
We study infinitesimal automorphisms of singular cubic hypersurfaces. The goal is to characterize the homaloidal cubic hypersurfaces in terms of the prolongations of the infinitesimal automorphisms. This study will be used to investigate the automorphism groups of projective Legendrian varieties.
2010 Mathematics Subject Classification: 14J50
Key Words and Phrases: cubic hypersurface, prolongation, Legendrian variety
- ⋅ 22nd-C-14:20 − 14:40 Existence of co-Higgs sheaves (Edoardo Ballico, Sukmoon Huh)
- Edoardo Ballico(Universita di Trento), 허석문*(성균관대)
Edoardo Ballico, Universita di Trento, Sukmoon Huh*, Sungkyunkwan University
A co-Higgs bundle is a generalized vector bundle on a complex manifold, considered as a generalized complex manifold, and it is introduced and developed by Hitchin and Gualtieri. In this talk, we report our recent results concerning (non)-existence of non-trivial nilpotent co-Higgs sheaves on smooth algebraic varieties. Then we introduce a logarithmic version of co-Higgs sheaves and describe some features of its stability.
2010 Mathematics Subject Classification: 14J60, 14D20, 53D18
Key Words and Phrases: co-Higgs bundle, logarithmic tangent bundle, nilpotent
- ⋅ 22nd-C-14:40 − 15:00 Bisecant curves on ruled surfaces (Insong Choe, Youngwook Choe, Seonja Kim, Euisung Park)
- 최인송*(건국대), 최영욱(영남대), 김선자(청운대), 박의성(고려대)
Insong Choe*, Konkuk University, Youngwook Choe, Yeungnam University, Seonja Kim, Chungwoon Universityy, Euisung Park, Korea University
The nef cone of a ruled surface $\mathbb{P} (E)$ associated to a stable bundle $E$ has extremal rays generated by classes of a fiber $\mathfrak f$ and a mnimal section $C_0$. We compute the minimal integer $b$ such that the class $2C_0 + b \mathfrak f$ contains a bisecant curve. Also we discuss its relation to the Lange stability of $Sym^2 E$.
2010 Mathematics Subject Classification: 14C20, 14J26, 14H60
Key Words and Phrases: ruled surfaces, moduli of vector bundles, bisecant curves, Segre invariants, elementary trasnformation
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Remarks on linear series on a reducible curve (Seonja Kim)
- 김선자(청운대)
Seonja Kim, Chungwoon University
One of interesting problems in algebraic curve theory is what or how many curves possess a specific projective morphisms. In this point of view, it is natural to study the sublocus $M_g(r,d)$ of the moduli space $M_g$ of genus $g$ curves whose general point corresponds to a smooth curve possessing a projective morphism to an $r$-dimensional projective space with degree $d$. The locus $M_g(r,d)$ is called a Brill-Noether locus of $M_g$.
If the Brill-Noether number $\rho :=g-(r+1)(g-d+r)$ is negative, then $M_g(r,d)$ has codimension at least one in the moduli space $ M_g$. In general, Brill-Noether loci are investigated by the theory of limit linear series on a reducible curve. In this talk, we consider the existence of smoothable limit linear series on a reducible curve which is given by two general moduli curves bridged by a chain of elliptic curves. This study can give some results concerning the relationship among Brill-Noether loci in the moduli space $M_g$.
2010 Mathematics Subject Classification: 14H51
Key Words and Phrases: linear series, Brill-Noether theory, reducible curve, moduli of curves
- ⋅ 22nd-D-16:05 − 16:25 Green's theorem and Gorenstein sequences (Jeaman Ahn, Juan C. Migliore, Yong-Su Shin)
- 안재만*(공주대), Juan C. Migliore(Univ. of Notre dame), 신용수(성신여대)
Jeaman Ahn*, Kongju National University, Juan C. Migliore, University of Notre dame, Yong-Su Shin, Sungshin Women's University
We study consequences, for a standard graded algebra, of extremal behavior in Green's Hyperplane Restriction Theorem. First, we extend his Theorem 4 from the case of a plane curve to the case of a hypersurface in a linear space. Second, assuming a certain Lefschetz condition, we give a connection to extremal behavior in Macaulay's theorem. We apply these results to show that $(1, 19, 17, 19, 1)$ is not a Gorenstein sequence, and as a result we classify the sequences of the form $(1, a, a-2, a, 1)$ that are Gorenstein sequences.
2010 Mathematics Subject Classification: Primary:13D40; Secondary:13H10, 14C20
Key Words and Phrases: Gorenstein sequence, Green's theorem, Hilbert function, Lefschetz condition, \linebreak Macaulay's Theorem
- ⋅ 22nd-D-16:25 − 16:45 Regularity and Multi-secant Lines (Eui Sung Park, Wan Seok Lee, Young Ho Woo)
- 박의성*(고려대), 이완석(부경대), 우영호(국가수리과학연구소)
Eui Sung Park*, Korea University, Wan Seok Lee, Pukyong National University, Young Ho Woo, NIMS
For a projective subscheme $X \subset \mathbb{P}^r$, we consider the invariants ${\rm reg}(X)$, $m(X)$ and $\ell (X)$ where ${\rm reg}(X)$ is the Castelnuovo-Mumford regularity of $X$, $m(X)$ is the maximal degree of a minimal generator of the homogeneous ideal of $X$ and $\ell (X)$ is the largest integer such that $X$ has a proper $\ell$-secant line. It holds always that
\begin{equation*}
{\rm reg}(X) \geq m(X) \geq \ell (X).
\end{equation*}
In this talk, I will speak about some interesting cases where the equality ${\rm reg}(X) = \ell (X)$. is attained. This is a report on a joint work with Wanseok Lee and Youngho Woo.
2010 Mathematics Subject Classification: 14N25
Key Words and Phrases: Castelnuovo-Mumford regularity, multisecant line, finite scheme
- ⋅ 22nd-E-17:00 − 17:20 Quantum super torus and super mirror symmetry (Hoil Kim)
- 김호일(경북대)
Hoil Kim, Kyungbook National University
(Noncommutaive) torus has been studied by several people including Mumford, Connes, Rieffel, Schwarz,
Manin and Polishchuk.
We extend their results to super torus.
More explicitly, we describe super Siegel domain, noncommutative super torus, symmetry group, Morita equivalence, super theta functions and super mirror symmetry.
2010 Mathematics Subject Classification: 14A22, 14K25, 58B34
Key Words and Phrases: super torus, Siegel domain, theta function, quantization, super mirror symmetry
- ⋅ 22nd-E-17:20 − 17:40 Classification of K3 surfaces of Picard number 20 over odd characteristic (Junmyeong Jang)
- 장준명(울산대)
Junmyeong Jang, University of Ulsan
Shioda and Inose classified the isomorphic classes of complex K3 surfaces of Picard number 20. They prove that, for any even positive definite integral lattice of rank 2, there exists a unique complex K3 surface of Picard number 20 whose transcendental lattice is isomorphic to it. In this talk, using the result of Shioda and Inose and the lifing technic, we classify K3 surfaces of Picard number 20 over an algebraically closed field of odd characteristic.
2010 Mathematics Subject Classification: 14J28, 14G17
Key Words and Phrases: K3 surface, crystalline cohomology, lifting of Neron-Severi group
- ⋅ 22nd-E-17:50 − 18:10 Deformations of weighted homogeneous surface singularities (Heesang Park, Dongsoo Shin)
- 박희상(건국대), 신동수*(충남대)
Heesang Park, Konkuk University, Dongsoo Shin*, Chungnam National University
We investigate the relation between certain partial resolutions of weighted homogeneous surface singularities admitting rational homology disk smoothings and their irreducible components of the reduced base spaces of the semi-universal deformation spaces. This is a joint work in progress with Heesang Park.
2010 Mathematics Subject Classification: 14B07
Key Words and Phrases: deformation, singularity, smooothing
- ⋅ 23rd-F-09:50 − 10:30 [Invited Talk] Gopakumar-Vafa invariants via vanishing cycles (Yukinobu Toda)
- Yukinobu Toda, Kavli IPMU, University of Tokyo
I propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. The proposal is a modification of a recent approach of Kiem-Li,
which is itself based on earlier ideas of Hosono-Saito-Takahashi. These invariants are conjectured to be equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. The main theorem is that, for local surfaces, our GV invariants agree with PT invariants for irreducible curve classes. I also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. This is a joint work with Davesh Maulik.
2010 Mathematics Subject Classification: 14N35
Key Words and Phrases: Gopakumar-Vafa invariants, Pandharipande-Thomas invariants, vanishing cycles
- ⋅ 23rd-G-10:45 − 11:05 Okounkov bodies associated to pseudoeffective divisors (Sung Rak Choi, Yoonsuk Hyun, Jinhyung Park, Joonyeong Won)
- 최성락*(연세대), 현윤석(삼성), 박진형(고등과학원), 원준녕(기초과학연구원)
Sung Rak Choi*, Yonsei University, Yoonsuk Hyun, Samsung Advanced Institute of Technology, Jinhyung Park, KIAS, Joonyeong Won, IBS
We will define a convex body (called the Okounkov body) in the Euclidean space for a given pseudoeffective divisor. This convex body encodes rich information of the positivity properties of the divisor. We will explain how to recover some of them. This is a report on the joint work with Yoonsuk Hyun, Jinhyung Park, and Joonyeong Won.
2010 Mathematics Subject Classification: 14C20
Key Words and Phrases: Okounkov bodies
- ⋅ 23rd-G-11:05 − 11:25 Local and log BPS numbers for del Pezzo surfaces (Jinwon Choi, Michel van Garrel, Sheldon Katz, Nobuyoshi Takahashi)
- 최진원*(숙명여대), Michel van Garrel(고등과학원), Sheldon Katz(Univ. of Illinois at Urbana-Champaign), Nobuyoshi Takahashi(Hiroshima Univ.)
Jinwon Choi*, Sookmyung Women's University, Michel van Garrel, KIAS, Sheldon Katz, University of Illinois at Urbana-Champaign, Nobuyoshi Takahashi, Hiroshima University
BPS numbers were introduced by physicists and have only recently been defined rigorously through the work of Kiem-Li. In this talk, we review local BPS numbers and then introduce log BPS numbers for a del Pezzo surface. The log BPS numbers are weighted counts of rational curves fully tangent to a smooth anticanonical divisor. We state and motivate the conjectural relationship between the two BPS numbers and explain how to prove it for all curve classes of arithmetic genus at most 2.
2010 Mathematics Subject Classification: 14N35
Key Words and Phrases: BPS numbers, Goparkumar-Vafa invariants, curve counting invariants
- ⋅ 23rd-G-11:35 − 12:15 [Invited Talk] Asymptotic and finite Hilbert stability of algebraic varieties (Donghoon Hyeon)
- 현동훈(서울대)
Donghoon Hyeon, Seoul National University
To polarize the Hilbert scheme, one should use the defining equations of high enough degree. Only then we are able to consider the Geometric Invariant Theory (GIT) stability of the Hilbert points of algebraic varieties, and this is called the asymptotic Hilbert stability. Asymptotic Hilbert stability has been used by David Gieseker in his GIT construction of the moduli space of stable curves. When we use equations of degree larger than or equal to the regularity, we can still define the Hilbert point although we may not be able to polarize the whole Hilbert scheme. In this talk, we shall explain how the GIT of Hilbert points of low degree (called the finite Hilbert stability) can be defined in a meaningful way, and how it can be applied to the study of the birational geometry of the moduli space of stable curves.
2010 Mathematics Subject Classification: 14
Key Words and Phrases: Geometric Invariant Theory
- Combinatorics
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] A necessary and sufficient condition for the existence of a properly colored $2$-factor (Kenta Ozeki)
- Kenta Ozeki, National Institute of Informatics \& JST, ERATO, Kawarabayashi Large Graph Project
Let $G$ be an {\em edge-colored} graph, which is a graph with each edge having a color.
(Note that the edge-coloring is not necessarily proper.)
A {\em properly-colored} cycle, or a PC cycle in short, in $G$ is a cycle with no two adjacent edges of the same color.
A {\em $2$-factor} of a graph is a spanning subgraph in which every vertex has degree two.
In this talk, we give a necessary and sufficient condition for the existence of a {\em PC $2$-factor} of an edge-colored graph $G$, i.e., a $2$-factor of $G$ such that each component is a PC cycle.
A PC $2$-factor was first consider as an ``intermediate structure'' to attack Bollob\'{a}s-Erd\H{o}s conjecture on a PC Hamiltonian cycle in an edge-colored complete graph.
Related to it, some sufficient conditions for the existence of a PC $2$-factor have been shown.
Our criterion gives another proofs to them.
This is a joint work with Michitaka Furuya (Kitasato University), Kenji Kimura (Ishinomaki Senshsu University), and Takamasa Yashima (Keio University).
2010 Mathematics Subject Classification: 05C38, 05C70
Key Words and Phrases: edge-colored graph, 2-factor, properly colored cycle
- ⋅ 22nd-B-11:35 − 11:55 A proof of Snevily's conjecture in set system (Younjin Kim, Kyung-Won Hwang)
- 김연진*(이화여대), 황경원(동아대)
Younjin Kim*, Ewha Womans University, Kyung-Won Hwang, Dong-A University
Let $K \!=\{ k_1,k_2, \ldots, k_r \}$ and $L \!= \{ l_1,l_2, \ldots, l_s \}$ be a set of nonnegative integers satisfying $\max l_j < \min k_i$. Let $\mathcal{F} = \{ F_1, F_2, \ldots, F_m \}$ be an $L$-intersecting family of subsets of $[n] = \{1,2, \ldots, n \}$ such that $|F| \in K$ for every $F\in \mathcal{F}$. In 1995, Snevily conjectured that if $K$ and $L$ are sets such that $\max l_j < \min k_i$, then $|\mathcal{F}| \leq {{n-1} \choose s} + { {n-1} \choose {s-1}}+ \cdots + { {n-1} \choose {s-r}}$. In this paper, we verify this conjecture.
2010 Mathematics Subject Classification: 05D05
Key Words and Phrases: Snevily's conjecture
- ⋅ 22nd-B-11:55 − 12:15 Normalized Laplacian eigenvalues and Randi\'c energy of graphs (Shaowei Sun)
- Shaowei Sun(성균관대)
Shaowei Sun, Sungkyunkwan University
Let $G$ be a connected graph of order $n$. Let $d_i$ be the degree of
the vertex $v_i$ in $G$. The Randi\'c matrix $R=R(G)=(b_{ij})_{n\times n}$ is
defined by $b_{ij}=1/\sqrt{d_i\,d_j}$ if the vertices $v_i$ and $v_j$ are adjacent,
and $b_{ij}=0$ otherwise. The Randi\'c energy $RE$ is the sum of absolute
values of the eigenvalues of $R$. The normalized Laplacian matrix is defined
as $\mathcal{L}(G)=I-R(G)$. We present necessary conditions under which the
graphs $G$ and $G+\{v_iv_j\}$ have cospectral normalized Laplacian matrices
and therefore equal Randi\'c energies. In addition, we characterize some
classes of graphs for which $G$ and $G+\{v_iv_j\}$ are not normalized-Laplacian
cospectral. We also report results on Randi\'c energy of edge-deleted
cycles and paths.
2010 Mathematics Subject Classification: 05C50
Key Words and Phrases: graph, normalized-Laplacian cospectral, randic energy, edge addition, path, cycle
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] List 3-dynamic coloring of graphs with small maximum average degree (Seog-Jin Kim, Boram Park)
- 김석진*(건국대), 박보람(아주대)
Seog-Jin Kim*, Konkuk University, Boram Park, Ajou University
An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring $\phi$ such that for any vertex $v$, $v$ has at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The {\em $r$-dynamic chromatic number} $\chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The {\em list $r$-dynamic chromatic number} of a graph $G$ is denoted by $ch_r^d(G)$.
Recently, Loeb, Mahoney, Reiniger, and Wise showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng, Lai, Lorenzen, Luo, Thompson, and Zhang studied the maximum average degree condition to have $\chi_3^d (G) \leq 4, \ 5$, or $6$.
In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $ch^d_3(G) \leq 6$ if $mad(G) < \frac{18}{7}$, and $ch^d_3(G) \leq 7$ if $mad(G) < \frac{14}{5}$, and both of the bounds are tight. This is joint work with Boram Park.
2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: dynamic coloring, list coloring, maximum average degree
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Dynamic colorings of planar graphs or graphs having no $K_5$ minor (Seog-Jin Kim, Younjin Kim, Lee, Sang June, Sang-il Oum, Won-Jin Park)
- 김석진(건국대), 김연진(이화여대), 이상준*(덕성여대), 엄상일(한국과학기술원), 박원진(서울대)
Seog-Jin Kim, Konkuk University, Younjin Kim, Ehwa Womans University, Lee, Sang June*, Duksung Women's University, Sang-il Oum, KAIST, Won-Jin Park, Seoul National University
A {\em dynamic coloring} of a graph $G$ is a proper coloring of the vertex set $V(G)$ such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. The {\em dynamic chromatic number} $\chi_d(G)$ is the smallest number $k$ such that there is a dynamic coloring of $G$ with $k$ colors. It is known that the gap $\chi_d (G) - \chi(G)$
could be arbitrarily large for some graphs.
Based on the Four Color Theorem and Wagner's theorem with proper colorings, one of the most
interesting problems about dynamic chromatic numbers is to find upper bounds of $\chi_d(G)$ for planar graphs $G$ or graphs having no $K_5$ minor. In this talk we introduce the following two results:
\begin{enumerate}
\item $\chi_d(G) \leq 4$ if
$G$ is a connected planar graph other than $C_5$. \\ (joint work with Seog-Jin Kim and Won-Jin Park)
\item $\chi_d(G) \leq 4$ if
$G$ is a connected $K_5$-minor-free graph other than $C_5$. \\(joint work with Younjin Kim and Sang-il Oum)
\end{enumerate}
2010 Mathematics Subject Classification: 15C15
Key Words and Phrases: dynamic coloring, planar graph, $K_5$-minor-free
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Plane partitions and a discrete integrable system (Shuhei Kamioka)
- Shuhei Kamioka, Kyoto University
Plane partitions are two-dimensional generalization of (integer) partitions which are visually depicted by three-dimensional Young diagrams. In this talk a close connection of plane partitions with a discrete integrable system, the discrete two-dimensional Toda molecule, is clarified. It is shown that a product formula which generalizes known generating functions such as MacMahon's triple product formula and Gansner's trace generating function can be derived from a specific solution to the discrete integrable system.
2010 Mathematics Subject Classification: 05A17
Key Words and Phrases: plane partitions, integrable systems, orthogonal polynomials
- ⋅ 22nd-D-16:05 − 16:25 A general bijection algorithm for trees and its application (Jin Yinglie)
- 김응렬(Nankai Univ.)
Jin Yinglie, Nankai University
Using a bijection to decompose a labelled rooted bipartite tree to several ones with smaller size and their exponential generating functions, this paper discusses the enumeration of trees. Moreover, gives a bijection between bicoloured ordered trees and RNA secondary structure.
2010 Mathematics Subject Classification: 05C05
Key Words and Phrases: tree, bijection, RNA
- ⋅ 22nd-D-16:25 − 16:45 Alternating partitions and their relationship with skew semi-standard Young, tableaux (Hyeonseo Hwang, Yonghoon Jo, Jangsoo Kim, Jongryul Lim)
- 황현서(한국과학영재학교), 조용훈(한국과학영재학교), 김장수(성균관대), 임종렬*(한국과학영재학교)
Hyeonseo Hwang, Korea Science Academy of KAIST, Yonghoon Jo, Korea Science Academy of KAIST, Jangsoo Kim, Sungkyunkwan University, Jongryul Lim*, Korea Science Academy of KAIST
In this talk, we define alternating partitions of a given positive integer $n$ and investigate their properties. Furthermore, we establish a bijection between alternating partitions and skew semi-standard Young tableaux and prove some identities related to them.
2010 Mathematics Subject Classification: 05A17
Key Words and Phrases: integer partitions, alternating partitions, skew semi-standard Young tableaux
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Counting permutations with or without patterns (Dongsu Kim)
- 김동수(한국과학기술원)
Dongsu Kim, KAIST
In combinatorics, permutations are counted with various conditions.
Many combinatorial objects can be regarded as permutations with or without certain patterns.
So counting permutations with patterns and counting permutations without patterns both
have been studied extensively.
This is a survey talk on the subject, with some new results on permutations avoiding $\{2413,4213\}$ patterns.
2010 Mathematics Subject Classification: 05A05
Key Words and Phrases: permutation, pattern avoiding permutations
- ⋅ 22nd-E-17:50 − 18:10 Construction of self-dual matrix codes over a field of characteristic 2 (Lucky Erap Galvez, Jon-Lark Kim )
- Lucky Erap Galvez*(서강대), Jon-Lark Kim (서강대)
Lucky Erap Galvez*, Sogang University, Jon-Lark Kim, Sogang University
We construct self-dual matrix codes over a finite field of characteristic
2 from a self-dual matrix code of smaller size. We show that every self-dual matrix code can be constructed using this building-up construction.
We classify all self-dual matrix codes of size $2\times 4$, $2\times 5$ over $GF(2)$, of size $2\times 3$, $2\times 4$ over $GF(4)$, and of size $2 \times 2$, $2 \times 3$ over $GF(8)$ which has been open
from Morrison's classification.
2010 Mathematics Subject Classification: 94B60. 11T71
Key Words and Phrases: matrix codes, equivalence, self-dual codes, building-up construction
- Algebra
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] A model structure approach to the Tate-Vogel cohomology (Jiangsheng Hu, Nanqing Ding)
- Jiangsheng Hu, Jiangsu University of Technology, Nanqing Ding*, Nanjing University
We study Tate-Vogel cohomology of complexes by applying the model structure induced by a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ of modules.
Vanishing of Tate-Vogel cohomology characterizes the finiteness of $\mathcal{A}$ dimension and $\mathcal{B}$ dimension of complexes defined in ``X. Y. Yang and N. Q. Ding, On a question of Gillespie, Forum Math. 27 (6) (2015), 3205-3231''. Applications go in three
directions. The first is to characterize when a left and right Noetherian ring is Gorenstein. The second is to obtain some criteria for the validity of the Finitistic Dimension Conjecture. The third is to investigate the relationships between flat dimension and Gorenstein flat dimension for complexes. This talk is a report on joint work with J. S. Hu.
2010 Mathematics Subject Classification: 16E05, 18G20, 18G35
Key Words and Phrases: Tate-Vogel cohomology, model structure, finitistic dimension, Gorenstein flat dimension
- ⋅ 22nd-B-11:35 − 11:55 The some properties of skew polynomial rings (Jin Hailan)
- 김해란(Yanbian Univ.)
Jin Hailan, Yanbian University
This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample; Then prove a necessary condition that skew polynomial ring constitutes Armendariz ring; We lastly investigate condition of skew polynomial ring is a (quasi)-Baer ring, and verify the conditions is necessary, but not sufficient by example and counterexample.
2010 Mathematics Subject Classification: 16S36
Key Words and Phrases: skew polynomial ring, (quasi)-Baer ring, Armendariz ring, Morita context ring, Morita invariance, nozero divisor ring
- ⋅ 22nd-B-11:55 − 12:15 On $\pi$-quasi-commutative rings (Zhelin Piao, Hyun-Min Kim, Dan Li)
- 박철림*(부산대), 김현민(부산대), Dan Li(부산대)
Zhelin Piao*, Pusan National University, Hyun-Min Kim, Pusan National University, Dan Li, Pusan National University
We study the quasi-commutativity in relation with powers of coefficients of polynomials. In the procedure we introduce the concept of $\pi$-quasi-commutative ring as a generalization of quasi-commutative rings. We show first that every $\pi$-quasi-commutative ring is Abelian and that a locally finite Abelian ring is $\pi$-quasi-commutative. The role of these facts are essential to our study in this note. The structures of various sorts of $\pi$-quasi-commutative rings are investigated to answer the questions raised naturally in the process, in relation to the structure of Jacobson and nil radicals.
2010 Mathematics Subject Classification: 16U70, 16U80, 16S36
Key Words and Phrases: $\pi$-quasi-commutative ring, center, quasi-commutative ring, idempotent, polynomial ring, matrix ring, Abelian ring, locally finite ring
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Catenary degrees of Krull domains and their generalizations (Andreas Reinhart)
- Andreas Reinhart, University of Graz
In this talk we discuss recent developments in factorization theory. We put special emphasis on invariants like the catenary degree and the monotone catenary degree, which measure how far factorizations are from being unique. After presenting what is known about these invariants in the case of Krull domains, we focus our attention on domains that are not completely integrally closed.
2010 Mathematics Subject Classification: 13A05, 13F15, 20M13
Key Words and Phrases: weakly Krull domains, non-principal orders, catenary degree
- ⋅ 22nd-C-14:20 − 14:40 Construction of non-Hopfian groups using 2-bridge link groups (Donghi Lee, Makoto Sakuma)
- 이동희*(부산대), Makoto Sakuma(Hiroshima Univ.)
Donghi Lee*, Pusan National University, Makoto Sakuma, Hiroshima University
A group $G$ is called Hopfian if every epimorphism $G \rightarrow G$ is an automorphism. Historically, not many have been known examples of finitely generated non-Hopfian groups with specific presentations. The earliest such example was found by Neumann in 1950 as follows: $\langle a,b \, | \, e_2=e_3=\cdots=1 \rangle$, where $e_i= a^{-1}b^{-1}ab^{-i}ab^{-1}a^{-1}b^ia^{-1}bab^{-i}aba^{-1}b^i$ for every integer $i \ge 2$. Soon after, the first non-Hopfian group with finite presentation was discovered by Higman as follows: $\langle a, s, t \, | \, a^s=a^2, \ a^t=a^2 \rangle$. Also a non-Hopfian group with the simplest presentation up to now was produced by Baumslag and Solitar as follows: $\langle a, t \, | \, (a^2)^t = a^3 \rangle$. Many other non-Hopfian groups with specific finite presentations have been obtained by generalizing Higman's group or Baumslag-Solitar's group.
Motivated by this background, we construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \dots$, where each $G_m$ has a specific presentation $G_m=\langle a, b \, | \, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \cdots =1 \rangle$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group with slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\langle m \rangle,m+1,m]$ in continued fraction expansion for every integer $i \ge 1$.
2010 Mathematics Subject Classification: 20F06
Key Words and Phrases: non-Hopfian group, 2-bridge link group, small cancellation condition
- ⋅ 22nd-C-14:40 − 15:00 Invariants of finite groups generated by generalized transvections in the modular case (Ji Zhu Nan)
- 남기수(Dalian Univ. of Technology)
Ji Zhu Nan, Dalian University of Technology
In this paper, we investigate the invariant rings of two classes of finite groups
$G \leq GL(n, F_q)$ that are generated by a number of generalized transvections with invariant
subspaces $H$ over a finite field $F_q$ in the modular case. We name these groups by generalized
transvection groups. The one class is concerned with a given invariant subspace which is involved several roots of unity. Constructing quotient group and tensor, we deduce the invariant
rings and study their Cohen-Macaulay and Gorenstein properties. The other one is concerned
with different invariant subspaces which have same dimension. We provide a classification of
these groups and illustrate their invariant rings in detail.
2010 Mathematics Subject Classification: 13A50, 20F55, 20F99
Key Words and Phrases: invariant, transvection, generalized transvection group
- ⋅ 22nd-D-15:15 − 15:35 An observation for the Riemann hypothesis (Sung Soo Kim)
- 김성수(한양대)
Sung Soo Kim, Hanyang University
In 1859, Bernhard Riemann proposed a hypothesis while studying on the distribution of prime numbers. The hypothesis was originally stated in terms of the Riemann zeta function. Later, in 1924, Franel and Landau found that the hypothesis can be equivalently stated in an elementary way using Farey series. In this talk, we visualize the Franel and Landau version of the hypothesis as an estimation of the area bounded by a function on the unit interval. This visualization enables us to approach the problem in some intuitive ways and gives an observation which seems to indicate that the hypothesis is true.
2010 Mathematics Subject Classification: 26E40
Key Words and Phrases: Riemann hypothesis, Farey series
- ⋅ 22nd-D-15:35 − 15:55 On well-posedness and computability in shape optimization (Chansu Park, Dongseong Seon, Martin Ziegler)
- 박찬수(서울대), 선동성(서울대), Martin Ziegler*(한국과학기술원)
Chansu Park, Seoul National University, Dongseong Seon, Seoul National University, Martin Ziegler*, KAIST
Computable Analysis is the synthesis of Mathematical Analysis and Computability Theory as rigorous algorithmic foundation to Reliable Numerics.
We apply it to example cases of Shape Optimization:
Recall that, on compact subsets of Euclidean space equipped with the Hausdorff metric, the volume functional lacks (lower semi) continuity, i.e., well-posedness;
and we strengthen that by constructing a computable closed subset of the unit interval with non-computable length (=1D volume).
On the other hand we show that, restricted to CONVEX compact subsets of Euclidean space, both volume and perimeter are continuous and computable.
2010 Mathematics Subject Classification: 49K40, 03D78
Key Words and Phrases: shape optimization, computable analysis
- ⋅ 22nd-D-16:05 − 16:25 Preservers of primitive symmetric matrices of exponent 2 (Seok-Zun Song)
- 송석준(제주대)
Seok-Zun Song, Jeju National University
Let $M(S)$ denote the set of $n\times n$ symmetric matrices over some semiring $S$. The set $P(S)$ of elements of $S$ that are a finite sum of squares of elements in $S$, is called the positive subsemiring of $S$. A matrix $B$, with entries in $P(S)$ is primitive if some power of $B$ has all nonzero entries. We characterize those linear operators on the set of symmetric matrices over $S$ that map the set of primitive matrices to itself and the set of nonprimitive matrices to itself. We also characterize those linear operators on the set of symmetric matrices over S that map the set of primitive matrices whose square has all nonzero entries to itself and the complement of that set to itself. The characterization of linear preservers of real symmetric primitive matrices follows as a special
case.
2010 Mathematics Subject Classification: 15A04, 15A86
Key Words and Phrases: semiring, double star matrix, linear operator, primitive matrix
- Functional Analysis
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk] Uncertainty relations in the framework of equalities (Tohru Ozawa)
- Tohru Ozawa, Waseda University
This talk is based on a recent joint work with Kazuya Yuasa, Department of Physics, Waseda University.
We study the Schrödinger–Robertson uncertainty relations in an algebraic framework.
Moreover, we show that some specific commutation relations imply new equalities,
which are regarded as equality versions of well-known inequalities such as Hardy's inequality.
2010 Mathematics Subject Classification: 81D05
Key Words and Phrases: uncertainty relations
- ⋅ 22nd-A-09:50 − 10:10 On a Gauge action on noncommutative sigma-model solitions (Hyun Ho Lee)
- 이현호(울산대)
Hyun Ho Lee, University of Ulsan
Noncommutative analogues of nonlinear sigma model were introduced by Dabrowski, Krajewski, and Landi around 2000. Later J.Rosenberg and V. Mathai gave the sophisticated definition of sigma model based on Connes' spectral triple language. The aim of sigma model is to classify all critical points of the action functional, or energy minimizing field maps. Depending on several time-spaces various Euler-Lagrange equation(s) are obtained for such critical points. An interesting observation is that once we fix our string world sheet as noncommutative torus we can give a gauge action on all known solution-spaces. However, there might be a bigger group action which works for a particular solution-space. We would like to discuss such s phenomenon using concrete examples.
2010 Mathematics Subject Classification: 58E20, 58B34, 46L60
Key Words and Phrases: noncommutative sigma-models, noncommuative tori, gauge action, strong Morita equivalence
- ⋅ 22nd-A-10:10 − 10:30 Pure infiniteness of labeled graph $C^*$-algebras (Ja A Jeong, Eun Ji Kang)
- 정자아(서울대), 강은지*(서울대)
Ja A Jeong, Seoul National University, Eun Ji Kang*, Seoul National University
Given a directed graph $E$ and a labeling map $\mathcal{L}$
which assigns a label in an alphabet $\mathcal{A}$ to each of the edges $E^1$,
Bates and Pask form a set of certain vertex subsets, denoted $\mathcal{E}$,
such that there exists a family of operators
$\{s_a, p_A: a\in \mathcal{L}(E^1),\ A \in \mathcal{E} \}$ which
is universal with respect to certain relations.
By $C^*(E,\mathcal{L},\mathcal{E})$ we denote the $C^*$-algebra generated by
the universal family $\{s_a, p_A\}$.
In this paper, we consider infiniteness of these
labeled graph $C^*$-algebras $C^*(E,\mathcal{L},\mathcal{E})$ and
show that $C^*(E,\mathcal{L},\mathcal{E})$ is infinite in the sense that
every nonzero hereditary subalgebra contains an infinite projection
if $(E, \mathcal{L},\mathcal{E})$ is disagreeable and every vertex connects to a loop.
(This property is abbreviated to (HP$_\infty$) in this paper.)
It is also proved that if every canonical quotient labeled space
$(E,\mathcal{L},\mathcal{E}/H)$ of a saturated hereditary subsets $H$ of $\mathcal{E}$
is disagreeable, then the $C^*$-algebra
$C^*(E,\mathcal{L},\mathcal{E})=C^*(p_A, s_a)$ is purely infinite
in the sense of Kirchberg and R\o rdam
if and only if every generating projection $p_A$, $A\in\mathcal{E}$, is
properly infinite, and again if and only if
every quotient of $C^*(E,\mathcal{L},\mathcal{E})$ satisfies the property (HP$_\infty$).
2010 Mathematics Subject Classification: 08193
Key Words and Phrases: labeled graph $C^*$-algebras, purely infinite $C^*$-algebras
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Maximal abelian subalgebras in purely infinite simple $C^*$-algebras (Jeong Hee Hong)
- 홍정희(한국해양대)
Jeong Hee Hong, Korea Maritime and Ocean University
In theory of operator algebras, MASAs (Maximal abelian subalgebras) have played very important role from the very beginnings, going back to Murray and von Neumann. In particular, the classification of MASAs and Cartan subalgebras in type $II_1$ factors is pivotal in the currently being developed deformation-rigidity program. By comparison with von Neumann algebras, much less is known about MASAs in $C^*$-algebras. This is partially due to lack of computable invariants
(like the Pukanszky invariant) and cutting-and-paste techniques (like the Popa's intertwining-by-bimodules technique). In addition to its intrinsic interest, classification of MASAs has a bearing on better understanding of the structure of the outer automorphism group of the algebra in question.
The aim of this talk is to report on the recently started and ongoing effort at making progress towards classification up to conjugacy of MASAs in some purely infinite simple C*-algebras, most notably the Cuntz algebras ${\mathcal O}_n$. For this, we begin by recalling the full and the restricted Weyl groups of ${\mathcal O}_n$, respectively, pointing out close connection of the latter with automorphisms of the two-sided shift. Then we provide examples of outer but not inner conjugate MASAs in ${\mathcal O}_n$. These examples underscore a very different pattern of behavior of MASAs of the Cuntz algebras by comparison with factor von Neumann algebras.
This talk is based on joint works with Roberto Conti, Tomohiro Hayashi and Wojciech Szyma\'{n}ski.
2010 Mathematics Subject Classification: 46L40
Key Words and Phrases: Cuntz algebra, MASA, automorphism
- ⋅ 22nd-B-11:35 − 11:55 Uniform Sobolev inequalities for Schr\"odinger operators and applications (Haruya Mizutani)
- Haruya Mizutani, Osaka University
The uniform Sobolev inequality due to Kenig--Ruiz--Sogge is a generalization of the Hardy--Littlewood--Sobolev inequality to the resolvent of the Laplacian. It is a powerful tool in the study of spectral and dispersive properties of Schr\"odinger equations. We discuss its generalization to the Schr\"odinger operator with potentials exhibiting scaling-critical singularities, as well as its applications to Keller type eigenvalues bounds for non-self-adjoint Schr\"odinger operators and global-in-time smoothing and Strichartz estimates. This talk is based on joint work with Jean-Marc Bouclet (Toulouse 3).
2010 Mathematics Subject Classification: 35J10
Key Words and Phrases: uniform Sobolev inequality, resolvent, Schr\"odinger operator
- ⋅ 22nd-B-11:55 − 12:15 Spatial realizations of KMS states on the $C^*$-algebras of higher-rank graphs (Astrid an Huef, Sooran Kang, Iain Raeburn)
- Astrid an Huef(University of Otago, New Zealand), 강수란*(성균관대), Iain Raeburn, (Univ. of Otago, New Zealand)
Astrid an Huef, University of Otago, New Zealand, Sooran Kang*, Sungkyunkwan University, Iain Raeburn, University of Otago, New Zealand
We review KMS states on $C^*$-algebras associated to directed graphs and higher-rank graphs. Then we discuss the work with Iain Raeburn and Astrid an Huef on spatial realizations of KMS states on $C^*$-algebras associated to higher-rank graphs.
2010 Mathematics Subject Classification: 46L05
Key Words and Phrases: Toeplitz algebra, higher-rank graph, gauge action, KMS state
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Recent developments of optimization in a Banach space related with the Bishop-Phelps(-Bollob\'{a}s) theorem (Yun Sung Choi)
- 최윤성(포항공대)
Yun Sung Choi, POSTECH
Some attention has been paid in recent papers to the Bishop-Phelps-Bollob\'{a}s property for operators between Banach spaces. This is a strong form of denseness of norm attaining operators, which further concerns approximating a point with norming points. In this talk we survey very recent results in this direction for linear or nonlinear mapping. In particular, the Bishop-Phelps-Bollob\'{a}s version of Lindenstrauss properties A and B is included.
2010 Mathematics Subject Classification: 46B20, 46B04, 46B22
Key Words and Phrases: Banach space, norm, operator, Bishop-Phelps theorem
- ⋅ 22nd-C-14:20 − 14:40 Stability of the Brascamp-Lieb constant and applications (Neal Bez)
- Neal Bez, Saitama University
The Brascamp--Lieb inequality is a natural and powerful generalisation of several fundamental inequalities, including the multilinear H\"older inequality, the Loomis--Whitney inequality and the Young convolution inequality. In this talk we discuss the stability of the Brascamp--Lieb inequality under perturbations of the underlying linear mappings, along with various applications. The talk is based on joint work with Jonathan Bennett, Michael Cowling, Taryn Flock and Sanghyuk Lee.
2010 Mathematics Subject Classification: 44A35
Key Words and Phrases: Brascamp--Lieb inequality, stability
- ⋅ 22nd-C-14:40 − 15:00 On the connected components of the conjugacy class of projectors on $ \ell_p\oplus\ell_q $ (Daniele Garrisi)
- Daniele Garrisi(인하대)
Daniele Garrisi, Inha University
We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of $ \ell_p\oplus\ell_q $ with $ p\neq q $.
2010 Mathematics Subject Classification: 46H05, 47L05
Key Words and Phrases: projectors, components, conjugation
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Topological full groups and $C^*$-algebras (Hiroki Matui)
- Hiroki Matui, Chiba University
From a topological dynamical system on a Cantor set $X$, we can construct a $C^*$-algebra with a Cartan subalgebra isomorphic to $C(X)$. Furthermore, we can define a topological full group associated with the dynamical system. This group is a countable infinite subgroup of the homeomorphism group on $X$. We will discuss relationships between the $C^*$-algebra and the topological full group. Various algebraic and analytic properties of topological full groups are also investigated.
2010 Mathematics Subject Classification: 46L35, 37B10
Key Words and Phrases: topological full groups, Cantor dynamical systems, $C^*$-algebras
- ⋅ 22nd-D-16:05 − 16:25 Generalization of algebraic properties of Toeplitz operators associated to some orthonormal basis for $L^2$ functions (Young-Bok Chung)
- 정영복(전남대)
Young-Bok Chung, Chonnam National University
We use a special orthonormal basis for the space of $L^2$ functions on a smooth bounded domain of the complex plane to generalize algebraic theory of Toeplitz operators on the unit circle. In particular, we classify Toeplitz operators in terms of Toeplitz matrices of arbitrary order associated to the orthonormal basis constructed above. This result will yield an expression of the symbol of the operator by means of the matrix representation. We also obtain a necessary condition for the product of two Toeplitz operators to be a Toeplitz operator which can be written as a concrete form in the case of unit circle and we finally solve commuting problems of two Toeplitz operators.
2010 Mathematics Subject Classification: 47B35
Key Words and Phrases: Toeplitz operator, Toeplitz matrices, Hardy space, Szego kernel
- ⋅ 22nd-D-16:25 − 16:45 On binormal operator matrices and their applications (Eungil Ko, Hyun-Kyoung Kwon, Ji Eun Lee)
- 고응일(이화여대), 권현경(Univ. of Alabama), 이지은*(세종대)
Eungil Ko, Ewha Womans University, Hyun-Kyoung Kwon, University of Alabama, Ji Eun Lee*, Sejong University
A bounded linear operator $T$ is {\it binormal} on a Hilbert space $\mathcal H$ if $T^{\ast}T$ and $TT^{\ast}$ commute where $T^{\ast}$ is the Hilbert adjoint of $T$.
In this paper, we characterize binormal operator matrices. As some applications, we focus on a characterization of binormal Toeplitz matrices on finite dimensional spaces.
Finally, we investigate some cases of nonnormal and binormal (Toeplitz) matrices and provide several examples of such matrices.
2010 Mathematics Subject Classification: Primary 47B35, 47B15
Key Words and Phrases: binormal operator, Toepltiz matrices, operator matrices
- ⋅ 22nd-E-17:00 − 17:20 Alternating resolvent algorithms for finding common zeros of two accretive operators in Banach spaces (Jong Kyu Kim)
- 김종규(경남대)
Jong Kyu Kim, Kyungnam University
The propose of this talk is to introduce a new iterative method by the combination of
the prox-Tikhonov regularization and the alternating resolvents for finding common
zeros of two accretive operators in Banach spaces. And we will give some applications and numerical examples. The results of this paper improve and extend the corresponding well-known results announced by many others.
2010 Mathematics Subject Classification: 47H06, 47H09, 47H10, 47J25
Key Words and Phrases: accretive operators, prox-Tikhonov method, alternating resolvent method, common zeros, accretive operators
- ⋅ 22nd-E-17:20 − 17:40 Classification of gapped Hamiltonians in quantum spin chains (Yoshiko Ogata)
- Yoshiko Ogata, The University of Tokyo
We consider the classification problem of gapped Hamiltonians in quantum spin chains. Two Hamiltonians are equivalent if they are connected by a continuous path without clothing the gap. We consider some subclass of Hamiltonians and show that all the Hamiltonians in the class are equivalent.
2010 Mathematics Subject Classification: 82
Key Words and Phrases: quantum spin chain
- ⋅ 22nd-E-17:50 − 18:10 A result on semi-weakly hyponormal weighted shifts (Chunji Li)
- 이춘길(Northeastern Univ.)
Chunji Li, Northeastern University
It is known that a semi-cubically hyponormal weighted shift need not satisfy the flatness property, in which equality of two weights forces all or almost all weights to be equal. So it is a natural question to describe all semi-cubically hyponormal weighted shifts $W_{\alpha }$ with first two weights equal. Let $\alpha :1,1,\sqrt{x},\left( \sqrt{u},\sqrt{v},\sqrt{w}\right) ^{\wedge }$ be a backward 3-step extension of a recursively generated weighted sequence with $1<x<u<v<w$ and let $W_{\alpha }$ be the associated weighted shift. In this paper, we characterize completely the semi-cubical hyponormal $W_{\alpha }$ satisfying the additional assumption of the positive determinant coefficient property, which result is parallel to results for quadratic hyponormality.
2010 Mathematics Subject Classification: 46, 47
Key Words and Phrases: hyponormal weighted shifts
- ⋅ 23rd-F-09:20 − 09:40 On $m$–isometric Toeplitz operators (Eungil Ko, Jongrak Lee)
- 고응일(이화여대), 이종락*(이화여대)
Eungil Ko, Ewha Womans University, Jongrak Lee*, Ewha Womans University
In this talk, we study $m$-isometric Toeplitz operators. In addition, we give a necessary and sufficient condition for Toeplitz operators to be $m$-expansive or $m$-contractive.
2010 Mathematics Subject Classification: 47A62, 47B15, 47B20
Key Words and Phrases: $m$-isometric operators, expansive operators, contractive operators, Toeplitz operators
- ⋅ 23rd-F-09:50 − 10:10 Analytic roots of hyponormal operators (Sungeun Jung, Eungil Ko)
- 정성은*(한국외대), 고응일(이화여대)
Sungeun Jung*, Hankuk University of Foreign Studies, Eungil Ko, Ewha Womans University
In this talk, we give a condition under which analytic roots of hyponormal operators are subscalar. As an application, we consider the invariant subspace problem for such operators. We also show that the product of some analytic root of a hyponormal operator and an algebraic operator which are commuting is subscalar.
2010 Mathematics Subject Classification: 47B20, 47A11
Key Words and Phrases: hyponormal operator, analytic root of a hyponormal operator, subscalar, property $(\beta)$, invariant subspace, algebraic operator
- ⋅ 23rd-F-10:10 − 10:30 On the iterated Duggal transforms of operators (Mee-Jung Lee, Eungil Ko)
- 이미정*(이화여대), 고응일(이화여대)
Mee-Jung Lee*, Ewha Womans University, Eungil Ko, Ewha Womans University
In this talk, we study properties of the iterated Duggal transforms of operators. In particular, we give the polar decomposition of the iterated Duggal transforms of invertible operators. We also consider the concept of ``almost commuting with a compact operator'' under the iterated Duggal transforms. Moreover, we show that, under suitable conditions, the algebraic and analytic cores are preserved by the iterated Duggal transforms. Finally, we study invariant subspaces of an operator when its iterated Duggal transforms are hyponormal.
2010 Mathematics Subject Classification: 47B20, 47A10
Key Words and Phrases: Duggal transform, invariant subspace, hyponormal opertaor
- ⋅ 23rd-G-10:45 − 11:05 The generalized Cauchy-Schwarz inequality for operators (Hanna Choi, Yoenha Kim, Eungil Ko)
- 최한나(이화여대), 김연하*(이화여대), 고응일(이화여대)
Hanna Choi, Ewha Womans University, Yoenha Kim*, Ewha Womans University, Eungil Ko, Ewha Womans University
In this talk, we introduce the generalized Cauchy-Schwarz inequality for an operator $T\in{\mathcal L(H)}$ and investigate various properties of operators which satisfy the generalized Cauchy-Schwarz inequality. In particular, every $p$-hyponormal operator satisfies this inequality. We also prove that if $T\in{\mathcal L(H)}$ satisfies the generalized Cauchy-Schwarz inequality, then $T$ is paranormal.
2010 Mathematics Subject Classification: 47A63
Key Words and Phrases: Cauchy-Schwarz inequality, p-hyponormal, paranormal
- ⋅ 23rd-G-11:05 − 11:25 Shape of the kernel of block Hankel operators (Dong-O Kang, Woo Young Lee, In Sung Hwang)
- 강동오*(충남대), 이우영(서울대), 황인성(성균관대)
Dong-O Kang*, Chungnam National University, Woo Young Lee, Seoul National University, In Sung Hwang, Sungkyunkwan University
Kernels of Hankel operators are known to be invariant subspaces for the unilateral shift. In this talk, the shape of kernels of block Hankel operators are presented. It is shown that the kernel of a block Hankel operator is decided by a certain type of independency of the column of the matrix-valued symbol function of the Hankel operator.
2010 Mathematics Subject Classification: 47B35
Key Words and Phrases: block inner functions, Hankel operators
- Harmonic Analysis
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Conjecture and improved extension theorems for paraboloids in the finite field setting (Doowon Koh)
- 고두원(충북대)
Doowon Koh, Chungbuk National University
In this talk we study the extension estimates for paraboloids in $d$-dimensional vector spaces over finite fields with $q$ elements. We use the connection between $L^2$ based restriction estimates and $L^p \to L^r$ extension estimates for paraboloids. As a consequence, we improve the $L^2 \to L^r$ extension results obtained by A. Lewko and M. Lewko in even dimensions $d \geq 6$, and odd dimensions $d = 4\ell + 3$ for $\ell \in \mathbb{N}.$ We also clarifies conjectures on finite field extension problems for paraboloids.
2010 Mathematics Subject Classification: 42B05
Key Words and Phrases: restriction theorem, extension theorem, paraboloid, finite field
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Estimates for the Fourier transforms of surface-carried measures (Joonil Kim)
- 김준일(연세대)
Joonil Kim, Yonsei University
We consider the Fourier transforms of some class of surface-carried measures.
In particular, for the cases of non-compact surfaces, we study their asymptotic behaviors of the integrals.
This estimate applies to the Strichartz estimates for
general dispersive equations
$u_t=iP(\partial_{x_1}/i,\ldots,\partial_{x_n}/i)u$ with $u(0,x)=u_0(x)$
where $P$ is a polynomial.
2010 Mathematics Subject Classification: 42B20
Key Words and Phrases: oscillatory integrals, Fourier transform of surface carried measure, Strichartz estimates
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Averaged decay estimates for Fourier transforms of measures (Yutae Choi, Seheon Ham, Sanghyuk Lee)
- 최유태(포항공대), 함세헌*(고등과학원), 이상혁(서울대)
Yutae Choi, POSTECH, Seheon Ham*, KIAS, Sanghyuk Lee, Seoul National University
For a positive Borel measure with compact support, we consider $L^2$ averaged decay estimates of its Fourier transform.
When the average is taken over the unit sphere, the decay estimates were studied extensively, in connection with the Falconer distance set problem, by Mattila, Sj\"{o}lin, Bourgain, Wolff, Erdo\u{g}an, and Luc\'{a}-Rogers.
In this talk, we study the case of space curves with non-vanishing torsion.
We extend the previous known results for the unit circle to higher dimensions.
Also we discuss sharpness of the estimates.
This is a joint work with Yutae Choi (Pohang University of Science and Technology) and Sanghyuk Lee (Seoul National University).
2010 Mathematics Subject Classification: 42B10
Key Words and Phrases: Fourier transform, averaged decay, measure
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Size estimate of divergence sets of wave equation (Chuhee Cho)
- 조주희(서울대)
Chuhee Cho, Seoul National University
In this talk, we consider the pointwise convergence of solutions for the wave equation and obtain estimates for the capacitary dimension of divergence sets.
2010 Mathematics Subject Classification: 35Q41
Key Words and Phrases: pointwise convergence, fractional dimension
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Sampling and interpolation (Yong-Kum Cho)
- 조영금(중앙대)
Yong-Kum Cho, Chung-Ang University
We give some explicit families of band-limited functions and discuss the related smapl\-ing
and interpolation problems.
2010 Mathematics Subject Classification: 42B10
Key Words and Phrases: sampling, interpolation, band-limited
- PDE
- ⋅ 22nd-A-09:00 − 09:20 Degree counting for Toda system of rank two: one bubbling (Youngae Lee)
- 이영애(국가수리과학연구소)
Youngae Lee, NIMS
In this talk, we study the degree counting formula of the rank two Toda system with simple singular sources. The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as one of parameters crosses $4\pi$. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with new difficulty arising from the new phenomena: blow up does not necessarily imply concentration of mass. This phenomena occurs due to the collapsing of singularities.
This talk is based on the joint works with Prof. Chang-Shou Lin, Prof. Juncheng Wei, Prof. Lei Zhang, and Dr. Wen Yang.
2010 Mathematics Subject Classification: 35J47
Key Words and Phrases: Toda system, shadow system, bubbling solutions
- ⋅ 22nd-A-09:20 − 09:40 Homogenization of elliptic and parabolic soft inclusions (Minha Yoo, Ki-Ahm Lee)
- 유민하*(국가수리과학연구소), 이기암(서울대)
Minha Yoo*, NIMS, Ki-Ahm Lee, Seoul National University
In this talk, we consider periodic Soft inclusions T-epsilon of elliptic and parabolic linear equation of non-divergence form. For each epsilon, the domain Omega is perforated by periodic balls T-epsilon with periodicity epsilon. Let u be the solution of elliptic and parabolic PDEs with Neumann data on the intersections between Omeag and the boundary of T-epsilon. The main object is to show the uniform convergence of u as epsilon tends to zero and to find the ``effective equation'' where the limit of u satisfy.
2010 Mathematics Subject Classification: 35B27
Key Words and Phrases: homogenization, elliptic pde, parabolic pde, soft inclusions
- ⋅ 22nd-A-09:50 − 10:10 Weighted regularity estimates in Orlicz spaces for fully nonlinear elliptic equations (Sun-Sig Byun, Mikyoung Lee, Jihoon Ok)
- 변순식(서울대), 이미경*(한국과학기술원), 옥지훈(고등과학원)
Sun-Sig Byun, Seoul National University, Mikyoung Lee*, KAIST, Jihoon Ok, KIAS
In this talk, we discuss a global $W^{2,p}$ estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations
$$F(D^{2}u, Du, u, x) = f(x) \textrm{ in } \Omega, \ \ \ u=0 \textrm{ on } \partial \Omega$$
to a more general function space. More precisely, given an $N$-function $\Phi$ and a Muckenhoupt weight $w$, we prove that if $f$ belongs to the associated weighted Orlicz space $L^{\Phi}_{w}(\Omega)$, then $D^2 u \in L^{\Phi}_{w}(\Omega)$ and $u$ satisfies a global $W^{2,\Phi}_{w}$ estimate, under a minimal regularity requirement on $F$ in the variable $x$ and a basic geometric assumption on $\partial \Omega$. The correct condition on the couple, $\Phi$ and $w$, is also addressed.
2010 Mathematics Subject Classification: 35J60, 35B45
Key Words and Phrases: fully nonlinear equation, viscosity solution, regularity, Hessian estimates, Muckenhoupt weight, Orlicz space
- ⋅ 22nd-A-10:10 − 10:30 Global $C^{1,\alpha}$ regularity and existence of multiple solutions for singular $p(x)$-Laplacian equations (Sun-Sig Byun, Eunkyung Ko)
- 변순식(서울대), 고은경*(울산과학기술원)
Sun-Sig Byun, Seoul National University, Eunkyung Ko*, UNIST
In this talk we consider the following singular
$p(x)$-Laplacian problem
\begin{eqnarray*}
\left\{\begin{array}{ll}
- \mbox{div} (|\nabla u|^{p(x)-2} \nabla u) =\frac{ \lambda}{u^{\beta(x)}}+u^{q(x)} , & \mbox{in}~ \Omega, \\
u>0, & \mbox{in}~ \Omega, \\
u=0, & \mbox{on}~ \partial \Omega,
\end{array}\right.
\end{eqnarray*}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 2$,
with smooth boundary $\partial \Omega$, $\beta \in C^1(
\bar{\Omega})$ with $ 0< \beta(x) <1$, $p\in C^1(\bar{\Omega})$, $q
\in C(\bar{\Omega})$ with $p(x)>1$, $p(x) \leq q(x) +1 <p^*(x)$ for
$x \in \bar{\Omega}$, where $ p^*(x)= \frac{Np(x)}{N-p(x)} $ for
$p(x) <N$ and $ p^*(x)= \infty $ for $ p(x) \geq N$. We present
$C^{1,\alpha}$ regularity of weak solutions of the problem and
strong comparison principle. Based on these two results, we prove
the existence of multiple (at least two) positive solutions for a
certain range of $\lambda$.
2010 Mathematics Subject Classification: 35J75
Key Words and Phrases: singular p(x)-Laplacian problem
- ⋅ 22nd-B-10:45 − 11:05 Nonrelativistic limit for standing waves of the semi-relativistic Schrodinger equations (Woocheol Choi, Younghun Hong, Jinmyoung Seok)
- 최우철*(고등과학원), 홍영훈(연세대), 석진명(경기대)
Woocheol Choi*, KIAS, Younghun Hong, Yonsei University, Jinmyoung Seok, Kyonggi University
In this talk, we are concerned with the nonrelativistic limit for standing waves of the semi-relativistic Schrodinger equations. The convergence result will be discussed along with the existence and regularity property of the solutions. This talk is based on joint work with Younghun Hong and Jinmyoung Seok.
2010 Mathematics Subject Classification: 35J61
Key Words and Phrases: nonrelativistic limit, standing waves
- ⋅ 22nd-B-11:05 − 11:25 Population dynamics with finite time extinction and compactly supported peak solutions (Oh Sang Kwon)
- 권오상(충북대)
Oh Sang Kwon, Chungbuk National University
The Allee effect is a phenomenon that a species get extinct if the population density is small. Allen-Cahn type bistable nonlinearities are often used to model such a phenomenon. However, such a model never gives an extinction in a finite time. In this talk, we consider a different approach of simply subtracting a small constant from a logistic model. This model will give us finite time extinction and compactly supported peak solution. This peak solution provide us a criterion for extinction or population expansion.
This is the joint work with Y.-J. Kim and X. Pan.
2010 Mathematics Subject Classification: 35B99
Key Words and Phrases: Population dynamics, Allee effect, peak solutions
- ⋅ 22nd-B-11:35 − 11:55 H\"{o}lder continuity of quasilinear parabolic $p$-Laplacian type equations (Sukjung Hwang, Gary M. Lieberman)
- 황숙정*(CMAC, 연세대), Gary M. Lieberman(Iowa State Univ.)
Sukjung Hwang*, CMAC, Yonsei University, Gary M. Lieberman, Iowa State University
Here we introduce the generalized quasilinear parabolic $p$-Laplacian type equations in the setting from Orlicz spaces. We provide uniform proof with a single geometric setting that a bounded weak solution is locaaly H\"{o}lder continuous with degree of commonality between degenerate and singular types. Recently, we continue our method of approach to porous media type equations.
2010 Mathematics Subject Classification: 35B45, 35K59
Key Words and Phrases: H\"{o}lder continuity, quasilinear parabolic equation
- ⋅ 22nd-B-11:55 − 12:15 Weighted $L_{p,q}$-estimates for higher order parabolic systems (Doyoon Kim, Jongkeun Choi)
- 김도윤(고려대), 최종근*(고려대)
Doyoon Kim, Korea University, Jongkeun Choi*, Korea University
We establish weighted $L_{p,q}$-estimates for higher order parabolic systems of divergence form in a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables. As an application, we prove the solvability in weighted Sobolev spaces for the systems in the whole space, on a half space, and on a bounded Reifenberge flat domains.
2010 Mathematics Subject Classification: 35B45
Key Words and Phrases: weighted $L_{p,q}$-estimates
- ⋅ 22nd-C-13:30 − 13:50 Spectral method for stokes equations in ball (Hi Jun Choe)
- 최희준(연세대)
Hi Jun Choe, Yonsei University
We develope a spectral method for Stokes equations in ball.
The characterization of solutions by spherical harmonics is
main idea. We show a Harnack type estimate for Stokes equations
which is surprising in view of system theory.
2010 Mathematics Subject Classification: 35c10
Key Words and Phrases: stokes equations, spherical harmonics, harnack
- ⋅ 22nd-C-13:50 − 14:10 On the fnite-time blowup of a 1D model for the 3D axisymmetric Euler equations (Kyudong Choi, Thomas Y. Hou, Alexander Kiselev, Guo Luo, Vladimir Sverak, Yao Yao)
- 최규동*(울산과학기술원), Thomas Y. Hou(California Institute of Technology), Alexander Kiselev(Rice Univ.), Guo Luo(City Univ. of Hong Kong), Vladimir Sverak(Univ. of Minnesota), Yao Yao(Georgia Institute of Technology)
Kyudong Choi*, UNIST, Thomas Y. Hou, California Institute of Technology, Alexander Kiselev, Rice University, Guo Luo, City University of Hong Kong, Vladimir Sverak, University of Minnesota, Yao Yao, Georgia Institute of Technology
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo $\&$ Hou PNAS (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.
2010 Mathematics Subject Classification: 76B03, 35Q35, 35B44
Key Words and Phrases: blow-up, 1d model, Euler, axisymmetry
- ⋅ 22nd-C-14:20 − 14:40 The Minkowski dimension of interior singular points in the incompressible Navier Stokes equations (Youngwoo Koh, Minsuk Yang)
- 고영우(고등과학원), 양민석*(고등과학원)
Youngwoo Koh, KIAS, Minsuk Yang*, KIAS
We study the possible interior singular points of suitable weak solutions to the three dimensional incom- pressible Navier–Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set, which is bounded by 95/63. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty.
2010 Mathematics Subject Classification: 35Q35
Key Words and Phrases: singular point, Minkowski dimension
- ⋅ 22nd-C-14:40 − 15:00 Regularity of the flow map of the Navier-Stokes equations (Hantaek Bae, Marco Cannone)
- 배한택*(울산과학기술원), Marco Cannone(Universit'e Paris-Est Marne-La-Vall'ee,, France)
Hantaek Bae*, UNIST, Marco Cannone, Universit'e Paris-Est Marne-La-Vall'ee, France
It is important to investigate regularity and long-time behavior of dissipative solutions in connection with coherent structure of solutions. Along this direction, we take the mild solution approach to the incompressible Navier-Stokes equations and we discuss the Log-Lipschitz regularity of the velocity field and H\"older regularity of the flow map of the 3D Navier-Stokes equations.
2010 Mathematics Subject Classification: 76D03, 76D05, 35Q35
Key Words and Phrases: Navier-Stokes equations, Log-Lipschitz regularity of the velocity field, H\"older regularity of the flow map
- ⋅ 22nd-D-15:15 − 15:35 Shear stress concentration occuring in stiff fiber-reinforced composites (KiHyun Yun)
- 윤기현(한국외대)
KiHyun Yun, Hankuk University of Foreign Studies
When two stiff fibers are are closely located in a fiber-reinforced composite, the shear stress tensor, the gradient of the solution to a conductivity equation, can be arbitrarily large as the distance between two inclusions tends to zero. It is crucial to precisely characterize the blow-up of the gradient of such an equation. Here, we show that the blow-up of the gradient is characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann-Poincare type operator, and we also review the recent progress on this subject.
2010 Mathematics Subject Classification: 35J05
Key Words and Phrases: shear stress tensor, fiber-reinforced composite, gradient estimate
- ⋅ 22nd-D-15:35 − 15:55 Wave maps from the hyperbolic plane. (Andrew Lawrie, Sung-Jin Oh, Sohrab Shahshahani)
- Andrew Lawrie(MIT), 오성진*(고등과학원), Sohrab Shahshahani(UMass Amherst)
Andrew Lawrie, MIT, Sung-Jin Oh*, KIAS, Sohrab Shahshahani, UMass Amherst
In this talk, we consider equivariant wave maps from the hyperbolic plane into two model rotationally symmetric targets, namely the two sphere ($\mathbb{S}^{2}$) and the hyperbolic plane itself ($\mathbb{H}^{2}$). Due to the non-Euclidean geometry of the domain, this problem exhibits markedly different phenomena compared to its Euclidean counterpart. For instance, there exist numerous stationary solutions to not only $\mathbb{S}^{2}$ but also $\mathbb{H}^{2}$, which has a negative constant curvature. Moreover, when the target is $\mathbb{S}^{2}$, the spectrum of the linearized operator about certain stationary solutions possesses a \emph{gap eigenvalue}, i.e., a simple eigenvalue in the gap $(0, 1/4)$ between $0$ and the essential spectrum.
2010 Mathematics Subject Classification: 35L71, 58J45
Key Words and Phrases: wave maps, geometric wave equation, energy critical, hyperbolic space
- ⋅ 22nd-D-16:05 − 16:25 Late-time behaviour of Israel particles in a FLRW spacetime (Ho Lee, Ernesto Nungesser)
- 이호*(경희대), Ernesto Nungesser(ICMAT)
Ho Lee*, Kyung Hee University, Ernesto Nungesser, ICMAT
In this talk we study the relativistic Boltzmann equation in a spatially flat FLRW spacetime. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions.
2010 Mathematics Subject Classification: 35Q20
Key Words and Phrases: Boltzmann, FLRW, Israel particle
- ⋅ 22nd-D-16:25 − 16:45 Uniform $\ell^p$-Stability of the Cucker-Smale model and its application to the mean-field limit (Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang)
- 하승열(서울대), 김정호(서울대), Xiongtao Zhang*(서울대)
Seung-Yeal Ha, Seoul National University, Jeongho Kim, Seoul National University, Xiongtao Zhang*, Seoul National University
We study uniform $\ell^p$-stability of the Cucker-Smale(C-S) flocking model with respect to initial data. When the coupling strength is sufficiently large compared to the size of initial data, it is well known that the global (mono-cluster) flocking configuration emerges asymptotically exponentially fast. However, global-in-time stability of such flocking states for the C-S model has not been treated in literature. In this paper, we show that $\ell^p$-distance between two solutions along the C-S flow is uniformly bounded by the $\ell^p$-distance between two initial data with total momentum. As a direct application of uniform $\ell^2$-stability, we obtain the uniform-in-time mean-field limit from the particle C-S model to the kinetic C-S model in Wasserstein-1 distance.
2010 Mathematics Subject Classification: 34Dxx, 34Gxx
Key Words and Phrases: $\ell^p$-stability, flocking, mean-field limit, the Cucker-Smale model
- ⋅ 22nd-E-17:00 − 17:20 Global existence and dynamics of solutions of a spatially inhomogeneous, cross-diffusion system (Jaywan Chung)
- 정재환(국가수리과학연구소)
Jaywan Chung, NIMS
When there is not enough resources, organisms may move faster to find more. Such a moving strategy is described by the starvation-driven dispersal (SDD) which is a spatially inhomogeneous diffusion. The SDD is a good moving strategy because it allows organisms use resources almost optimally, unlike the random dispersal (RD). In this talk, we consider a triangular cross-diffusion system which models the weak-strong compeition between SDD and RD. The global existence of solutions will be given and the competitive superiority of SDD over RD will be discussed analyzing the linear stability. This is a joint work with Ohsang Kwon.
2010 Mathematics Subject Classification: 35K51, 35A01, 35B35
Key Words and Phrases: cross-diffusion, spatially inhomogeneous diffusion, triangular system, global existence, linear stability
- ⋅ 22nd-E-17:20 − 17:40 Remarks on Chern-Simons equations in one space dimension (Hyungjin Huh)
- 허형진(중앙대)
Hyungjin Huh, Chung-Ang University
Chern-Simons-Higgs and Chern-Simons-Dirac equations are considered under the \linebreak gauge condition $A_0=0$. We find solitary wave solutions and prove global existence of solutions.
2010 Mathematics Subject Classification: 35Q40
Key Words and Phrases: Chern-Simons-Higgs, Chern-Simons-Dirac, Gauge condition
- ⋅ 22nd-E-17:50 − 18:10 Transonic shocks and vorticity (Myoungjean Bae)
- 배명진(포항공대)
Myoungjean Bae, POSTECH
Unless a shock has a special geometric structure, irrotational flow is very likely to generate vorticity across a shock. So it is important to analyze multi-dimensional compressible flow with nonzero vorticity. In this talk, recent results on 2-D, 3-D solutions to Euler/Euler-Poisson system with nonzero vorticity will be presented. This talk is based on joint works with several collaborators, including B. Duan, Y. Park and C. Xie among many others.
2010 Mathematics Subject Classification: 35M10, 76L05
Key Words and Phrases: transonic, shocks, free boundary, vorticity, mixed type PDEs
- ⋅ 22nd-E-18:10 − 18:30 Infinite iteration of normal form for nonlinear dispersive equations (Soonsik Kwon)
- 권순식(한국과학기술원)
Soonsik Kwon, KAIST
We will begin with explaining the Poincare-Dulac normal form idea for canonical nonlinear dispersive equations. Taking an infinite iteration of normal form, we change a nonlinear dispersive equation to an equation with infinite series nonlinear terms, which contain near-resonant interactions. We will present several results in various setting. Also, a relation to canonical transform will be discussed. Works in the talk are joint works with Tadahiro Oh, Zihua Guo, Jaywan Chung, and Haewon Yoon.
2010 Mathematics Subject Classification: 35Q55
Key Words and Phrases: normal form, unconditional well-posedness, nonlinear dispersive equations
- ⋅ 23rd-F-09:00 − 09:20 Pointwise behavior of the Navier-Stokes flow with slowly decreasing initial data (Bum Ja Jin, Tongkeun Chang)
- 진범자*(목포대), 장통근(연세대)
Bum Ja Jin*, Mokpo National University, Tongkeun Chang, Yonsei University
In this paper we study the spatial and temporal decay estimate of the Navier-Stokes flow corresponding to an uniformly but slowly decaying initial velocity.
We show the local solvability of the Navier-Stokes equations with
\[
|u(x,t)|\leq C_0(1+|x|+\sqrt{t})^{-\min\{\alpha,n\}}\Big(ln{(1+|x|+\sqrt{t})}\Big)^{\delta_{\alpha n}}
\]
when $(1+|x|)^\alpha e^{-tA}h\in L^\infty({\mathbb R}^n_+\times (0,\infty))$.
We also show that the solution exists globally in time for $1\leq \alpha$ when $\|(1+|x|)^\alpha e^{-tA}h\|_{L^\infty({\mathbb R}^n_+\times (0,\infty))}$ is small enough.
2010 Mathematics Subject Classification: 35K61
Key Words and Phrases: pointwise behavior, spatial decay, temporal decay
- ⋅ 23rd-F-09:20 − 09:40 Small data scattering for fractional Hartree equations (Yonggeun Cho)
- 조용근(전북대)
Yonggeun Cho, Chonbuk National University
In this talk we will consider scattering problem of the fractional
Schr\"{o}dinger equations with Hartree type potential
$\mu|x|^{-\gamma}$. The nonexistence of scattering for $0 < \gamma\le 1$
and small data scattering for $2 < \gamma < d$ will be presented
briefly. The main argument for scattering relies on the time decay of linear solution. In order to get a good time decay it is useful to use vector field $\mathbf{J} = x + i\alpha|\nabla|^{\alpha-2}\nabla$, which enables us to show
a small data cattering when $\frac{6-2\alpha}{4-\alpha} < \gamma < 2$. The main difficulty
is caused by the nonlocality of $|\nabla|^\alpha$ and weak dipserson of
$e^{it|\nabla|^\alpha}$. These will be settled down by commutator estimates
via Balakrishnan's formula.
2010 Mathematics Subject Classification: 35Q55
Key Words and Phrases: fractional NLS, Hartree type potential, short-range interaction
- ⋅ 23rd-F-09:50 − 10:10 Existence of weak solutions in Wasserstein space for a chemotaxis model coupled to fluid equations (Kyungkeun Kang, Hwa Kil Kim)
- 강경근(연세대), 김화길*(고등과학원)
Kyungkeun Kang, Yonsei University, Hwa Kil Kim*, KIAS
We consider a coupled system of Keller-Segel type equations and the
incompressible Navier-Stokes equations in spatial dimension two and
three. We first establish existence of a weak solution of a
Fokker-Plank equation in the Wasserstein space under the assumption
that initial mass is bounded and integrable. As a result, we
construct the density of biological organism in the Wasserstein
space for Keller-Segel-Navier-Stokes equations, in case that its
initial mass is just assumed to be bounded and integrable.
2010 Mathematics Subject Classification: 35K55
Key Words and Phrases: chemotaxis, Navier-Stokes equations, Fokker-Plank equations, Wasserstein space, splitting method
- ⋅ 23rd-F-10:10 − 10:30 Derivation of quasi-geostrpic equation from the rotational compressible magnetohydrodynamic flows (Chingsaio Cheng, Young-Sam Kwon, Cheng-Fang Su )
- Chingsaio Cheng(National Central Univ.), 권영삼*(동아대), Cheng-Fang Su(National, Central Univ.)
Chingsaio Cheng, National Central University, Young-Sam Kwon*, Dong-A University, Cheng-Fang Su, National Central University
In this talk, we derive the quasi-geostropic equation governed by magnetic field from compressible magnetohydrodynamic flows. The method is based on the relative entropy method.
It covers two results: the existence of a global strong solution of quasigeostropic equations and convergence of velocity, density, and magnetic field.
2010 Mathematics Subject Classification: 35
Key Words and Phrases: asymptotic limit, compressible magnetohydrodynamic flows
- ⋅ 23rd-G-10:45 − 11:05 On the generalized Kirchhoff type equation in the presence of past and finite history (Daewook Kim)
- 김대욱(서원대)
Daewook Kim, Seowon University
In this talk we concern on an aspect of decay rate of the generalized Kirchhoff type energy of the viscoelastic system in the presence of past and finite history. We get its proof by using the smallness condition functions with respect to generalized Kirchhoff coefficient, the relaxation function and internal time-varying delay. In fact, the difference of the energy consist in Kirchhoff type potential energy and finite and infinite delay.
2010 Mathematics Subject Classification: 35L70
Key Words and Phrases: generalized Kirchhoff type energy, viscoelastic system, relaxation function, internal time-varying delay
- ⋅ 23rd-G-11:05 − 11:25 A weak solution for a class of the square growth nonlinear elliptic systems (Tacksun Jung, Q-Heung Choi)
- 정택선(군산대), 최규흥*(인하대)
Tacksun Jung, Kunsan National University, Q-Heung Choi*, Inha University
This paper is devoted to investigate existence of nontrivial weak solutions for a class of nonlinear elliptic systems with Dirichlet boundary condition. We get a theorem which shows existence of at least one nontrivial solutions for a class of square growth nonlinear elliptic systems. We obtain this result by combining critical point theory and topological methods. Among the topological methods we use generalized mountain pass theorem.
2010 Mathematics Subject Classification: 35J50, 35J55
Key Words and Phrases: class of nonlinear elliptic systems, variational method, critical point theory, $(P.S.)$ condition, generalized mountain pass theorem
- ⋅ 23rd-G-11:35 − 11:55 Existence of Infinitely many solutions for a class of the elliptic systems with even functionals (Tacksun Jung, Q-Heung Choi)
- 정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University
We get a result that shows the existence of infinitely many solutions for a class of the elliptic systems involving subcritical Sobolev exponents nonlinear terms with even functionals on the bounded domain with smooth boundary. We get this result by variational method and critical point theory induced from invariant subspaces and invariant functional.
2010 Mathematics Subject Classification: 35J47, 35J50, 35J57
Key Words and Phrases: elliptic system, deformation lemma, $(P.S.)^{*}$ condition, subcritical Sobolev exponents, variational method, critical point theory, invariant functional, invariant subspaces
- ⋅ 23rd-G-11:55 − 12:15 Existence of nontrivial weak solution for the fractional p-Laplacian problems with discontinuous nonlinearities (In Hyoun Kim, Yun-Ho Kim)
- 김인현(인천대), 김연호*(상명대)
In Hyoun Kim, Incheon National University, Yun-Ho Kim*, Sangmyung University
In this talk, we are interested in the existence of a nontrivial solution to nonlinear problem
\begin{equation*}
\begin{cases}
(-\Delta)_p^su = \lambda f(x,u)
\quad \text{a.e.}\ &\text{in} \quad \Omega, \\
u = 0 &\text{in} \ \ \mathbb R^{N}\setminus\Omega,
\end{cases}
\end{equation*}
where the operator
\begin{equation*}
(-\Delta)^s_p u(x)=2\lim_{\varepsilon\searrow0} \int_{\mathbb{R}^{N}\backslash
B_\varepsilon(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\,dy, \quad x\in \mathbb R^{N}
\end{equation*}
is the fractional $p$-Laplacian. Here $\Omega$ is a bounded domain in $\mathbb R^{N}$ with Lipschitz boundary, $B_\varepsilon(x):=\{y\in \mathbb{R}^N: |x-y|<\varepsilon\}$, and $f:\Omega\times\mathbb{R}\to \mathbb{R}$ is a possibly discontinuous function. The aim of this talk is to show the existence of at least one nontrivial weak solution for the problem above without
the Ambrosetti and Rabinowitz condition, by using a critical point theorem which is a generalization of an abstract result of Bonnano.
2010 Mathematics Subject Classification: 35R11, 35A15, 35J60
Key Words and Phrases: fractional p-Laplacian, critical points of nonsmooth functions, variational method
- Probability and Stochastic Differential Equation
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Stability of estimates for fundamental solutions under Feynman-Kac perturbations for symmetric Markov processes (Kazuhiro Kuwae, Daehong Kim)
- Kazuhiro Kuwae*, Fukuoka University, Daehong Kim, Kumamoto University
We give a necessary and sufficient condition for the stability of the global estimates for fundamental solution of generalized Feynman-Kac semigroup of symmetric Markov processes under the stability of the global heat kernel estimates by bounded perturbations. As an application, a weak type global estimate holds for generalized Feynman-Kac semigroup with (extended) Kato conditions for measures appeared in the perturbation under the stability of the given heat kernel estimates by bounded perturbations. This generalizes the all known results on the stability of global integral kernel estimates under symmetric Feynman-Kac perturbations by Kato class measures in the framework of symmetric Markov processes.
2010 Mathematics Subject Classification: 31C25, 60J45, 60J57
Key Words and Phrases: Feynman--Kac perturbation, symmetric Markov processes, Dirichlet forms, heat kernel, spectral function, continuous additive functional of zero energy, Kato class, (semi-)Green-tight measures, \linebreak (semi-)
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Remark on unavoidable sets of some time inhomogeneous diffusion processes in ergodic random media (Daehong Kim)
- Daehong Kim, Kumamoto University
A set is said to be unavoidable with respect to a stochastic process if the process hits the set almost surely. In this talk, we discuss the unavoidability of sets for certain time inhomogeneous diffusion processes in stationary ergodic random envrionments and give several examples. Tanaka's celebrated result on the recurrence of a diffusion process in a multidimensional Brownian environment will be reviewed in the present setting.
2010 Mathematics Subject Classification: 31C25, 60K37, 60J60
Key Words and Phrases: ergodic random environments, unavoidable sets, time inhomogeneous diffusion processes, Dirichlet forms, simulated annealing, recurrence
- ⋅ 22nd-D-16:05 − 16:25 Comparison principle and positivity for the stochastic fractional heat equation (Le Chen, Kunwoo Kim)
- Le Chen(Univ. of Kansas), 김건우*(포항공대)
Le Chen, University of Kansas, Kunwoo Kim*, POSTECH
In this talk, we first consider a sample-path comparison principle for the stochastic fractional heat equation with measure-valued initial data which includes delta measure and measures with heavier tails than linear exponential growth. Using the comparison principle, we also show that the solution is strictly positive as long as the initial data is positive somewhere (i.e., the stochastic fractional heat equation also has infinite speed of propagation).
2010 Mathematics Subject Classification: 60H15, 60G60, 35R60
Key Words and Phrases: stochastic fractional heat equation, comparison principle, positivity, measure-valued initial data
- ⋅ 22nd-D-16:25 − 16:45 Application of elliptic regularity results: n-regularized Liouville Brownian motion (Jiyong Shin)
- 신지용(고등과학원)
Jiyong Shin, KIAS
In this talk we apply elliptic regularity results to a concrete symmetric Dirichlet form.
Given the symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we
show weak existence of the corresponding singular stochastic differential equation for any starting
point in $R^d$. As an application of our approach we can show the existence of n-regularized
Liouville Brownian motion only via Dirichlet form theory starting from all points in $R^2$.
2010 Mathematics Subject Classification: 31C25, 60J60, 47D07
Key Words and Phrases: Dirichlet forms, Liouville Brownian motion, elliptic regularity
- ⋅ 22nd-E-17:00 − 17:20 An $L_p$-boundedness of stochastic singular integral operators and its application to SPDEs (Ildoo Kim, Kyeonghun Kim)
- 김일두*(고등과학원), 김경훈(고려대)
Ildoo Kim*, KIAS, Kyeonghun Kim, Korea University
In this talk we introduce a stochastic counterpart of the H\"ormander condition on the kernel $K(r,t,x,y)$:
there exists a pseudo-metric $\rho$ on $(0,\infty)\times \mathbf{R}^d$ and a positive constant $C_0$ such that
for $X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty) \times \mathbf{R}^d$,
\begin{align}
\label{sto sin con}
\sup_{X,Y}\int_{0}^\infty \left[ \int_{\rho(X,Z) \geq C_0 \rho(X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right]^2 dr <\infty.
\end{align}
We prove that the stochastic singular integral of the type
\begin{align}
\label{sto sin}
\mathbb{T} g(t,x) :=\int_0^{t} \int_{\mathbf{R}^d} K(t,s,x,y) g(s,y)dy dW_s
\end{align}
is a bounded operator on $\mathbb{L}_p=L_p(\Omega \times (0,\infty);
L_{p}(\mathbf{R}^d))$ for any $p\geq 2$ if it is bounded when $p=2$ and the stochastic H\"ormander condition holds. Here $\Omega$ is a probability space and $W_t$ is a Wiener process on $\Omega$. Proving the $L_p$-boundedness of such integral operators is the key step in constructing an $L_p$-theory for linear stochastic partial differential equations (SPDEs in short). As a byproduct of our result on stochastic singular operators we obtain the maximal $L_p$-regularity result for a very wide class of SPDEs.
2010 Mathematics Subject Classification: 60H15, 42B20, 35S10, 35K30, 35B45
Key Words and Phrases: stochastic singular integral operator, $L_p$-estimate, pseudo-differential operator, stochastic partial differential equation
- ⋅ 22nd-E-17:20 − 17:40 Poisson kernel estimates for subordinate Brownian motions in Lipschitz domains (Jaehoon Kang)
- 강재훈(서울대)
Jaehoon Kang, Seoul National University
In this talk, we discuss Poisson kernel estimates for subordinate Brownian motions in bounded Lipschitz domains. To derive Poisson kernel estimates, we use lower bound of ratio of harmonic functions near boundary, which can be obtained by considering lower bound of exit time of narrow cone and factorization of harmonic functions.
2010 Mathematics Subject Classification: 60J45
Key Words and Phrases: Poisson kernel, subordinate Brownian motion, Lipschitz domain, harmonic function
- ⋅ 22nd-E-17:50 − 18:10 The pricing of options with default risk under stochastic volatility (Min-Ku Lee, Jeong-Hoon Kim, Sung-Jin Yang)
- 이민구*(군산대), 김정훈(연세대), 양승진(연세대)
Min-Ku Lee*, Kunsan National University, Jeong-Hoon Kim, Yonsei University, Sung-Jin Yang, Yonsei University
In this paper, we derive the closed-form pricing formula of the option with the default risk under stochastic volatility. The Fourier transform method is applied to solve the option pricing problem in the form of a partial differential equation under the well-known Heston's stochastic volatility model. Additionally, we investigate the effects of the stochastic volatility on the option in comparison with the constant volatility.
2010 Mathematics Subject Classification: 91G80
Key Words and Phrases: vulnerable option, Heston model, option pricing, default risk
- Analysis
- ⋅ 22nd-B-10:45 − 11:05 Best proximity point of generalized varphi-weak contraction mapping in metric space (Kyung Soo Kim)
- 김경수(경남대)
Kyung Soo Kim, Kyungnam University
The purpose of this talk, we consider the existence of a unique best proximity point $x^*$ in $A$ such that $d(x^*, Tx^*) = dist(A,B)=\inf\{d(a,b): a \in A,~ b \in B\}$ for generalized varphi-weak contraction mapping $T: A \rightarrow B$, where $A$, $B$ are nonempty subsets of a metric space $(X,d)$.
2010 Mathematics Subject Classification: 54H25, 47H09, 47H10, 41A65
Key Words and Phrases: metric space, generalized varphi-weak contraction mapping, fixed point, P-property, optimal solution, best proximity point
- ⋅ 22nd-B-11:05 − 11:25 Boundedness for the general semilinear duffing equations via the twist theorem (Daxiong Piao)
- 박대웅(Ocean Univ. of China)
Daxiong Piao, Ocean University of China
We consider the boundedness of all solutions for the periodic semilinear equation
$$x'' + {\omega}^2 x + \psi (x, t) = p(t)$$
where $\psi(x, t)$ does not necessarily satisfy the so called polynomial-like growth condition:
$$\lim_{|x|\rightarrow \infty} x^m \psi^{(m)} (x, t) = 0$$
for some finite $m$. Usually this condition is needed in the references about boundedness problems of semilinear Duffing equations. Two cases of resonance and non-resonance are considered respectively. This is a joint work with Yiqian Wang, Zhiguo Wang, Lei Jiao and Xiao Ma.
2010 Mathematics Subject Classification: 34C15, 37E40, 37J40
Key Words and Phrases: Hamiltonian systems, boundedness of solutions, canonical transformation, Moser's small twist theorem
- ⋅ 22nd-B-11:35 − 11:55 Boundedness of solutions in periodic continuous linear systems (Dohan Kim, Rinko Miyazaki, Jong Son Shin)
- 김도한(서울대), Rinko Miyazaki(Shizuoka Univ.), 신정선*(Hosei Univ.)
Dohan Kim, Seoul National University, Rinko Miyazaki, Shizuoka University, Jong Son Shin*, Hosei University
We give criteria on the existence of bounded solutions to the perturbed $\tau$-periodic continuous linear system $x'(t) = (A+B(t))x(t) + f(t),\ x(0)=w$ by using our recent results. The proof is based on the eigenvalues and eigenspaces of the matrix $A$ and of the periodic operator $V(0,-\tau)$ generated from solution operators $V(t,s)$ of the $\tau$-periodic linear system
$
x'(t)=B(t)x,
$
provided that $A$ and $B(t)$ are commutative. Moreover, we consider the set of initial values $w$ at $t=0$ for the system to have bounded solutions on $[0,\infty)$.
2010 Mathematics Subject Classification: 34A30, 34C11
Key Words and Phrases: periodic continuous linear systems, spectrum, bounded solution
- ⋅ 22nd-C-13:30 − 13:50 Change of scale transformations with a drift on an analogue of Wiener space (Dong Hyun Cho)
- 조동현(경기대)
Dong Hyun Cho, Kyonggi University
Let $C[0,T]$ denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval $[0,T]$. Let $ a$ be in $C[0,T]$ and let $h$ be of bounded variation with $h\neq 0$ a.e. on $[0,T]$. Define a stochastic process $Z:C[0,T]\times[0,T]\to\mathbb R$ by
\begin{eqnarray*}
Z(x,t)=(h\chi_{[0,t]},x)+x(0)+a(t)
\end{eqnarray*}
for $x\in C[0,T]$ and $t\in [0,T]$, where $(\cdot,\cdot)$ denotes the Paley-Wiener-Zygmund stochastic integral. For a partition $0=t_0<t_1<\cdots<t_n<T$ of $[0,T]$, define a random vector $Z_n:C[0,T]\to\mathbb R^{n+1}$ by
\begin{eqnarray*}
Z_n(x)=(Z(x,t_0),Z(x,t_1),\ldots,Z(x,t_n)).
\end{eqnarray*}
With the conditioning function $Z_n$ which does not contain present position $Z(x,T)$ of the path $Z(x,\cdot)$, we evaluate conditional expectations, that is, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by
\begin{eqnarray*}
\int_{L_2[0,T]}\exp\{i(v,Z(x,\cdot))\}d\sigma (v)
\text{ and }
f((v_1,Z(x,\cdot)),\ldots,(v_r,Z(x,\cdot))),
\end{eqnarray*}
where $\sigma$ is a complex Borel measure of bounded variation on $L_2[0,T]$, $f\in L_p(\mathbb R^r)$ with $1\le p\le\infty$ and $\{v_1,\ldots,v_r\}$ is an orthonormal subset of $L_2[0,T]$. We then show that the $L_p$-analytic conditional Fourier-Feynman transform $T_q^{(p)}[[(F_Z*G_Z)_q|Z_n](\cdot,\vec\xi_{n})|Z_{n}] $ of the conditional convolution product for the functions $F_Z$ and $G_Z$ which are described above, can be expressed by the formula
\begin{align*}
&\ T_q^{(p)}[[(F_Z*G_Z)_q|Z_n](\cdot,\vec\xi_{n})|Z_{n}](y,\vec\eta_{n})\\
=&\
\biggl[T_q^{(p)}[F_Z|Z_{n}]\biggl(\frac{1}{\sqrt{2}}y+(\sqrt{2}-1)a, \frac{1}{\sqrt{2}}(\vec\eta_{n}+\vec\xi_{n})-(\sqrt{2}-1)\vec a_n\biggr)\biggr]\\
&\ \times
\biggl[T_q^{(p)}[G_Z|Z_{n}]\biggl(\frac{1}{\sqrt{2}}y- a, \frac{1}{\sqrt{2}}(\vec\eta_{n}-\vec\xi_{n})+\vec a_n\biggr)\biggr]
\end{align*}
for a nonzero real $q$, almost surely $y\in C[0,T]$, and $P_{Z_n}$ almost surely $\vec\xi_{n}, \vec\eta_{n}\in\mathbb R^{n+1}$, where $\vec a_n=(a(t_0),a(t_1),\ldots,a(t_n))$ and $P_{Z_n}$ is the probability distribution of $Z_n$ on the Borel class of $\mathbb R^{n+1}$. Compared with the previous results, the conditioning function $Z_n$ in this talk does not contain the present position $Z(x,T)$ of $Z(x,\cdot)$ and the effects of drift $a$ depend on the polygonal function of $a$ so that the results of this talk do not depend on a particular choice of the initial distribution of the paths.
2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: analogue of Wiener space, change of scale formula, conditional Fourier-Feynman transform, conditional convolution product, conditional expectation, Wiener space
- ⋅ 22nd-C-13:50 − 14:10 Challenge for unsolvable problem for the change of scale formula about unbounded cylinder function (Young Sik Kim)
- 김영식(한양대)
Young Sik Kim, Hanyang University
We challenge and prove the unsolvable problem for the change of scale formula for the Wiener integral about the unbounded cylinder function on the Wiener space.
2010 Mathematics Subject Classification: 26D10, 42B35, 42C4
Key Words and Phrases: Wiener space
- Geometric Group Theory and Dynamics of Group Actions
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk] A cohomological obstruction to the existence of Clifford--Klein forms (Yosuke Morita)
- Yosuke Morita, The University of Tokyo
A Clifford--Klein form is a quotient of a homogeneous space $G/H$ by a discrete subgroup $\Gamma$ of $G$ acting properly and freely on $G/H$. It naturally admits a structure of a manifold locally modelled on $G/H$. Comparing relative Lie algebra cohomology and de Rham cohomology of a Clifford--Klein form, we give a new obstruction to the existence of compact Clifford--Klein forms of a given homogeneous space. As a corollary, we see that every complete pseudo-Riemannian manifold of signature $(p,q)$ with positive constant sectional curvature is noncompact if $p, q > 0$, $q$: odd.
2010 Mathematics Subject Classification: Primary 57S30; Secondary 17B56, 22E40, 22F30
Key Words and Phrases: Clifford--Klein form, discontinuous group, pseudo-Riemannian manifold, relative Lie algebra cohomology
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Primitive stable representations in higher rank semisimple Lie groups (Inkang Kim, Sungwoon Kim)
- 김인강(고등과학원), 김성운*(제주대)
Inkang Kim, KIAS, Sungwoon Kim*, Jeju National University
We define primitive stable representations of free groups into higher rank semisimple Lie groups. Then we show that positive representations on a compact surface with one boundary component, which are introduced by Fock and Goncharov, are primitive stable. Furthermore we prove that the holonomies of convex projective structures on a compact surface with one boundary component are primitive stable.
2010 Mathematics Subject Classification: 32G15, 22F30, 20E05
Key Words and Phrases: primitive stable, free group, convex projective structure
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] The asymptotic geometry of statistically convex-cocompact actions (Wenyuan Yang)
- Wenyuan Yang, Peking University
In this talk, I will describe a notion of statistically convex-cocompact actions of groups with a contracting element. This could be thought of as a statistical version of convex-cocompact Kleinian groups. Moreover, this notion also includes relatively hyperbolic groups, CAT(0) groups with rank-1 elements, mapping class groups. Our main result shows that this class of groups have purely exponential growth, and ``almost every'' elements are contracting elements, etc. This gives new results in the above class of groups and recovers some known results via different methods. For instance, one corollary is Maher's result that pseudo-Anosov elements are generic in mapping class groups. Another is Knieper's result that the number of non-rank 1 geodesics is exponentially small in compact rank-1 manifolds.
2010 Mathematics Subject Classification: 20F65, 20F67
Key Words and Phrases: contracting element, purely exponential growth, convex-compact actions
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Two aperiodic mono-tile tilings and their dynamics (Jeong-Yup Lee, Robert V. Moody)
- 이정엽*(가톨릭관동대), Robert V. Moody(Univ. of Victoria)
Jeong-Yup Lee*, Catholic Kwandong University, Robert V. Moody, University of Victoria
It had been a long-standing question whether there exists a single prototile which tiles the plane only aperiodically. We call it ``aperiodic mono-tile". Some years ago, Taylor-Socolar came up with an aperiodic mono-tile under the assumption that the reflection of the prototile is allowed to use to tile the plane. After the discovery of Taylor-Socolar tile, Penrose also introduced another new aperiodic mono-tile under the same assumption of allowing the reflection of the prototile. We construct two tilings by these tiles and consider tiling dynamics with these tilings. It is shown in [Baake-Gaehler-Grimm, 2012] that the tiling dynamics are actually different. However both of these tiles are based on hexagons and there are some relations between the two tilings. In this talk, we show the relations between the two tilings. The two dynamical systems are closely related with nested triangulations which are the geometrical realization of Q-adic integers.
2010 Mathematics Subject Classification: 52C23
Key Words and Phrases: aperiodic tiling, aperiodic mono-tile
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk, The winner of 2016 Excellent Research Paper Award] Obstruction for a virtual $C^2$ action on the circle (Hyungryul Baik, Sang-hyun Kim, Thomas Koberda)
- 백형렬(Max Planck Institute), 김상현*(서울대), Thomas Koberda(Univ. of Virginia)
Hyungryul Baik, Max Planck Institute, Sang-hyun Kim*, Seoul National University, Thomas Koberda, University of Virginia
When does a group virtually admit a faithful $C^2$ action on the circle? We provide an obstruction using a RAAG. Examples include all (non-virtually-free) mapping class groups, Out(Fn) and Torelli groups. This answers a question by Farb. (Joint work with Hyungryul Baik and Thomas Koberda)
2010 Mathematics Subject Classification: 37E30
Key Words and Phrases: diffeomorphism group, mapping class group, Artin group
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Rotation number and actions of 2-orbifold groups on the circle (Yoshifumi Matsuda)
- Yoshifumi Matsuda, Aoyama Gakuin University
A Hyperbolic structure on a 2-orbifold induces an action of a 2-orbifold group on the circle. Such a action is called a Fuchsian action. We show that the semi-conjugacy class of a Fuchsian action of certain 2-orbifold groups is characterized by rotation number of several elements. We also show that some lifts of Fuchsian actions admits a similar characterization.
2010 Mathematics Subject Classification: 37E45, 37C85
Key Words and Phrases: rotation number, group actions on the cirlcle
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Bounded orbits in homogeneous spaces (Jinpeng An)
- Jinpeng An, Peking University
For certain choices of subgroups $H$ of $SL(n,R)$, bounded orbits of the translation actions of $H$ on the homogeneous space $SL(n,R)/SL(n,Z)$ are closely related to bad approximation properties of numbers, vectors, and matrices. We will discuss recent progresses for the case that $H$ is a one-parameter diagonalizable subgroup and related results on badly approximable vectors in Diophantine approximation.
2010 Mathematics Subject Classification: 22F30
Key Words and Phrases: homogeneous space, bounded orbit
- ⋅ 22nd-D-16:05 − 16:25 Equidistribution with an error rate and Diophantine approximation over function fields (Sanghoon Kwon, Seonhee Lim)
- 권상훈*(고등과학원 난제연구센터), 임선희(서울대)
Sanghoon Kwon*, KIAS-Center for Mathematical Challenge, Seonhee Lim, Seoul National University
Let $K=\mathbb{F}_q(\!(t^{-1})\!)$ and $Z=\mathbb{F}_q[t]$. For a certain proper subgroup $H$ of the horospherical group for one-parameter diagonalizable groups, we prove pointwise equidistribution with an error rate of each $H$-orbit in $SL(d,K)/SL(d,Z)$ for compactly supported locally constant functions. We sketch the proof of equidistribution theorem and present related result for certain unbounded functions.
2010 Mathematics Subject Classification: 28A33, 37A15, 22E40
Key Words and Phrases: field of formal series, effective equidistribution, diophantine approximation
- ⋅ 22nd-D-16:25 − 16:45 Asymptotic distribution of values of isotropic quadratic forms at S-integral points (Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai)
- 한지영*(서울대), 임선희(서울대), Keivan Mallahi-Karai(Jacobs Univ.)
Jiyoung Han*, Seoul National University, Seonhee Lim, Seoul National University, Keivan Mallahi-Karai, Jacobs University
We prove a generalization of a theorem of Eskin-Margulis-Mozes in an $S$-arithmetic setup: suppose we are given a finite set of places $S$ over $\mathbb Q$ containing the archimedean place, an irrational isotropic form $q$ of rank $n\geq 4$ on $\mathbb Q_S$, a product of
$p$-adic intervals $I_p$, and a product $\Omega$ of star-shaped sets.
We show that unless $n=4$ and $q$ is split in at least one place, the number of $S$-integral vectors $v \in T \Omega$ satisfying simultaneously $q(v) \in I_p$ for $p \in S$ is asymptotically given by $$
\lambda(q, \Omega) |I| \cdot \| T \|^{n-2},$$ as $T$ goes to infinity, where $|I |$ is the product of Haar measures of the $p$-adic intervals $I_p$.
2010 Mathematics Subject Classification: 37L60
Key Words and Phrases: homogeneous dynamics, ergodic theory, S-arithmetic Lie group
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Rigidity for group actions (Nhan-Phu Chung, Yongle Jiang, Keonhee Lee)
- 종인부*(성균관대), Yongle Jiang(State Univ. of New York at Buffalo), 이건희(충남대)
Nhan-Phu Chung*, Sungkyunkwan University, Yongle Jiang, State University of New York at Buffalo, Keonhee Lee, Chungnam National University
In this talk, we will present certain rigidity results for group actions on compact spaces. In the first part, we will provide a new characterization of one end groups via cocycle superrigidity of their full shifts. As a consequence, we have an application in continuous orbit equivalence rigidity. In the second part, we prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is $C^0$ local rigid. A new characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property is also mentioned. The first part is joint with Yongle Jiang and the second part is joint work with Keonhee Lee.
2010 Mathematics Subject Classification: 37A20, 37C85
Key Words and Phrases: cocycle superrigidity, groups with one end, subshifts of finite type, pseudo-orbit tracing property, expansiveness, local rigid
- ⋅ 23rd-F-09:00 − 09:40 [Invited Talk] Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex (Eiko Kin, Hyunshik Shin)
- Eiko Kin(Osaka Univ.), 신현식*(한국과학기술원)
Eiko Kin, Osaka University, Hyunshik Shin*, KAIST
Let $M$ be a hyperbolic fibered 3-manifold and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(S_n, \psi_n)$ of fibers whose projective classes coverge to $[(S,\psi)]$ in the fibered face such that the asymptotic translation length of $\psi_n$ on the curve complex $\mathcal{C}(S_n)$ converges to $0$ at a rate of $1/|\chi(S_n)|^2$. As an application, we show that minimal asymptotic translation lengths of mapping class group, hyperelliptic mapping class group, and hyperelliptic handlebody group of a closed surface of genus $g$ are bounded below and above by $C/g^2$ and $D/g^2$ for some constants $C$ and $D$, respectively.
2010 Mathematics Subject Classification: 57M50, 37E30
Key Words and Phrases: mapping class groups, pseudo-Anosov, curve complex, translation length
- ⋅ 23rd-F-09:50 − 10:10 Finite index subgroups of right-angled Artin groups (Hyo Won Park)
- 박효원(서울대)
Hyo Won Park, Seoul National University
Right-angled Artin groups are the graph product whose vertex groups are infinite cyclic groups,
which are defined by finite simple graphs.
A finite simple graph is called thin-chordal if it has no induced subgraphs that are isomorphic
to either the cycle with 4 vertices or the path with 4 vertices.
We will discuss group properties related to right-angled Artin groups from thin-chordal graphs.
We show that a right-angled Artin group is defined by a thin-chordal graph if and only if
every finite index subgroup of the group is a right-angled Artin group.
2010 Mathematics Subject Classification: 20F36
Key Words and Phrases: right-angled Artion groups, finite index subgroups, thin-chordal graph
- ⋅ 23rd-F-10:10 − 10:30 On the volume and the Chern-Simons invariant for the $2$-bridge knot orbifolds (Ji-Young Ham, Joongul Lee, Alexander Mednykh, Aleksey Rasskazov)
- 함지영*(홍익대), 이준걸(홍익대), Alexander Mednykh(Sobolev Institute of Mathematics, Novosibirsk State University, Chelyabinsk University), Aleksey Rasskazov(Sobolev Institute of Mathematics, Novosibirsk State University, Chelyabinsk University)
Ji-Young Ham*, Hongik University, Joongul Lee, Hongik University, Alexander Mednykh, Sobolev Institute of Mathematics, Novosibirsk State University,, Chelyabinsk University, Aleksey Rasskazov, Sobolev Institute of Mathematics, Novosibirsk State University,, Chelyabinsk University
We extend some part of the unpublished paper written by Mednykh and Rasskazov. Using the approach indicated in this paper we derive the Riley-Mednykh polynomial for some family of the $2$-bridge knot orbifolds. As a result we obtain explicit formulae for the volume of cone-manifolds and the Chern-Simons invariant of orbifolds of the knot with Conway's notation $C(2n,4)$.
2010 Mathematics Subject Classification: 57M27, 57M25
Key Words and Phrases: fundamental set, volume, Chern-Simons invariant, cone-manifold, orbifold, explicit formula, $2$-bridge knot, knot with Conway's notation $C(2n,4)$, Riley-Mednykh polynomial
- Differential Geometry I
- ⋅ 22nd-A-09:00 − 09:20 On embeddings of the Grassmannian $Gr(2,m)$ into the Grassmannian $Gr(2,n)$ (Minhyuk Kwon)
- 권민혁(서울대)
Minhyuk Kwon, Seoul National University
In this paper, we consider holomorphic embeddings of $Gr(2,m)$ into $Gr(2,n)$. We can study such embeddings by finding all possible total Chern classes of the pull-back of the universal bundles under these embeddings. To do this, we use the relations between Chern classes of the universal bundles and Schubert cycles together with properties of complex vector bundles of rank $2$ on Grassmannians. Consequently, we find a condition on $m$ and $n$ for which any holomorphic embedding of $Gr(2,m)$ into $Gr(2,n)$ is linear.
2010 Mathematics Subject Classification: 32M10, 57T15, 14M15
Key Words and Phrases: Grassmannians, holomorphic embeddings, Chern classes
- ⋅ 22nd-A-09:20 − 09:40 Cartan-Kaehler theory by means of reduction of Pfaffian systems (Chong-Kyu Han, Hyeseon Kim)
- 한종규*(서울대), 김혜선(고등과학원 난제연구센터)
Chong-Kyu Han*, Seoul National University, Hyeseon Kim, KIAS-Center for Mathematical Challenge
Given a Pfaffian system, to construct an equivalent involutive system seems the main problem of the classical Cartan-Kaehler theory, where the reduction of the system to where torsion vanishes is among the main ideas. Without following the Cartan-Kaehler algorithm the authors present here a method of reduction to invariant submanifolds of Pfaffian systems and obtain eventually a system in involution so that the existence can be proved in non-analytic categories as well. As an application, the authors present their recent results on controllability for affine control systems.
2010 Mathematics Subject Classification: 37C10, 57R27, 58A17, 93B05, 93C15
Key Words and Phrases: integrability, torsion, invariant submanifolds, first integrals, controllability
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Zeta-determinant and BFK-gluing formula (Yoonweon Lee)
- 이윤원(인하대)
Yoonweon Lee, Inha University
Zeta-determinant is a global spectral invariant, which plays an important role in the theory of the analytic torsion. In this talk, we review the definition of the zeta-determinant and analytic torsion, and give some computations and geometric applications of the zeta-determinants. We next discuss the BFK-gluing formula of the zeta-determinant of Laplacians given by Burghelea, Friedlander and Kappeler and give some applications of BFK-gluing formula to the relative zeta-determinants on a manifold with cylindrical ends or cusps and the gluing formula of the Dolbeault Laplacian on a compact Riemann surface discussed by Wentworth.
2010 Mathematics Subject Classification: 58J52
Key Words and Phrases: zeta function, zeta-determinant, Laplacian, gluing formula, heat kernel
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] The Ricci flow on four-manifolds and the Seiberg-Witten equations (Masashi Ishida)
- Masashi Ishida, Tohoku University
A solution to the normalized Ricci flow is called non-singular if the solution exists for all
time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton
introduced this notion as a nice class of solutions and classified 3-dimensional non-singular
solutions. On the other hand, in 1994, new invariants of smooth 4-manifolds were introduced by Edward Witten. The invariants are constructed from nonlinear partial differential equations which are called the Seiberg-Witten equations. In this talk, we shall study properties of 4-dimensional non-singular solutions by using the Seiberg-Witten equations. In particular, we shall see that gauge theoretical invariants associated with the Seiberg-Witten equations give us obstructions to the existence of 4-dimensional non-singular solutions.
2010 Mathematics Subject Classification: 53C44, 57R57
Key Words and Phrases: Ricci flow, the Seiberg-Witten invariants
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Scalar curvature and Einstein metrics on noncompact complete manifolds (Seongtag Kim)
- 김성택(인하대)
Seongtag Kim, Inha University
Conformal changes of Riemannian metrics is an important tool for the
construction of a model space from a given manifold. This methods was successful for the
uniformization of compact 2-dimensional manifolds. In higher dimensional case, the natural
candidates of the model space are space of constant sectional
curvature or Einstein spaces. However, controlling all the sectional curvatures by
choosing only a conformal factor is highly over determined. Instead,
scalar curvature and conformally invariant quantities, e.g. Bach tensor, have
been extensively studied using conformal methods. Important results have been made in the study of the Yamabe problem in Riemannian case and CR case, and the rigidities of Bach-flat
metrics. This method is also used for the construction of an Einstein metric as
a solution of the Einstein constraint equations which comes from the
Cauchy problem in Relativity. In this talk, we will review the
developments of these problem and introduce recent results on noncompact complete case.
2010 Mathematics Subject Classification: 53C21
Key Words and Phrases: scalar curvature and Einstein metrics, noncompact complete manifolds, Einstein constraint equations, conformal metrics
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Min-max theory for free boundary minimal hypersurfaces (Martin Man-chun Li)
- Martin Man-chun Li, The Chinese University of Hong Kong
Given a Riemannian manifold $M$ with boundary, consider the area functional on the space of hypersurfaces $\Sigma \subset M$ with $\partial \Sigma \subset \partial M$, the critical points are called free boundary minimal hypersurfaces. Such critical points are minimal hypersurfaces meeting the boundary $\partial M$ orthogonally. In this talk, we give a general existence theory for free boundary mimimal hypersurfaces in compact Riemannian manifolds with boundary. If the ambient manifold has nonnegative Ricci curvature and strictly convex boundary, we show that there exist infinitely many properly embedded free boundary minimal hypersurfaces. This is joint work with Xin Zhou.
2010 Mathematics Subject Classification: 53A10, 53C42
Key Words and Phrases: min-max theory, free boundary problem, minimal surfaces
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Plateau's problem for $C^{1,\alpha}$ tangentially immersed boundaries (Leobardo Rosales)
- Leobardo Rosales, Keimyung University
The classic Plateau's problem asks if given a smooth simple closed curve in space, does there exist a smooth orientable (embedded) surface-with-boundary spanning that curve, and having least area among all surfaces-with-boundary spanning the given curve? While the answer is in the affirmative, to solve Plateau's problem requires developing the theory of currents; these are generalized orientable surfaces-with-boundary. In this talk we study Plateau's problem for $C^{1,\alpha}$ tangentially immersed boundaries.
2010 Mathematics Subject Classification: 28A75, 49Q05, 49Q15
Key Words and Phrases: Plateau, currents, area-minimizing
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Archimedes' gravestone and some characterizations of spheres (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\mathbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surface area of regions between parallel hyperplanes.
We also establish two characterizations of elliptic paraboloids in the $(n+1)$-dimensional Euclidean space
${\mathbb E}^{n+1}$ with the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume of regions between parallel hyperplanes.
For further study, we suggest a few open problems.
2010 Mathematics Subject Classification: 53A05, 53A07
Key Words and Phrases: Archimedes, Gaussian curvature, surface area, hypersphere, hypersurface, Gauss-Kronecker curvature
- ⋅ 22nd-D-16:05 − 16:45 [Invited Talk] Transversally harmonic and biharmonic maps on foliated manifolds (Seoung Dal Jung)
- 정승달(제주대)
Seoung Dal Jung, Jeju National University
Let $(M,\mathcal F)$ and $(M',\mathcal F')$ be two foliated Riemannian manifolds and $\phi:M\to M'$ be a smooth foliated map, i.e., leaf preserving map. In this talk, we give the properties of transversally harmonic and biharmonic map. In fact, transversally harmonic map can be considered as a generalization of harmonic map. Specially, we study the Liouville type theorems for transversally harmonic and biharmonic maps.
2010 Mathematics Subject Classification: 53C12
Key Words and Phrases: transversally harmonic map, transversally biharmonic map
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Projective spaces and 7-sphere (Jae-Hyouk Lee)
- 이재혁(이화여대)
Jae-Hyouk Lee, Ewha Womans University
In this talk, we consider invariant polynomials on complex and quaternionic projective spaces and discuss hypersurface geometry on them and related spheres via Hopf fibrations. As an application, we sutdy the hypersurface geometry of a 7-sphere.
2010 Mathematics Subject Classification: 53C30, 53A20
Key Words and Phrases: complex projective space, quaternionic projective space, hypersurface, focal set
- ⋅ 22nd-E-17:50 − 18:10 The gap theorem on four-dimensional compact Ricci solitons (Jong Taek Cho)
- 조종택(전남대)
Jong Taek Cho, Chonnam National University
In this talk, we prove a fundamental gap theorem on four-dimensional compact Ricci solitons.
2010 Mathematics Subject Classification: 53C20
Key Words and Phrases: Einstein manifold, Ricci soliton, gap theorem
- ⋅ 22nd-E-18:10 − 18:30 Singularities of spacelike surfaces in Anti de Sitter 4-space (Donghe Pei)
- 배동하(Northeast Normal Univ.)
Donghe Pei, Northeast Normal University
In this talk, we study the singularities of spacelike surfaces in Anti de Sitter 4-space by the lightlike Gauss-Kronecker curvature. And we find the equivalent conditions of inflection points of real type, imaginary type or flat type from the view point of lightlike geometry.
2010 Mathematics Subject Classification: 53C21
Key Words and Phrases: singularities of spacelike surfaces in Anti de Sitter 4-space
- Differential Geometry II
- ⋅ 22nd-D-15:15 − 15:35 A real hypersurface in complex Grassmannians of rank two with Reeb parallel shape operator (Hyunjin Lee, Young Jin Suh )
- 이현진*(경북대), 서영진(경북대)
Hyunjin Lee*, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this talk I will introduce the notion of Reeb parallel shape operator for a real hypersurface in Riemannian manifolds. And we give a classification theorem on Hopf hypersurfaces in complex Grassmannians with rank two.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: Reeb parallelism, shape operator, Hopf hypersurface, complex Grassmannian with rank two
- ⋅ 22nd-D-15:35 − 15:55 Real hypersurfaces in complex hyperbolic two-plane Grassmannians with respect to Ricci tensor (Gyu Jong Kim, Young Jin Suh)
- 김규종*(경북대), 서영진(경북대)
Gyu Jong Kim*, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this paper, first we introduce a new notion of pseudo anti
commuting for real hypersurfaces in complex hyperbolic two-plane Grassmannians.
And give a classification theorem that such a hypersurface must be a tube over a totally real totally geodesic, a horosphere whose center at the infinity is singular, or an exceptional case.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: complex hyperbolic two-plane Grassmannians, hypersurface, pseudo anti commuting
- ⋅ 22nd-D-16:05 − 16:25 Classification results of real hypersurfaces in complex Grassmannians with rank two (Changhwa Woo, Young Jin Suh)
- 우창화*(경북대), 서영진(경북대)
Changhwa Woo*, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this talk, we introduce a new commuting condition between the Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_{X}\phi T=TR_{X}\phi$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex (hyperbolic) two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex (hyperbolic) two-plane Grassmannians with commuting condition respectively.
2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C15
Key Words and Phrases: complex two-plane Grassmannian, real hypersurface, Hopf hypersurface, generalized Tanaka-Webster connection, parallelism, Reeb parallelism, Ricci tensor
- Symplectic Topology
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk] Non-displaceable toric fibers on compact toric manifolds via tropicalizations (Jaeho Lee)
- 이재호(기초과학연구원)
Jaeho Lee, IBS-CGP
We give a combinatorial way to locate non-displaceable Lagrangian toric fibers on compact
toric manifolds. By taking the intersection of certain tropicalizations coming from combinatorial
data of a moment polytope, we locate all strongly bulk-balanced fibers introduced by Fukaya, Oh, Ohta, and Ono.
2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Lagrangian toric fiber, tropicalization
- ⋅ 22nd-A-09:50 − 10:30 [Invited Talk] Fragmented Hofer's geometry and $C^0$-symplectic topology (Morimichi Kawasaki)
- Morimichi Kawasaki(기초과학연구원)
Morimichi Kawasaki, IBS-CGP
For an open subset of a symplectic manifold, we define the fragmented Hofer's norm which is Hofer's norm controlled by its fragmentation norms.
We give some observation on fragmented Hofer's norms.
For example, fragmented Hofer's norms are equivalent to the original Hofer's norm on compact symplectic manifolds, but not equivalent on the Euclidean space.
We also consider fragmented Hofer's geometry on Hamiltonian homeomorphism \linebreak groups.
As its application, we prove that the Hamiltonian homeomorphism group of an exact symplectic manifold is non-simple if the positive answer of Muller's problem holds.
2010 Mathematics Subject Classification: 53D35
Key Words and Phrases: Hofer's geometry, $C^0$-symplectic topology
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Lipschitz-exact Lagrangian submanifolds and Tonelli Hamiltonian (Lino Amorim, Yong-Geun Oh, Joana Oliveira Dos Santos)
- Lino Amorim(Univ. of Oxford), 오용근*(기초과학연구원 \& 포항공대), Joana Oliveira Dos Santos(Oxford Brooks Univ.)
Lino Amorim, University of Oxford, Yong-Geun Oh*, IBS-CGP \& POSTECH, Joana Oliveira Dos Santos, Oxford Brooks University
In this talk, we will introduce the notion of Lipschitz-exact
Lagrangian submanifolds and prove that any such Larangian admits
a graph selector. Then we explain How this can be used to generalize
Arnaud's result to the class of Lipschitz-exact Lagrangians:
any such Lagrangian submanifold must be a graph provided it is invariant
under a Tonelli Hamiltonian. This is based on the joint work with
Amorim and Oliveira Dos Santos.
2010 Mathematics Subject Classification: 53D05, 53D35, 53D40, 37E30
Key Words and Phrases: exact Lagrangian submanfiolds, graph selectors, Tonelli Hamiltonian, Aubry-Mather theory
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Lagrangian Floer theory - generation criterion for Fukaya category and related topics (Kaoru Ono)
- Kaoru Ono, Research Institute for Mathematical Sciences, Kyoto University
I will present ``generation criterion'' for Fukaya category and
discuss some related topics such as (super)heaviness of Lagrangian
submanifolds. This talk is based on a joint work with
M. Abouzaid, K. Fukaya, Y.-G. Oh and H. Ohta and also
other joint works with K. Fukaya, Y.-G. Oh and H. Ohta.
2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Lagrangian submanifold, Floer cohomology, quantum cohomology
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Symplectic displacement energy for exact Lagrangian immersions (Manabu Akaho)
- Manabu Akaho, Tokyo Metropolitan University
Displaceability or non-displaceability is an interesting phenomenon in symplectic topology. For example, the Arnold conjecture asks the non-displaceability of Lagrangian submanifolds under Hamiltonian isotopies. In this talk, we discuss the displacement energy for exact Lagrangian immersions. More precisely, we give a lower bound of the displacement energy by the symplectic area of teardrop discs. Our approach is based on some version of Floer homology of Lagrangian immersions and Chekanov's homotopy technique of continuations.
2010 Mathematics Subject Classification: 58F05, 58E05, 53D40
Key Words and Phrases: Lagrangian immersion, Floer homology, displacement energy
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] On the minimal number of singular fibers in Lefschetz fibrations over the torus (Andras Stipsicz, Ki-Heon Yun)
- Andras Stipsicz(Renyi institute of Mathematics), 윤기헌*(성신여대)
Andras Stipsicz, Renyi Institute of Mathematics, Ki-Heon Yun*, Sungshin Women's University
We show that the minimal number of singular fibers $N(g,1)$ in a
genus-$g$ Lefschetz fibration over the torus is at least $3$. As an
application, we show that $N(g, 1) \in \{ 3, 4\}$ for $g\ge 5$, $N(g, 1) \in \{3,
4,5 \}$ for $g= 3, 4$ and $N(2,1) = 7$.
2010 Mathematics Subject Classification: 57N13, 53D35
Key Words and Phrases: Lefschetz fibration, mapping class group
- ⋅ 23rd-F-09:00 − 09:40 [Invited Talk] Equivariant symplectic homology of Brieskorn manifolds (Myeonggi Kwon, Otto van Koert)
- 권명기*(서울대), Otto van Koert(서울대)
Myeonggi Kwon*, Seoul National University, Otto van Koert, Seoul National University
In this talk, we show that every rational number can be realized as the mean Euler characteristic of some contact structures of the 5-sphere. The mean Euler characteristic of $S^1$-equivariant symplectic homology is an invariant of contact structure, which takes values in real numbers. Our proof computes this invariant explicitly for some Brieskorn manifolds and uses a formula under the boundary connected sum. The main tool for computation is a version of Morse-Bott spectral sequence which converges to the equivariant symplectic homology groups. This is joint work with Otto van Koert.
2010 Mathematics Subject Classification: 53D10
Key Words and Phrases: symplectic homology, contact structure, Brieskorn manifold
- ⋅ 23rd-F-09:50 − 10:30 [Invited Talk] Higher rank vector bundles in Fukaya category (Hanwool Bae)
- 배한울(서울대)
Hanwool Bae, Seoul National University
The objects of Fukaya category of a symplectic manifold are said to be Lagrangian submanifolds equipped with flat line bundles. We extend the set of objects so that they contain not only line bundles but also flat vector bundles of any finite rank.
For this purpose, we modify the usual A-infinity structures of Fukaya category using the flat connections and prove that they satisfy the A-infinity relations. Then we also prove that every vector bundle on Lagrangian torus is isomorphic to a certain twisted complex in Fukaya category.
2010 Mathematics Subject Classification: 53D37
Key Words and Phrases: Fukaya category
- ⋅ 23rd-G-10:45 − 11:25 [Invited Talk] Integrable systems on Grassmannians and mirror symmetry (Yuichi Nohara)
- Yuichi Nohara, Meiji University
For each triangulation of a planar $n$-gon, one can associate a completely integrable system on the complex Grassmannian of two-planes in $\mathbb{C}^n$, which we call a bending system. We study Floer theoretic properties of Lagrangian fibers of bending systems and relation to mirror symmetry of the Grassmannian.
2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: integrable system, mirror symmetry
- Knots and Low Dimensional Manifolds
- ⋅ 22nd-A-09:00 − 09:20 Arc presentation of spatial graphs (Sungjong No, Seungsang Oh, Minjung Lee)
- 노성종*(이화여대), 오승상(고려대), 이민정(고려대)
Sungjong No*, Ewha Womans University, Seungsang Oh, Korea University, Minjung Lee, Korea University
Bae and Park found an upper bound on the arc index of prime links in terms of
the minimal crossing number. We extend the definition of the arc presentation to
spatial graphs and find an upper bound on the arc index $\alpha (G)$ of any spatial graph
$G$ such $\alpha (G) \leq c(G) + e + b$, where $c(G)$ is the minimal crossing number of $G$, $e$ is
the number of edges, and $b$ is the number of bouquet cut-components. This upper
bound is lowest possible.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: arc index, spatial graph
- ⋅ 22nd-A-09:20 − 09:40 Dehn surgeries on Seifert/Seifert twisted torus knots (Sungmo Kang)
- 강성모(전남대)
Sungmo Kang, Chonnam National University
Twisted torus knots are constructed basically by two torus knots and lie in a genus two Heegaard surface of the 3-sphere bounding two handlebody. In each handlebody a twisted torus knot is a curve and it is called Seifert curve if 2-handle addition along the curve yields Seifert-fibered manifold. According to J. Dean's paper, there are three types of Seifert curves on the boundary of a genus two handlebody. This enables us to consider several types of Seifert/Seifert twisted torus knots which admit Seifert-fibered manifold or graph manifod Dehn surgeries. In this talk, we show the classification of some types of Seifert/Seifert twisted torus knots with Dehn surgeries.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: Dehn surgery, twisted torus knots, Seifert curves
- ⋅ 22nd-A-09:50 − 10:10 The restoring argument and some intrinsically knotted graphs with 22 edges (Hyoungjun Kim, Thomas Mattman, Seungsang Oh)
- 김형준*(이화여대), Thomas Mattman(California State Univ., Chico), 오승상(고려대)
Hyoungjun Kim*, Ewha Womans University, Thomas Mattman, California State University, Chico, Seungsang Oh, Korea University
A graph is called intrinsically knotted if every embedding of the graph contains a non-trivially knotted cycle. Robertson and Seymour proved that there are only finitely many minor minimal intrinsically knotted graphs, but finding the complete set of them is still an open problem. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Lee, Kim, Lee, and Oh found the complete set of minor minimal intrinsically knotted graphs with 21 edges. It is also shown by Barsotti and Mattman, independently. Since Y-$\nabla Y$ move preserve intrinsic knotting, every intrinsically knotted graph has at least one cousin that is triangle-free intrinsically knotted. This means that finding the set of triangle-free intrinsically knotted graphs is the first step for classifying the complete set of minor minimal intrinsically knotted graphs. The restoring argument is the constructing operation which helps us to determine the given graph is IK or not. By using operation, I will show that there are five triangle-free intrinsically knotted graphs with 22 edges and a single degree 5 vertex. This work is collaborated with Thomas Mattman and Seungsang Oh.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: spatial graph, knot, intrinsic
- ⋅ 22nd-A-10:10 − 10:30 Knot mosaic tabulation (Hwa Jeong Lee, Lewis D. Ludwig, Joseph S. Paat, Amanda Peiffer)
- 이화정*(대구경북과학기술원), Lewis D. Ludwig(Denison Univ.), Joseph S. Paat(Johns Hopkins), Amanda Peiffer(Denison Univ.)
Hwa Jeong Lee*, DGIST, Lewis D. Ludwig, Denison University, Joseph S. Paat, Johns Hopkins, Amanda Peiffer, Denison University
In this presentation, we determine the mosaic number for all eight-crossing or fewer prime knots.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: knot, Mosaic number
- ⋅ 22nd-C-13:30 − 13:50 The complex of Haken spheres (Sangbum Cho, Yuya Koda)
- 조상범*(한양대), Yuya Koda(Hiroshima Univ.)
Sangbum Cho*, Hanyang University, Yuya Koda, Hiroshima University
When a Heegaard splitting admits Haken spheres, the complex of Haken spheres for the splitting is defined in a natural way, whose vertices are equivalence classes of the Haken spheres. In this talk, we describe the combinatorial structure of this complex for each of the reducible Heegaard splittings of genus two.
2010 Mathematics Subject Classification: 57N10
Key Words and Phrases: Heegaard splitting, Haken sphere, simplicial complex
- ⋅ 22nd-C-13:50 − 14:10 Reduction of bridge positions along bridge disks (Jung Hoon Lee)
- 이정훈(전북대)
Jung Hoon Lee, Chonbuk National University
Suppose a knot in a $3$-manifold is in $n$-bridge position.
We consider a reduction of the knot along a bridge disk $D$ and
show that the result is an $(n - 1)$-bridge position if and only if
there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair.
We also consider a reduction along two disjoint bridge disks.
2010 Mathematics Subject Classification: 57M99
Key Words and Phrases: bridge splitting, bridge disk, cancelling pair
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Twisted torus knots (Sangyop Lee)
- 이상엽(중앙대)
Sangyop Lee, Chung-Ang University
Twisted torus knots are obtained by adding full twists to some adjacent strands of torus knots. We will discuss some properties of these knots.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: twisted torus knots, Dehn surgery, incompressible surfaces
- ⋅ 22nd-D-15:15 − 15:35 Rack homology groups of decompositions of finite quandles (Yongju Bae, Seonmi Choi)
- 배용주(경북대), 최선미*(경북대)
Yongju Bae, Kyungpook National University, Seonmi Choi*, Kyungpook National University
A quandle is a set equipped with a binary operation satisfying three quandle axioms and it also can be expressed as a sequence of permutations of the underlying set satisfying two conditions. A decomposition of finite quandles was studied by Nelson, Wong, Ehrman, Gurpinar, Thibault and Yetter. For two finite quandles $Q$ and $Q'$, one can define a new operation $*$ on $Q\cup Q'$ whose restrictions on $Q$ and $Q'$ are the original quandle operations on $Q$ and $Q'$, respectively. In this talk, we will study a rack homology group of $(Q\cup Q',*)$.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: finite quandles, rack homology groups
- ⋅ 22nd-D-15:35 − 15:55 A quandle and its intrinsic topology (Yongju Bae, Byeorhi Kim)
- 배용주*(경북대), 김벼리(경북대)
Yongju Bae*, Kyungpook National University, Byeorhi Kim, Kyungpook National University
Let $Q$ be a quandle and $A$ a subset of $Q$. Let $c(A)$ denote the smallest connected subquandle of $Q$ containing $A$. Then $c$ satisfies Kuratowski's closure axioms and hence induces a topology $\mathfrak{T}$ for $Q$. In this talk, we will study properties of the topological space $(Q,\mathfrak{T})$.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: quandle
- ⋅ 22nd-D-16:05 − 16:45 [Invited Talk] Epimorphisms between knot groups (Masaaki Suzuki)
- Masaaki Suzuki, Meiji University
Recently, epimorphisms between knot groups are studied by many researchers. In this talk, we will survey several (mainly the speaker's) results on this topic. Namely, all the pairs of prime knots with up to $11$ crossings which admit meridional epimorphisms between knot groups are determined completely. Here a homomorphism is called to be meridional if it preserves meridians. On the other hand, we can construct a non-meridional epimorphism too. Finally, we consider the relationship between epimorphisms of two-bridge knot groups and their crossing numbers.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: epimorphism, knot group
- ⋅ 23rd-F-09:00 − 09:20 Rectangle condition and a family of alternating 3-bridge knots (Bo-hyun Kwon)
- 권보현(고려대)
Bo-hyun Kwon, Korea University
In this talk, we define the rectangle condition for n-bridge presentation of knots whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of 3-manifolds. We show that the satisfaction of the rectangle condition for n-bridge presentation is greater than or equal to 2. Especially, we gives a family of alternating 3-bridge knots by using the rectangle condition and a modified train track argument.
2010 Mathematics Subject Classification: 57M27
Key Words and Phrases: rectangle condition, 3-bridge knots, essential rational n-tangles, train track
- ⋅ 23rd-F-09:20 − 09:40 Unknotted gropes and Whitney towers (Jae Choon Cha, Taehee Kim)
- 차재춘(포항공대), 김태희*(건국대)
Jae Choon Cha, POSTECH, Taehee Kim*, Konkuk University
Gropes and Whitney towers are primary tools for the study of 4-manifolds. In this talk, we analyze the exterior of gropes and Whitney towers in a 4-manifold, and introduce the notion of unknotted gropes and Whitney towers. We show that this new notion is well-suited to transformations on gropes and Whitney towers. As an application, by taking a slice of unknotted gropes and Whitney towers we construct bi-filtrations of knots in 3-space, and show that these bi-filtrations are highly nontrivial using amenable signatures. This is joint work with Jae Choon Cha.
2010 Mathematics Subject Classification: 57N13
Key Words and Phrases: 4-manifold, grope, Whitney tower, knot, amenable signature
- ⋅ 23rd-F-09:50 − 10:30 [Invited Talk] The slice-ribbon conjecture and related topics (Tetsuya Abe)
- Tetsuya Abe, Osaka City University Advanced Mathematical Institute (OCAMI)
We survey the recent progress on the slice-ribbon conjecture,
which is one of the biggest conjectures in knot concordance theory.
In this talk, we will give some potential counterexamples of the slice-ribbon conjecture
using annulus twist, which is a certain operation on knots.
Also, we will discuss related topics.
2010 Mathematics Subject Classification: 57M25, 57R65
Key Words and Phrases: slice-ribbon conjecture, annulus twist, fibered knot, handle decomposition, knot concordance group
- ⋅ 23rd-G-10:45 − 11:05 Octahedral developing of knot complement and the character variety (Seonhwa Kim, Hyuk Kim, Insung Park, Seokbeom Yoon)
- 김선화*(기초과학연구원), 김혁(서울대), 박인성(Indiana Univ.), 윤석범(서울대)
Seonhwa Kim*, IBS-CGP, Hyuk Kim, Seoul National University, Insung Park, Indiana University, USA, Seokbeom Yoon, Seoul National University
It has been turned out that a knot complement can be decompose into ideal octahedra along a knot diagram as inspired by several quantum invariants. A solution to the Thurston's gluing equation gives a pseudo-developing map of the knot complement, called octahedral developing, which reflects both geometric properties of the knot complement and combinatorial information of the knot diagram. Furthermore, we see a relation between the set of octahedral shapes and character variety of the knot group.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: knot, PSL(2,C) representation
- ⋅ 23rd-G-11:05 − 11:25 Hyperbolic structures on knotted trivalent graphs (Jinseok Cho, Roland van der Veen)
- 조진석*(부산교대), Roland van der Veen(Leiden Univ.)
Jinseok Cho*, Busan National University of Education, Roland van der Veen, Leiden University
For a boundyar-parabolic representation $\rho : \pi_1(K) \rightarrow PSL(2,\mathbb C)$ of a knot $K$, the developing map of $\rho$ can be constructed combinatorially using the shadow-colorings and it gives explicit volume formula. In this talk, we generalize this method to knotted trivalent graphs. Especially, this method determines the hyperbolic structures with parabolic meridians on the graph complement manifolds. Furthermore, the explicit volume formulas can be obtained combinatorially.
2010 Mathematics Subject Classification: 57M50, 57M05, 57M27
Key Words and Phrases: hyerbolic structure, trivalent graph, volume
- ⋅ 23rd-G-11:35 − 12:15 [Invited Talk] Topological concordance versus smooth concordance (Se-Goo Kim)
- 김세구(경희대)
Se-Goo Kim, Kyung Hee University
A slice knot $K$ in the 3-sphere bounds a 2-disk, called a slice disk, in the 4-ball, whose boundary is the 3-sphere. Depending on whether a slice disk of $K$ can be embedded locally flatly or smoothly, $K$ is called a \emph{topologically} or \emph{smoothly} slice knot, respectively. Smoothly slice knots are topologically slice. However, there are many topologically slice knots that are not smoothly slice. For example, the existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to four-manifolds. Presented is the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two, as an application of Ozsv\'{a}th and Szab\'{o}'s Heegaard Floer theory.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: topological concordance, smooth concordance, slice knot
- Topology
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Symmetric commutator quotient groups of links and link towers in 3-manifolds (Fuquan Fang, Fengchun Lei, Jie Wu)
- Fuquan Fang, Capital Normal University, China, Fengchun Lei*, Dalian University of Technology, China, Jie Wu, National University of Singapore, Singapore
We introduce the intersection subgroup of the normal closures of the meridians of $n$-links in 3-manifolds and consider the quotient group of the intersection subgroup modular the symmetric commutator subgroup. We show that when the link is strongly nonsplittable and $n\geq 3$, the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians can be measured by the $n$th homotopy groups of the 3-manifold. This might be of particular interesting when the links are lying in the 3-sphere. Some other related topics will be discussed. This is a joint work with Fuquan Fang and Jie Wu.
2010 Mathematics Subject Classification: 55Q40, 57M25
Key Words and Phrases: symmetric commutator quotient groups, homotopy groups, Link tower
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Infra-nilmanifolds modeled on the generalized Heisenberg group $\mathrm{Nil}^{m+1}$ (Jong Bum Lee)
- 이종범(서강대)
Jong Bum Lee, Sogang University
The group $\mathrm{Nil}^{m+1} = \mathbb{R}^m\rtimes_\sigma\mathbb{R}$ generalizes the classical Heisenberg group $\mathrm{Nil}^3$. It is a connected and simply connected $m$-step
nilpotent Lie group. We take a subset of points in $\mathrm{Nil}^{m+1}$ with integer components which forms a lattice $\Gamma_{m+1}$. We prove that for $m\ge 3$
there is no infra-nilmanifold that is essentially covered by the nilmanifold $\Gamma_{m+1}\backslash\mathrm{Nil}^{m+1}$.
2010 Mathematics Subject Classification: 57S30
Key Words and Phrases: almost Bieberbach group, generalized Heisenberg group, infra-nilmanifold, nilpotent Lie group
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Topology of robot motion planning (Younggi Choi)
- 최영기(서울대)
Younggi Choi, Seoul National University
We introduce how to solve the problem of finding the minimum number of instructions for robot motion by using topological tools such as filtration length of spectral sequences, zero divisors cup length, stable cohomology operations, and we discuss some open problems.
2010 Mathematics Subject Classification: 55M99
Key Words and Phrases: robot motion planning, filtration length of spectral sequences, topological complexity
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Fixed point theory for non-Hausdorff spaces and its applications (Sang-Eon Han)
- 한상언(전북대)
Sang-Eon Han, Chonbuk National University
In this talk, we studies the fixed point theory from the viewpoint of non-Hausdorff topology.
Motivated by the ordinary Banach contraction principle [1] and Lefschetz theorem [9],
we can study the fixed point property for non-Hausdorff topological spaces [2-8, 10].
Unlike the ordinary research into the fixed point property, we observe some interesting results in non-Hausdorff topological versions of fixed point theorems.
This approach can be used in many areas in applied sciences.
\begin{thebibliography}{99}
\bibitem{1} L. J. Brouwer, Uber Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97--115.
\bibitem{2} S.-E. Han, Non-product property of the digital fundamental group, Information Sciences, Vol. 71 (2005), 73--91.
\bibitem{3} S.-E. Han, The $k$-homotopic thinning and a torus-like digital image in ${\bf Z}^{n}$, Journal of Mathematical Imaging and Vision, 31(1) (2008), 1--16.
\bibitem{4} S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Sciences and Applications, 9(3) (2016), 895--905.
\bibitem{5} S.-E. Han, Contractibility and Fixed point property: The case of Khalimsky topological spaces, Fixed point theory and Applications, (2016) 75 DOI. 10.1186/s13663-016-0566-8.
\bibitem{6} S.-E. Han, Euler characteristics of digital wedge sums and their applications, Topological Methods in Nonlinear Analysis, (2016), in press.
\bibitem{7} S.-E. Han, The fixed point property of an $M$-retract and its applications, Topology and its Applications, (2016), in press.
\bibitem{8} S.-E Han, Wei Yao, Homotopies based on Marcus Wyse topology and their applications, Topology and its applications, in press.
\bibitem{9} S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1) (1926), 1-49.
\bibitem{10} A. Rosenfeld, Digital topology, Amer. Math. Monthly 86 (1979), 76--87.
\end{thebibliography}
2010 Mathematics Subject Classification: 47H10, 54E35, N35, 68U10
Key Words and Phrases: Khalimsky topology, Marcus-Wyse topology, fixed point property, Alexandroff topology, almost fixed point property, digital space, digital image, digital topology
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] On flagified Bott and Bott-Samelson varieties (Dong Youp Suh)
- 서동엽(한국과학기술원)
Dong Youp Suh, KAIST
A Bott tower is a particular type of iterated $\mathbb CP^1$-fibrations which has many interesting properties.
In particular, every Bott tower is a nonsigular toric variety. A Bott--Samelson variety is a nonsingular algebraic variety constructed from a complex
semi-simple Lie group $G$ and its Borel subgroup $B$ as a resolution of singularities of a Schubert subvariety of the
flag manifold $G/B$. Even though every Bott--Samelson variety $M$ is not necessarily toric, as a smooth manifold, it is diffeomorphic
to a Bott tower $N$. In fact, the Bott tower $N$ is a toric degeneration of the Bott--Samelson variety $M$.
In toric topology, the notion of Bott tower is extended to generalized Bott tower as a particular type of iterated $\mathbb CP^{n_i}$-fibrations.
Like Bott tower, every generalized Bott tower is a nonsingular toric variety, and it satisfies many interesting properties.
However, there does not exist the notion of generalized Bott--Samelson variety. Namely, we can not use complex semi-simple Lie group to define
a generalized Bott--Samelson variety having generalized Bott tower as its toric degeneration.
In this sense, the notion of generalized Bott tower might not be a \lq true' generalization of Bott tower.
In this lecture, we introduce a flagified Bott tower and a flagified Bott--Samelson variety as a \lq true' generalization of a Bott tower and a Bott-Samelson variety, respectively.
A flagified Bott tower is a particular type of an iterated flag-manifold fibrations, and flagified Bott--Samelson variety is accordingly defined from a complex
semi-simple Lie group $G$ and its Borel subgroup $B$ which provides a different resolution of singularities of a Schubert variety from a Bott--Samelson vairety.
Even though a flagified Bott tower is not necessarily toric, it can be shown they are GKM-manifolds.
Moreover many of the properties Bott and Bott--Samleson varieties satisfy also hold for the flagified cases.
We will discuss some of these properties in this lecture. The current lecture is based on an on-going joint project with Shintaro Kuroki, Eunjeong Lee, and Jongbaek Song.
2010 Mathematics Subject Classification: 55N91, 57S25, 14M15, 14M25
Key Words and Phrases: Bott tower, Bott-Samelson variety, generalized Bott tower, flagified Bott tower, flagified Bott-Samelson variety
- ⋅ 22nd-D-16:05 − 16:25 A generalized Bott tower and its associated flagified Bott tower (Eunjeong Lee, Jongbaek Song, Dong Youp Suh)
- 이은정*(한국과학기술원), 송종백(한국과학기술원), 서동엽(한국과학기술원)
Eunjeong Lee*, KAIST, Jongbaek Song, KAIST, Dong Youp Suh, KAIST
A Bott tower is a sequence of $\mathbb{C}P^1$-fiber bundles such that each stage of which is a toric manifold. The notion of Bott tower can be extended to a generalized Bott tower, which is also a toric manifold. Namely, a generalized Bott tower is a sequence of $\mathbb{C}P^{n_j}$-fiber bundles, which are projectivization of Whitney sum of complex line bundles. Recently, another extended notion has been introduced, called flagified Bott tower, which is a sequence of $\mathcal{F}lag(n_j+1)$-fiber bundles. A flagified Bott tower is not a toric manifold but it has a torus action. For a given generalized Bott tower $M$, one can define the associated flagified Bott tower $N$ by using the splitting principle of the flag bundle. In this talk, we study the relation between a generalized Bott tower $M$ and its associated flagified Bott tower $N$. More precisely, when we consider the torus orbit closure of an appropriate point of the associated flagified Bott tower $N$, it can be understood as a sequence of blow-ups of the generalized Bott tower $M$. This talk is based on an on-going project with Jongbaek Song and Dong Youp Suh.
2010 Mathematics Subject Classification: 57R22, 57S25, 14M25
Key Words and Phrases: Bott tower, generalized Bott tower, flagified Bott tower, torus orbit, blow-up
- ⋅ 22nd-D-16:25 − 16:45 Weighted Stanley-Reisner ring of a simplicial fan (Jongbaek Song, Bahri Anthony, Soumen Sarkar)
- 송종백*(한국과학기술원), Bahri Anthony(Rider Univ.), Soumen Sarkar(Indian Institute Technology - Madras)
Jongbaek Song*, KAIST, Bahri Anthony, Rider University, Soumen Sarkar, Indian Institute Technology - Madras
Given an abstract simplicial complex $K$, the corresponding Stanley-Reisner ring $R[K]$ is obtained from the polynomial ring with coefficient ring $R$ by quotienting certain ideal which encodes the face structure of $K$. The Stanley-Reisner ring $R[\Sigma]$ of a simplicial fan $\Sigma$ can be naturally defined by considering its underlying simplicial complex. It is well-known that the equivariant cohomology ring of a smooth toric variety is isomorphic to the Stanley-Reisner ring of the corresponding fan which is smooth. However, in general, if the toric variety is singular, then its equivariant cohomology ring can be identified with the Stanley-Reisner ring over the rational coefficients. In this talk, we define a weighted Stanley-Reisner ring with integer coefficients and explain how it is related to the equivariant cohomology ring of a singular toric variety.
2010 Mathematics Subject Classification: 55N25, 13F55
Key Words and Phrases: Stanley-Reisner ring, equivariant cohomology, toric variety
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] The Browder-Dupont invariant: A tool to distinguish stably tangent bundles in odd dimensions (Yanghyun Byun)
- 변양현(한양대)
Yanghyun Byun, Hanyang University
We consider the so-called projective Stiefel manifold $X_n$ of type $(n,2)$ for each integer $n>2$. The dimension of $X_n$ is $2n-3$. There is a natural construction of a vector bundle $\theta_n$ of rank $2n-3$ over $X_n$ which is stably isomorphic to the tangent bundle $TX_n$. It is well-known that there are at most two vector bundles over an odd dimensional manifold up to isomorphism which are stably isomorphic to the tangent bundle and are of the rank same as the dimension. The only known invariant which may distinguish between the two is the Browder-Dupont invariant. We will actually calculate the invariant of $\theta_n$ to find that it is indeed isomorphic to $TX_n$.
2010 Mathematics Subject Classification: 57R22
Key Words and Phrases: stably tangent bundle, Browder-Dupont invariant
- ⋅ 22nd-E-17:50 − 18:30 [Invited Talk] Structure of factor maps in symbolic dynamical systems (Uijin Jung)
- 정의진(아주대)
Uijin Jung, Ajou University
A topological dynamical system (TDS) is a compact metric space together with a self-homeomorphism. Symbolic dynamical systems are TDSs where phase spaces are discretized, and can be used to investigate the properties of general TDSs using extensions given by generating partitions. It is well known from 70's in symbolic dynamics that given an entropy preserving (finite-to-one) factor map $ \pi : X \to Y$ from an irreducible shift of finite type $X$ onto a sofic shift $Y$, almost all points in $Y$ have the same number of preimages in $X$. This number is called the degree of a map, and widely studied and proved useful in the study of equal entropy factor maps between irreducible shifts of finite type and sofic shifts. To find a natural generalization of degree for non entropy-preserving maps (i.e., infinite-to-one maps), Allahbakhshi and Quas (2013) introduced the notion of class degree: If we define a certain equivalence relation on the fiber of reach point in $Y$, then almost all points in $Y$ have the same number of equivalence classes (called transition classes), and if a map $\pi$ is finite-to-one then class degree equals to degree. Many results have been established after the introduction of the class degree, and now class degree can be regarded as a generalization of degree in many aspect. In this talk, I will give a survey on the known results on the structure of fibers of factor maps between symbolic dynamical systems in terms of class degree and present recent progress on the topic.
2010 Mathematics Subject Classification: 37B10
Key Words and Phrases: degree, class degree, transition class, shift of finite type, factor map, sofic shift
- ⋅ 23rd-F-09:50 − 10:30 [Invited Talk] Hyperbolic structure of closed invariant sets for differentiable dynamical systems (Manseob Lee)
- 이만섭(목원대)
Manseob Lee, Mokwon University
In this talk, we introduce a main research problem in
differentiable dynamical systems which is called the $C^1$ density
conjecture, that is, diffeomorphisms of a compact smooth manifold
$M$ exhibiting either a homoclinic tangency or heterodimensional
cycle $C^1$ dense in the complement of the $C^1$ closure of
hyperbolic systems. For the problem, we consider a relation
between a closed invariant set and hyperbolic structure. Then we
are going to use the robust and generic diffeomorphism of various
dynamic properties.
2010 Mathematics Subject Classification: 37C10, 37C20
Key Words and Phrases: hyperbolic, homoclinic class, chain component, shadowing, expansive
- ⋅ 23rd-G-10:45 − 11:05 Several results on chaos in topological dynamics (Jiandong Yin, Bin Ling)
- Jiandong Yin*, Nanchang University, Bin Ling, Nanchang University
The aim of this note is to present several chaotic results in topological dynamics. Let F be a Furstenberg family, we introduce firstly a new conception of F-chaos and prove some sufficient conditions for a topological dynamics to be F-chaotic. Finally, as an application of one result, we give an example in which it is shown additionally that there exists a topological dynamics with proper positive Banach upper density recurrent points.
2010 Mathematics Subject Classification: 37B
Key Words and Phrases: chaos, F-chaos, topological dynamics
- ⋅ 23rd-G-11:05 − 11:25 Chain transitive sets of discrete dynamical systems on compact $C^\infty$-manifolds (Seunghee Lee)
- 이승희(국가수리과학연구소)
Seunghee Lee, NIMS
Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold $M$, and let $\Lambda\subset M$ be a chain transitive set of $f$. In this talk, we prove that if a diffeomorphism $f$ belongs to the $C^1$ interior of the set of $\Lambda$-topologically stable
diffeomorphisms then $\Lambda$ is hyperbolic.
2010 Mathematics Subject Classification: 37C75, 37C15
Key Words and Phrases: topological stability, chain transitive set, hyperbolicity
- ⋅ 23rd-G-11:35 − 11:55 Positively weak measure expansive systems (Jumi Oh)
- 오주미(성균관대)
Jumi Oh, Sungkyunkwan university
In this talk, we introduce the notion of positively weak measure expansive homeomorphisms and flows with the shadowing property on a compact metric space $X.$ And we prove that if a homeomorphism (or flow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive. Furthermore, we show that if a continuous map $f$ has a positive entropy with respect to an ergodic invariant measure, then $f$ is positively weak measure expansive.
2010 Mathematics Subject Classification: 37C50, 34D10, 37D20
Key Words and Phrases: shadowing property, measure expansive, positively weak measure expansive, topological entropy
- Numerical Analysis
- ⋅ 22nd-A-09:00 − 09:20 A new development of immersed finite element methods (Do Young Kwak)
- 곽도영(한국과학기술원)
Do Young Kwak, KAIST
We give a survey of new class of discretization methods, called IFEM, for elliptic interface problems using structured grids then present a new development also. For scalar problems we consider using
Lagrangian type $P_1$, Crouzeix-Raviart nonconforming $P_1$, bilinear and Rannacher Turek element
for rectangle grids. We point out that the earlier version of IFEMs does not yield optimal order of convergence. As a remedy for this, we consider two variants. One is to add line integrals to the bilinear forms to make the scheme consistent. Another is to us CR (or RT) nonconforming $P_1$($Q_1$) basis functions. We note that the convergence of
IFEM with nonconforming elements are guaranteed with an additional regularity assumption that Darcy velocity
belongs to $H^1$. Applications to mixed methods, and multigrid convergence is also discussed.
We also discuss the IFEM for elasticity problems, using CR nonconforming $P_1$ basis functions. In this case, the bilinear
forms are modified by adding the stability terms to guarantee the Korn's inequality. Several numerical examples are
provided which support the theory.
2010 Mathematics Subject Classification: 65N30, 74S05
Key Words and Phrases: immersed finite element method, Crouzeix-Raviart finite element, elasticity problems
- ⋅ 22nd-A-09:20 − 09:40 A hybrid two-step finite element method for flux approximation (JaEun Ku, Young Ju Lee, Dongwoo Sheen)
- 구자연(Oklahoma State Univ.), 이영주(Texas State Univ.), 신동우*(서울대)
JaEun Ku, Oklahoma State University, Young Ju Lee, Texas State Universiyt, Dongwoo Sheen*, Seoul National University
We present a new two--step method based on the
hybridization of mesh sizes in the traditional mixed finite element
method.
On a coarse mesh, the primary variable is approximated by a standard
Galerkin method, whose computational cost is very low.
Then, on a fine mesh, an $H({\rm div})$ projection of the dual variable
is sought as an accurate approximation for the flux variable.
Our method does not rely on the framework of traditional mixed
formulations,
the choice of pair of finite element spaces is, therefore, free from the
requirement of inf-sup stability condition. More precisely, our method is
formulated in a fully decoupled manner, still achieving an optimal error
convergence order.
This leads to a computational strategy much easier and wider to implement than
the mixed finite element method. Additionally, the independently posed
solution strategy allows to use different meshes as well as different
discretization schemes in the calculation of the primary and flux variables.
We show that the finer mesh size $h$ can be taken as {the} square
of the coarse mesh size $H$, or {a higher order power} with a proper choice of
parameter $\delta$. This means
that the computational cost
for the coarse-grid solution is negligible compared to that for the
fine-grid solution. {In fact, numerical experiments show an advantage of
using our strategy compared to the mixed finite element method.}
{Some guidelines to choose an optimal parameter $\delta$ are also given.}
In addition,
our approach is shown to provide an asymptotically exact a posteriori error
estimator for the primary variable $p$ in $H^1$ norm.
2010 Mathematics Subject Classification: 65N30, 65N15
Key Words and Phrases: two--step method, the primary and flux variables
- ⋅ 22nd-A-09:50 − 10:10 Hybrid DG finite element methods (Eun-Jae Park, Dong-wook Shin, Youngmok Jeon)
- 박은재*(연세대), 신동욱(연세대), 전영목(아주대)
Eun-Jae Park*, Yonsei University, Dong-wook Shin, Yonsei University, Youngmok Jeon, Ajou University
In this talk we present a priori and posteriori error estimators for hybrid discontinuous Galerkin (HDG) methods for elliptic equations \cite{JP-sinum10, JP-hybrid2}.
First, we present arbitrary-order HDG methods to solve the Poisson problem
and propose residual type error estimators. Next, we present guaranteed type error estimators by postprocessing scalar and flux unknowns.
Then, we consider diffusion problems with discontinuous coefficients.
Some numerical examples are presented to show the performance of the methods.
This is supported in part by NRF-2015R1A5A1009350.
\begin{thebibliography}{99}
\bibitem{JP-sinum10} Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 48 (2010), no. 5, 1968--1983.
\bibitem{JP-hybrid2} Y. Jeon and E.-J. Park, New locally conservative finite element methods on a rectangular mesh, Numer. Math. 123 (2013), 97--119.
\bibitem{JPS} Y. Jeon, E.-J. Park and D. Sheen, A hybridized finite element method for the Stokes problem, Computers and Mathematics with Applications, vol. 68, no. 12 (2014), 2222--2232.
\bibitem{SJP} D.-w. Shin, Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for advection--diffusion--reaction problems, Applied Numerical Mathematics, vol. 95 (2015), 292--303.
\end{thebibliography}
2010 Mathematics Subject Classification: 65
Key Words and Phrases: DG, adaptivity
- ⋅ 22nd-A-10:10 − 10:30 The hybrid difference methods (Youngmok Jeon)
- 전영목(아주대)
Youngmok Jeon, Ajou University
In this presentation we introduce the hybrid difference methods(HDM) for the elliptic and Navier-Stokes equations. The HDM is a finite difference version of the hybridized discontinuous Galerkin method. The HDM is comparable with the finite difference method (FDM). The main difference between the FDM and HDM is that the FD formula of a single type is deployed for all interior nodes in the FDM, while the cell finite difference and the interface difference are combined in the HDM.
The HDM is as easy to implement as the FDM, and it apparently possess several advantages over the FDM.
\begin{itemize}[leftmargin=5mm]
\item[1)] The method can be applied to nonuniform grids, retaining the optimal order of convergence.
\item[2)] Problems on a complicated geometry can be treated reasonably well, and the boundary condition can be imposed exactly on the exact boundary (no variational crime).
\item[3)] Stability problems when solving the Stokes/Navier-Stokes problem can be resolved without introducing a staggered grid.
\item[4)] Numerical analysis is based on a discrete divergence theory on each cell.
\item[5)] The flux conservation property holds in each cell and flux continuation holds across intercell boundaries.
\item[6)] The embedded static condensation property of the HDM reduces degrees of freedom a lot.
\end{itemize}
2010 Mathematics Subject Classification: 65N30, 65N38, 65N50
Key Words and Phrases: hybrid difference, cell finite difference, interface finite difference
- ⋅ 22nd-B-10:45 − 11:05 Finite element methods for domain singularity using the stress intensity factor (Seokchan Kim)
- 김석찬(창원대)
Seokchan Kim, Changwon National University
We consider the Poisson equation with homogeneous Dirichlet boundary
conditions, on a polygonal domain with one reentrant corner. The solution of the Poisson equation with a concave corner yields a singular decomposition, $u = w + \lambda \eta s$, where $w$ is regular, $s$ is a singular function, and the coefficient $\lambda$ is the so called stress intensity factor. Kim and Lee suggested an efficient way to compute an accurate numerical solution for this kind of domain singularity, which exploit a good approximated value of the stress intensity factor in 2016.
We try to apply this idea to other problems with other boundary conditions or with multiple singularities, etc.
2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: domain singularity, stress intensity factor
- ⋅ 22nd-B-11:05 − 11:25 A compact difference scheme for numerical solutions of the coupled Schr\"{o}dinger-KdV equations (Su-Cheol Yi)
- 이수철(창원대)
Su-Cheol Yi, Changwon National Uiniversity
A compact finite difference scheme is proposed to numerically solve the coupled Schr\"{o}dinger-KdV equations with initial and periodic boundary conditions. This scheme is second- and fourth-order accurate in time and space, respectively. Some conservation properties and error estimates are investigated. The proposed scheme well preserves and simulates the conservation properties and solitary solutions, and supports the theoretical results, and hence, our scheme can be considered as a stable and practical numerical method to solve the coupled equations.
2010 Mathematics Subject Classification: 65M06, 65M12, 65M15
Key Words and Phrases: Schr\"{o}dinger-KdV equations, compact difference scheme, stability, error estimate
- ⋅ 22nd-B-11:35 − 11:55 Optimal control problem for Maxwell's equations with pointwise state constraints (Imbo Sim)
- 심임보(국가수리과학연구소)
Imbo Sim, NIMS
In this presentation, we treat the optimal control problem of Maxwell's equations with the pointwise state constraints. To cope with the lack of regularity of the control to state mapping, we regularize the optimal control problem using the penalizing methods. This provides the high regularity of the optimality system, so it requires that fractional order regularity estimates for Maxwell's equations. For the error analysis of the regularized optimality system, we apply the Brezzi-Rappaz-Raviart (BRR) theory and the fractional order regularity estimates reflecting the effect of the variational coefficients. Finally, the numerical experiments are shown by using the N\'ed\'elec's curl-conforming edge elements for verifications of the theoretical results.
2010 Mathematics Subject Classification: 65N22
Key Words and Phrases: Optimal Control, Maxwell's Equations
- ⋅ 22nd-B-11:55 − 12:15 Computational approaches for random PDE optimization problems based on different matching functionals (Hyung-Chun Lee)
- 이형천(아주대)
Hyung-Chun Lee, Ajou University
In this talk, we consider an optimal control problem for partial differential equations with random inputs. To determine an applicable deterministic control $\hat{f}(x)$, we consider the three cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the three cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.
2010 Mathematics Subject Classification: 65N55, 65N30, 65Y10
Key Words and Phrases: stochastic optimal control, finite element method, stochastic sparse collocation method
- ⋅ 22nd-C-13:30 − 13:50 Estimation of the effective reproduction number from influenza outbreak data using Kalman filter (Hyunjoong Kim, Hee-Dae Kwon, Jeehyun Lee)
- 김현중(Univ. of Utah), 권희대(인하대), 이지현*(연세대)
Hyunjoong Kim, University of Utah, Hee-Dae Kwon, Inha University, Jeehyun Lee*, Yonsei University
The reproduction number, the number of secondary infections that arise from a typical primary case, measures the transmission potential. The effective reproductive number is often more practical when an infection is spreading through a population. However, estimation of time-dependent parameter is very challenging due to many factors including non-reproducible and incomplete epidemic data. In order to overcome difficulties in a standard approach, ideas of Kalman filter (KF) is employed. The Kalman filter is a recursive algorithm which calculates the optimal state of the system by taking a weighted average of the probability distribution from the model prediction and the measurement. In this research, KF is tailored to the frame involving nonlinearity, continuous-time dynamics and discrete-time measurement, and parameter estimation. The numerical results demonstrate that KF is much more efficient and robust than LSM in the aspects of the perturbation of initial values, the timing of sampling when the data is insufficient to represent complete dynamics and model discrepancy. The proposed technique is also applied to Novel influenza A (H1N1) 2009 data in Korea to estimate the real-time effective reproduction number.
2010 Mathematics Subject Classification: 65C99
Key Words and Phrases: parameter estimation, Kalman filter, effective reproduction number
- ⋅ 22nd-C-13:50 − 14:10 The Cahn-Hilliard equation for the growth of tissue (Junseok Kim, Darae Jeong)
- 김준석*(고려대), 정다래(고려대)
Junseok Kim*, Korea University, Darae Jeong, Korea University
In this talk, we consider the Cahn-Hilliard model and its finite difference method for the growth of tissue. The model is based on a phase-field equation. The proposed numerical algorithm is accurate and robust. We present the numerical experiments which show good agreement between theoretical prediction and numerical result.
2010 Mathematics Subject Classification: 65M06
Key Words and Phrases: Cahn-Hilliard equation, finite difference method, tissue growth
- ⋅ 22nd-C-14:20 − 14:40 Superconvergence of mixed finite element methods on triangular meshes (Kwang-Yeon Kim)
- 김광연(강원대)
Kwang-Yeon Kim, Kangwon National University
We deal with the superconvergence of the low-order Raviart-Thomas mixed finite element methods on triangular meshes for second-order elliptic equations. Superconvergence for low-order primal conforming finite element methods was already known and rigorously explained on uniform meshes in 1970s. Furthermore, through efforts in the past decades, it is now established that this superconvergence result is valid on mildly structured meshes satisfying the so-called Condition $(\alpha,\sigma)$, which means that most pairs of triangles form approximate parallelograms. For the mixed finite element methods, Brandts (1994 and 2000) and Dupont and Keenan (1998) independently established the superconvergence result on uniform meshes, but there is no known result on non-uniform meshes. In this talk we discuss how this superconvergence result can be extended to mildly structured meshes satisfying Condition $(\alpha,\sigma)$.
2010 Mathematics Subject Classification: 65N30, 65N15
Key Words and Phrases: superconvergence, Raviar-Thomas mixed finite element method
- ⋅ 22nd-C-14:40 − 15:00 Conservative upwind correction method for scalar linear hyperbolic equations (Sang Dong Kim, Yong Hun Lee)
- 김상동(경북대), 이용훈*(전북대)
Sang Dong Kim, Kyungpook National University, Yong Hun Lee*, Chonbuk National University
A conservative method for scalar hyperbolic equations
is presented using a quadrature rule and an ODE solver.
This numerical scheme consists of the upwind part plus a correction part,
which is derived by introducing a new variable for a given hyperbolic equation.
We provide stability and accuracy for the derived algorithm with
numerous computations as well.
2010 Mathematics Subject Classification: 65M55, 65N30, 49J20, 49K20
Key Words and Phrases: conservative method, hyperbolic scalar equation, ODE solver, quadrature rule, upwind method
- ⋅ 22nd-D-15:15 − 15:35 Axial green function methods (Do Wan Kim)
- 김도완(인하대)
Do Wan Kim, Inha University
We are going to talk about axial Green's function methods (AGMs) on free grids called axial lines. These are novel approaches in numerical computation. AGMs that we have developed for elliptic boundary value problems and the steady Stokes flows in complicated geometry use axial lines for discretization. These axial lines are parallel to axes and there is no restriction on their distribution. The salient feature of the methods is that not only one- dimensional Green's function for the axially split differential operators is sufficient to solve the multi-dimensional problems but also the free grids are available. In this talk, short introduction to AGMs is presented and then we show that the localization of axial lines enables us to enforce Neumann boundary condition, and refinement of axial lines on separated regions are readily available as well. We also discuss about the relationship between AGMs and FDMs.
2010 Mathematics Subject Classification: 65N99
Key Words and Phrases: Axial Green function, Complicated geometry, Localization
- ⋅ 22nd-D-15:35 − 15:55 High order stable operator splitting methods for the phase field equations (June-Yub Lee, Hyun Geun Lee, Jaemin Shin)
- 이준엽*(이화여대), 이현근(이화여대), 신재민(이화여대)
June-Yub Lee*, Ewha Womans University, Hyun Geun Lee, Ewha Womans University, Jaemin Shin, Ewha Womans University
The phase-field method has recently emerged as a powerful computational approach
for modeling and predicting mesoscale morphological and microstructure evolution
in materials. The basic idea in the phase-field model
is to introduce conserved or non-conserved order
parameters $\phi({\bf x},t)$ that vary continuously over thin interfacial layers
and are mostly uniform in the bulk phases.
Numerous numerical algorithms have been developed to improve accuracy
and numerical stability of the phase-field method.
In order to remove the time step constraint and guarantee the accuracy in time
for a sufficiently large time step, we present a first and a second order
semi-analytical Fourier spectral (SAFS) methods for solving the Allen--Cahn equation.
The core idea of the methods is to decompose the original equation into linear
and nonlinear subequations, which have closed-form solutions in the Fourier and
physical spaces, respectively.
We also propose a simple and stable second order operator splitting
method for Allen--Cahn (AC) type equations with nonlinear source terms
and for solving the phase field crystal equation.
We present numerical experiments to show
the accuracy and efficiency of the proposed methods.
2010 Mathematics Subject Classification: 65M12, 65M70, 35Q99
Key Words and Phrases: Allen-Cahn Equation, Fourier spectral method, phase field crystal equation
- ⋅ 22nd-D-16:05 − 16:25 A stable and convergent method for Hodge decomposition of fluid-solid interaction (Chohong Min)
- 민조홍(이화여대)
Chohong Min, Ewha Womans University
Our discussion in this work is restricted to the interaction of viscous incompressible fluid flow and a rigid body. We take the monolithic approach by Gibou and Min [22] that results in an extended Hodge projection. The projection updates not only the fluid vector field but also the solid velocities. We derive the equivalence of the extended Hodge projection to the Poisson equation with non-local Robin boundary condition. We prove the existence, uniqueness, and regularity for the weak solution of the Poisson equation, through which the Hodge projection is shown to be unique and orthogonal. Also, we show the stability of the projection in a sense that the projection does not increase the total kinetic energy of fluid and solid. Also, we discuss a numerical method as a discrete analogue to the Hodge projection, then we show that the unique decomposition and orthogonality also hold in the discrete setting. As one of our main results, we prove that the numerical solution is convergent with at least the rst order accuracy. We carry out numerical experiments in two and three dimensions, which validate our analysis and arguments.
2010 Mathematics Subject Classification: 74F10
Key Words and Phrases: fluid-solid interaction, convergence analysis, finite difference method
- ⋅ 22nd-D-16:25 − 16:45 Domain decomposition methods for the total variation minimization with $L^1$ fidelity term (Chang-Ock Lee, Changmin Nam)
- 이창옥*(한국과학기술원), 남창민(한국과학기술원)
Chang-Ock Lee*, KAIST, Changmin Nam, KAIST
In this talk, we propose domain decomposition methods for solving total variation minimization problem with $L^1$ fidelity term. We decompose the domain into rectangular subdomains, in which the local total variation problems are solved. We decouple the total variation terms defined on the interface so that the whole problem can be parallelized at the subdomain level. We apply our algorithms to the denoising, inpainting, and deblurring problems.
2010 Mathematics Subject Classification: 65N55, 65Y05, 49M27, 68U10
Key Words and Phrases: total variation, domain decomposition, parallel computation, $L^1$ fidelity term, image processing
- ⋅ 22nd-E-17:00 − 17:20 Mathematical modeling of the circulatory system with the beating heart (Wanho Lee, Eunok Jung)
- 이완호*(국가수리과학연구소), 정은옥(건국대)
Wanho Lee*, NIMS, Eunok Jung, Konkuk University
In this talk, we present a mathematical and computational model of the circulatory system with the beating heart to understand the complex hemodynamics of blood circulation. The immersed boundary method has been introduced to describe the interaction between the moving two-dimensional heart and intracardiac blood flow. The whole-heart model is governed by the Navier-Stokes system; this system is combined with a multi-compartment model of circulation using pressure-flow relations and the linearity of the discretized Navier-Stokes system. We investigate the velocity field, flowmeters, and pressure- volume loop in normal case. Simulation results show qualitatively good agreements with others found in the literature. This model, combining the heart and circulation, is useful for understanding the complex, hemodynamic mechanisms involved in normal circulation and cardiac diseases.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: circulatory system, beating heart model, lumped parameter model, immersed boundary method
- ⋅ 22nd-E-17:20 − 17:40 Boundary layer solutions of charge converving PB equations (YunKyong Hyon)
- 현윤경(국가수리과학연구소)
YunKyong Hyon, NIMS
For multispecies ions, we study boundary layer solutions of charge conserving Poisson–Boltzmann (CCPB) equations (with a small parameter) over a finite one-dimensional (1D) spatial domain, subjected to Robin type boundary conditions with variable coefficients. Hereafter, 1D boundary layer solutions mean that as approaches zero, the profiles of solutions form boundary layers near boundary points and become flat in the interior domain. These solutions are related to electric double layers with many applications in biology and physics. We rigorously prove the asymptotic behaviors of 1D boundary layer solutions at interior and boundary points. The asymptotic limits of the solution values (electric potentials) at interior and boundary points with a potential gap (related to zeta potential) are uniquely determined by explicit nonlinear formulas (cannot be found in classical Poisson–Boltzmann equations) which are solvable by numerical computations.
2010 Mathematics Subject Classification: 76A05, 76M99, 65C30
Key Words and Phrases: charge conserving Poisson–Boltzmann equations, boundary layer, multispecies ions
- ⋅ 22nd-E-17:50 − 18:10 Contractile ring-dependent cytokinesis model combining IB and PF methods (Seunggyu Lee)
- 이승규(국가수리과학연구소)
Seunggyu Lee, NIMS
A mathematical model of contractile ring-dependent cytokinesis is presented by using both phase-field and immersed-boundary methods in a three-dimensional domain. It is one of the powerful hypotheses that cytokinesis happens driven by the contractile ring; however, there are only few mathematical model following the hypothesis in author's knowledge. I consider a hybrid method to model the phenomenon. First, a cell membrane is represented by a zero-contour of a phase-field implicitly because of its topological change. Otherwise, immersed-boundary particles represent a contractile ring explicitly based on the author's previous work. Here, the multi-component phase-field equation is considered to avoid the emerging of each cell membranes right after their divisions. The numerical convergence of contractile ring to cell membrane is also proved.
2010 Mathematics Subject Classification: 35K35, 65M06, 92B05
Key Words and Phrases: cytokinesis, contractile ring, phase-field, immersed-boundary
- ⋅ 22nd-E-18:10 − 18:30 A study for mesh generation based on partial differential equations (Hyea Hyun Kim, Hee Jun Yang, Kiwan Jeon)
- 김혜현(경희대), 양희준(경희대), 전기완*(국가수리과학연구소)
Hyea Hyun Kim, Kyung Hee University, Hee Jun Yang, Kyung Hee University, Kiwan Jeon*, NIMS
We provide a triangular mesh generation method based on elliptic partial differential equations. We first configure initial triangles generated by the body centered rectangular lattice, which are close to the target object. We obtain a deformation field by solving the proposed PDE to move the initial triangles in order to fit the boundary shape and guarantee regular shaped mesh. We discuss the benefit of the proposed algorithm in view of parallel computation.
2010 Mathematics Subject Classification: 65N50
Key Words and Phrases: mesh generation, PDE based, parallel computation
- ⋅ 23rd-F-09:00 − 09:20 High order discontinuous Galerkin methods for hyperbolic conservation laws (Mi-Young Kim, Eun-Jae Park, Jaemin Shin)
- 김미영*(인하대), 박은재(연세대), 신재민(연세대)
Mi-Young Kim*, Inha University, Eun-Jae Park, Yonsei University, Jaemin Shin, Yonsei University
In this work, we
present a novel high-order discontinuous Galerkin method with
Lagrange multiplier (DGLM) for hyperbolic conservation laws.
Lagrange multipliers are introduced on the inter-element
boundaries via the concept of weak divergence. Static condensation
on element unknowns considerably reduces globally coupled degrees
of freedom, resulting in the stiffness equations in the Lagrange
multipliers only. We establish stability results and provide
conditions on the stabilization parameter, which plays a role in
capturing shocks and discontinuities as well. The error estimates
are derived in energy norm. Accuracy tests are performed, which
shows optimal convergence in L2 norms. Numerical results indicate
that the DGLM has potentials in delivering high order accurate
information for various problems in hyperbolic conservation laws.
Numerical examples include inviscid Burgers' equations, shallow
water equations (subcritical flow and supercritical upstream,
subcritical downstream flow), and compressible Euler equations
(Sod's Shock Tube and Intersection of Mach 3).
2010 Mathematics Subject Classification: 65M12
Key Words and Phrases: high order discontinuous Galerkin methods
- ⋅ 23rd-F-09:20 − 09:40 A sixth order WENO schemes based on exponential polynomials (Youngsoo Ha, Chang Ho Kim, Hyoseon Yang, Jungho Yoon)
- 하영수(서울대), 김창호*(건국대), 양효선(이화여대), 윤정호(이화여대)
Youngsoo Ha, Seoul National University, Chang Ho Kim*, Konkuk University, Hyoseon Yang, Ewha Womans University, Jungho Yoon, Ewha Womans University
In this presentation, We present a sixth-order weighted essentially nonoscillatory \linebreak (WENO) finite difference scheme. To design new WENO weights, we present two important measurements: a discontinuity detector (at the cell boundary) and a smoothness indicator. The interpolation method is implemented by using exponential polynomials with tension parameters such that they can be tuned to the characteristics of the given data, yielding better approximation near steep gradients without spurious oscillations, compared to the WENO schemes based on algebraic polynomials at lower computational cost. A detailed analysis is performed to verify that the proposed scheme provides the required convergence order of accuracy. Some numerical experiments are presented and compared with other sixth-order WENO schemes to demonstrate the new algorithm's ability.
2010 Mathematics Subject Classification: 65M12, 65M70, 41A10, 42C05
Key Words and Phrases: hyperbolic conservation laws, Euler equation, WENO scheme, convergence order, smoothness indicator, nonlinear weights
- ⋅ 23rd-F-09:50 − 10:10 A fast and accurate redistancing based on the Hopf-Lax formula (Byungjoon Lee, Jerome Darbon, Stanley Osher, Myungjoo Kang)
- 이병준*(서울대), Jerome Darbon(CNRS/CMLA), Stanley Osher(UCLA), 강명주(서울대)
Byungjoon Lee*, Seoul National University, Jerome Darbon, Ecole Normale Superieure de Cachan, Stanley Osher, University of California at Los Angeles, Myungjoo Kang, Seoul National University
This article presents a fast new numerical method for redistancing objective functions based on the Hopf--Lax formula.The algorithm suggested here is a special case of the previous work in Darbon, Osher (2016) and an extension that applies the Hopf–Lax formula for computing the signed distance to the front. We propose the split Bregman approach to solve the minimization problem as a solution of the eikonal equation obtained from Hopf–Lax formula. Our redistancing procedure is expected to be generalized and widely applied to many fields such as computational fluid dynamics, the
minimal surface problem, and elsewhere.
2010 Mathematics Subject Classification: 35F21, 46N10
Key Words and Phrases: reinitialization, level set method, Hopf-Lax formula, Hamilton-Jacobi equations, Split Bregman method
- ⋅ 23rd-F-10:10 − 10:30 Solving the Euler and ideal magnetohydrodynamics equations using central-upwind schemes with a multidimensional limiting process (Youngsoo Ha, M. Kang, S. Do, K. Kim)
- 하영수*(서울대), 강명주(서울대), 도성주(서울대), 김창호(건국대)
Youngsoo Ha*, Seoul National University, M. Kang, Seoul National University, S. Do, Seoul National University, K. Kim, Konkuk University
In this talk, we present semi-discrete central-upwind difference schemes with a modified multi-dimensional limiting process (MLP) to solve the Euler and ideal magnetohydrodynamics equations. In general, high-order central difference schemes for conservation laws involve no Riemann solvers or characteristic decompositions but have a tendency to smear linear discontinuities. To overcome this drawback of central-upwind schemes, we use a multi-dimensional limiting process that uses multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications.
This approach is based on directly solving the equations with a new limiting strategy to control the oscillations at nonsmooth regions and to yield decreasing numerical dissipation. Some numerical results are provided to demonstrate the performance of the proposed scheme.
2010 Mathematics Subject Classification: 64M12, 65M70
Key Words and Phrases: central-upwind scheme, MLP limiter, hyperbolic conservation laws, Euler equation
- ⋅ 23rd-G-10:45 − 11:05 Image deblurring under impulse noise via sparse representation (Miyoun Jung, Myeongmin Kang, Myungjoo Kang)
- 정미연*(한국외대), 강명민(서울대), 강명주(서울대)
Miyoun Jung*, Hankuk University of Foreign Studies, Myeongmin Kang, Seoul National University, Myungjoo Kang, Seoul National University
In this paper, we propose a new model for restoring images corrupted by blur and impulse noise.
The model consists of the $l_0$ data-fidelity term, the sparse representation prior and total variation (TV) regularization. TV regularization is effective to restore cartoon images while sparse representation prior is well adapted to textures and details. An alternating minimization algorithm is employed to solve the proposed minimization problem. Numerical results are reported to demonstrate that the proposed model performs better than the state-of-the-art methods.
2010 Mathematics Subject Classification: 68U10, 65K10
Key Words and Phrases: image deblurring, sparse representation, total variation, $l_0$ term
- ⋅ 23rd-G-11:05 − 11:25 Computation of eigenfunctions (Yunho Kim)
- 김윤호(울산과학기술원)
Yunho Kim, UNIST
In this talk, we propose a new formulation of an eigenvalue problem. We begin with a symmetric and positive definite matrix and discuss how to find an eigenvector corresponding to the least eigenvalue. Conventional methods to find an eigenvector corresponding to the least eigenvalue first estimate an eigenvalue and then estimate a corresponding eigenvector by solving a linear system, i.e., matrix inversion is implicitly done. However, our method finds an eigenvector of the least eigenvalue by only matrix multiplications without estimating the eigenvalue. Our proposed algorithm guarantees convergence to such an eigenvector almost independent of an initial guess. The same formulation applies to general eigenvalue problems, finding eigenfunctions of symmetric elliptic operators on manifolds, etc. We will discuss such formulations and convergence as well.
2010 Mathematics Subject Classification: 15A18, 65F15, 65K10
Key Words and Phrases: eigenvalue, eigenvector, optimization, spectral decomposition
- ⋅ 23rd-G-11:35 − 11:55 The total generalized variation based variational models for ultrasound image denoising (Myeongmin Kang, Myungjoo Kang, Miyoun Jung)
- 강명민*(서울대), 강명주(서울대), 정미연(한국외대)
Myeongmin Kang*, Seoul National University, Myungjoo Kang, Seoul National University, Miyoun Jung, Hankuk University of Foreign Studies
In this article, we introduce a class of variational models for the restoration of ultrasound images corrupted by noise. The proposed models involve the convex or nonconvex total generalized variation regularization. The total generalized variation regularization removes the staircasing artifacts that appear in the restored images of total variation based models. Incorporating total generalized variation regularization with nonconvex function enables the restored images to have well-preserved edges. To realize the proposed convex model, we adopt the alternating direction method of multipliers, and the iteratively reweighted $\ell_1$ algorithm is employed to deal with the nonconvex model. These methods result in fast and efficient optimization algorithms for solving our models. Numerical results show that the proposed models are superior to other state-of-the-art models.
2010 Mathematics Subject Classification: 62H35, 68U10, 94A08
Key Words and Phrases: ultrasound imaging, total generalized variation, nonconvex regularization
- ⋅ 23rd-G-11:55 − 12:15 Image analysis and deep learning (Myungjoo Kang)
- 강명주(서울대)
Myungjoo Kang, Seoul National University
Since 1990, many mathematicians studied the image processing based on partial differential equations and variational methods. Using TV(total variation) and some optimization techniques, there were a lot of improvement in image processing areas. Until deep learning coming out, those were the state of art methods in these areas. But once using deep learning techniques, it turns out that in almost every areas in image processing fields, deep learning is the best method. I will compare the results between the variational image processing techniques and deep learning. Also, I will give a short introduction about the deep learning.
2010 Mathematics Subject Classification: 65Y99
Key Words and Phrases: image analysis, total variation, deep learning
- Inverse Problems
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Size estimate in the complex conductivity equation (Catalin Carstea, Tu Nguyen, Jenn-Nan Wang)
- Catalin Carstea, NCTS Mathematics Division, Taiwan, Tu Nguyen, Vietnam Academy of Science and Technology, Jenn-Nan Wang*, National Taiwan University
In this talk I am going to discuss quantitative uniqueness estimates for solutions of second order elliptic equations with anisotropic complex coefficients. Our method relies on a delicate Carleman estimate. Using this estimate, we can derive three-ball inequalities and the Lipschitz propagation of smallness. We then study the problem of estimating the size of an inclusion embedded inside of a conductive body with anisotropic complex admittivity by one boundary measurement.
2010 Mathematics Subject Classification: 35Q60
Key Words and Phrases: Carleman estimates, size estimate
- ⋅ 22nd-B-11:35 − 11:55 Stability of the electromagnetic scattering from a large cavity (KiHyun Yun)
- 윤기현(한국외대)
KiHyun Yun, Hankuk University of Foreign Studies
In this talk, we consider a time harmonic scattering problem of electromagnetic wave from a two-dimensional open cavity embedded in the infinite ground place. A variational formulation reduces the problem into a bounded domain problem. The problem has been challenging both mathematically and computationally due to highly oscillatory nature of the solution. We establish the stability estimates for the solution that provide the explicit dependence on a high wave number.
2010 Mathematics Subject Classification: 35J05
Key Words and Phrases: high wave number, cavity, electromagnetic scattering, stability
- ⋅ 22nd-B-11:55 − 12:15 Industrial mathematics in medical ultrasound imaging (Jaeseong Jang, Chi Young Ahn)
- 장재성*(연세대), 안치영(국가수리과학연구소)
Jaeseong Jang*, Yonsei university, Chi Young Ahn, NIMS
Medical ultrasound imaging has been widely applied for various assessments in obstetrics/gynecology, cardiology, and radiology. With recent developments in hardware and software of ultrasound systems, medical ultrasound imaging expands its applications. Especially, mathematics contributes to the development by providing tools for modelling and analysis. In this presentation, the contribution of mathematics is explained by giving examples of inverse problems which arise from the applications in diagnostic fields. Moreover, we describe challenging issues in ultrasound imaging with recent technologies of domestic medical ultrasound products.
2010 Mathematics Subject Classification: 68U10, 92C50
Key Words and Phrases: industrial mathematics, ultrasound imaging, ultrasound examinations, image processing, technology advances, mathematical modelling
- ⋅ 22nd-C-13:30 − 13:50 A direct approach to robust image reconstruction method for lung imaging in electrical impedance tomography (Kyounghun Lee, Eung Je Woo, Jin Keun Seo)
- 이경훈*(연세대), 우응제(경희대), 서진근(연세대)
Kyounghun Lee*, Yonsei University, Eung Je Woo, Kyung Hee University, Jin Keun Seo, Yonsei University
Electrical impedance tomography (EIT) for lung imaging has been difficulties at dealing with the forward modeling error caused by the chest movements during lung ventilation. The inherent ill-posed nature of EIT combined with the boundary geometry uncertainties produces severe boundary artifacts when using standard image reconstruction methods. In this talk, I propose a robust reconstruction method that effectively handles the boundary geometry uncertainties using the correlations between the columns of the sensitivity matrix and the EIT-data, and that does not demand to solve the linearized EIT system.
2010 Mathematics Subject Classification: 49N45
Key Words and Phrases: inverse problem, electrical impedance tomography
- ⋅ 22nd-C-13:50 − 14:10 Incompatibility streaking artifacts in quantitative susceptibility mapping (Liangdong Zhou, Jae Kyu Choi, Yi Wang, Jin Keun Seo)
- Liangdong Zhou*(연세대), 최재규(Shanghai Jiao Tong Univ.), Yi Wang(Cornell Univ.), 서진근(연세대)
Liangdong Zhou*, Yonsei University, Jae Kyu Choi, Shanghai Jiao Tong University, Yi Wang, Cornell University, Jin Keun Seo, Yonsei University
This paper provides a mathematical ground for sources of streaking artifacts in quantitative susceptibility mapping (QSM) due to the dipole-incompatible field data and the singularity of the zero cone. QSM is to visualize a susceptibility distribution from measured local field perturbations associated with magnetization induced in the human body inside an MRI scanner. The incompatibility artifacts are the inherent nature of the QSM model, which is due to the zero cone in k-space. To identify the source of streaking artifacts, we decompose the measured data into a compatible part and a compatibility violator part and use special characteristics of error propagation. Two types of streaking artifacts are investigated in theoretical point of view; one comes from the loss of the data on the zero cone and the other comes from compatibility violators from the measured data. The major sources of streaking artifacts are due to the mismatch of the measured data with the model, and these can be probed indirectly from the reconstructed image outside of the region of interest. We perform numerical simulations to validate our theoretical findings.
2010 Mathematics Subject Classification: 65N21, 35R30, 47N40
Key Words and Phrases: quantitative susceptibility mapping, MRI, inverse problem, streaking artifacts
- ⋅ 22nd-C-14:20 − 14:40 Characterization and quantification of metal artifacts in X-ray computed tomography (Hyoung Suk Park)
- 박형석(국가수리과학연구소)
Hyoung Suk Park, NIMS
Metal artifacts reduction in X-ray computed tomography (X-ray CT) is becoming increasingly important as artificial prostheses and metallic implants become more widespread in aging population. Despite the rapid advances in CT technologies and various works seeking to reduce metal artifacts, it remains a challenging issue due to the difficulties in analyzing X-ray data. In this presentation, we provide a rigorous characterization of metal artifacts in X-ray CT using the notion of the Wavefront set from microlocal analysis. Based on this characterization, we propose a new metric for the quantitative evaluation of metal artifacts from X-ray data that does not require any artifacts-free image to be compared.
2010 Mathematics Subject Classification: 47G10
Key Words and Phrases: computed tomography, metal artifacts, wavefront set
- ⋅ 22nd-D-15:15 − 15:35 Application of MUSIC for anomaly detection in microwave imaging (Won-Kwang Park, Hwapyung Kim, Kwang-Jae Lee, Seong-Ho Son, Jin Keun Seo)
- 박원광*(국민대), 김화평(연세대), 이광재(한국전자통신연구원), 손성호(한국전자통신연구원), 서진근(연세대)
Won-Kwang Park*, Kookmin University, Hwapyung Kim, Yonsei University, Kwang-Jae Lee, ETRI, Seong-Ho Son, ETRI, Jin Keun Seo, Yonsei University
It is well-known that MUltiple SIgnal Classification (MUSIC) is a famous non-iterative detection/imaging technique in inverse scattering problem. In this contribution, we apply MUSIC for detecting location of small inhomogeneity whose permittivity is differ from the background medium. Experimental results through the S-parameters at 1 and 3 Ghz frequencies shows both the effectiveness and limitation of MUSIC in the real-world microwave imaging application.
2010 Mathematics Subject Classification: 78A46
Key Words and Phrases: MUltiple SIgnal Classification (MUSIC), inverse scattering problem, microwave imaging
- ⋅ 22nd-D-15:35 − 15:55 Exponential decay estimate of the eigenvalues for the Neumann-Poincar\'{e} operator (Kazunori Ando)
- Kazunori Ando, Ehime University
We show that the eigenvalues of the Neumann-Poincar\'{e} (NP) operator on analytic boundaries of simply connected bounded planar domains tend to zero exponentially fast, and the exponential convergence rate is determined by the maximal Grauert radius of the boundary. To show it, we use 1) the symmetrization technique of the NP operator which enables us to estimate its eigenvalues by the operator norm by the min-max prinriple and 2) the analytic extension of the integral kernel which leads us to the Paley-Wiener type theorem. We present a few examples of boundaries to show that our extimate is optimal.
2010 Mathematics Subject Classification: 35R30, 35C20
Key Words and Phrases: Neumann-Poincar\'{e} operator, eigenvalues, analytic boundary, exponential decay, maximal Grauert radius
- ⋅ 22nd-D-16:05 − 16:25 Eigenvalues and eigenfunctions of double layer potentials (Yoshihisa Miyanishi, Takashi Suzuki)
- Yoshihisa Miyanishi*, Center for Mathematical Modeling and Data Science, Osaka University, Takashi Suzuki, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University
Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let $\Omega$ be a $C^2$ bounded region in ${\mathbf{R}}^n$ ($n=2, 3$). The double layer potential $K: L^2(\partial \Omega) \rightarrow L^2(\partial \Omega) $ is defined by
$$
(K \psi)(x) \equiv \int_{\partial \Omega} \psi(y)\cdot \nu_{y} E(x, y) \; ds_y,
$$
where
$$
E(x, y)=
\begin{cases}
\frac{1}{\pi} \log\frac{1}{|x-y|}, \quad \;\mbox{if}\; n=2, \\
\frac{1}{2\pi} \frac{1}{|x-y|}, \quad\hspace{4.5mm}\;\mbox{if} \; n=3,
\end{cases}
$$
$ds_y$ is the line or surface element and $\nu_y$ is the outer normal derivative on $\partial \Omega$. It is known that $K$ is a compact operator on $L^2(\partial \Omega)$ and consists of at most a countable number of eigenvalues, with $0$ as the only possible limit point. We intend to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of $\partial\Omega$.
Reference
Y. Miyanishi and T. Suzuki, Eigenvalues and eigenfunctions of double layer potentials, arXiv:1501.03627, Trans. Amer. Math., to appear.
2010 Mathematics Subject Classification: Primary 47G40; Secondary 34L20
Key Words and Phrases: double layer potential, eigenvalues, eigenfunctions, nodal sets
- Cryptography
- ⋅ 22nd-A-09:00 − 09:20 On the hardness of generalized small integer solution problem (Yongha Son)
- 손용하(서울대)
Yongha Son, Seoul National University
Ajtai's SIS is one of the main problems in lattice cryptography, which has emerged as a very attractive foundation due to a worst-case to average-case reduction.
SIS concerns random lattices related to the group $G=\mathbb{Z}_q^n.$ Gama \textit{et al.} proposed Generalized SIS (GSIS) which uses wide-range of random lattices related to any sufficient large finite abelian group and prove hardness of GSIS.
In this talk, we propose another approach for hardness of GSIS using different method from Gama's work.
We expect to reduce the size of abelian groups required to prove hardness, and it will derive various applications of GSIS.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: lattice, SIS, worst-case to average-case reduction
- ⋅ 22nd-A-09:20 − 09:40 Minimal condition for shortest vectors in lattices of low dimension (Seunghwan Chang, Taewan Kim, Hyang-Sook Lee, Juhee Lee, Seongan Lim)
- 장승환*(이화여대 수리과학연구소), 김태완(한국전자통신연구원 초연결통신연구소), 이향숙(이화여대), 이주희(이화여대 수리과학연구소), 임선간(이화여대 수리과학연구소)
Seunghwan Chang*, Institute of Mathematical Sciences, Ewha Womans University, Taewan Kim, Hyper-connected Communication Research Laboratory, ETRI, Hyang-Sook Lee, Ewha Womans University, Juhee Lee, Institute of Mathematical Sciences, Ewha Womans University, Seongan Lim, Institute of Mathematical Sciences, Ewha Womans University
For a lattice in the Euclidean space, finding a nonzero shortest vector is
in general a difficult computational problem. In this article, we investigate
bases for lattices of low dimension that contain a shortest vector as a member
of them. We prove a sufficient condition for a basis to have this property
for lattices of dimension $\le 5$. We also prove that the sufficient condition is
minimal.
2010 Mathematics Subject Classification: 12E20, 68W40
Key Words and Phrases: lattices, shortest vector problem, Minkowski-reduced bases, greedy-reduced bases
- ⋅ 22nd-A-09:50 − 10:10 FHE over the integers and modular arithmetic circuits (Eunkyung Kim, Mehdi Tibouchi)
- 김은경*(이화여대), Mehdi Tibouchi(NTT Secure Platform Laboratories)
Eunkyung Kim*, Ewha Womans University, Mehdi Tibouchi, NTT Secure Platform Laboratories
In 2015, Nuida-Kurosawa constructed the first fully homomorphic encryption (FHE)
over the integers supporting arbitrary prime fields of constant size.
As a result, they let us have two ways of homomorphically evaluating
modular arithmetic circuits: for any prime $Q$, evaluate mod-$Q$ arithmetic circuits
directly using their FHE scheme with the message space ${\mathbb Z}/Q{\mathbb Z}$,
or first convert the arithmetic circuits to Boolean ones doing the same things and
then evaluate that Boolean circuits using an FHE scheme with the binary message space,
e.g., using the NK scheme with $Q=2$.
At a first glance, the former looks more efficient
since it encrypts mod-$Q$ ($\log Q$ bits) message at once.
Then, we show a counterintuitive result that the binary approach
is often preferable to the mod $Q$ approach.
Based on our concrete estimates, we conclude that the Nuida-Kurosawa
approach is not competitive for any $Q > 3$ (and possibly not even $Q = 3$) as soon
as one needs to carry out ciphertext refresh operations|and if one does not need
bootstrapping, other somewhat homomorphic schemes with large message space
are certainly preferable.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: fully homomorphic encryption, modular arithmetic circuits
- ⋅ 22nd-A-10:10 − 10:30 On the fast enumeration algorithm on lattice reduced bases (Younjin Kim, Hyang-Sook Lee)
- 김연진*(이화여대), 이향숙(이화여대)
Younjin Kim*, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University
The NP-hardness for the Shortest Vector Problem (SVP) has been conjecutred. In 1998, Ajtai proved that the SVP is NP-hard under random reduction. In the lattice based cryptography, it has been very important to find efficient algorithms for solving the Shortest Vector Problem. The enumeration algorithm for the SVP is one of the main algorithms for finding the shortest vector in the lattice. For the improved enumeration algorithm, the main idea is to find a good basis. In this paper, we show that if the basis satisfies some reduction then the running time of enumeration drops down to $d2^{O(d)}$.
2010 Mathematics Subject Classification: 05A16
Key Words and Phrases: lattice based cryptography, enumeration algorithm
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] On the lattice reductions for NTRU variants (Soonhak Kwon)
- 권순학(성균관대)
Soonhak Kwon, Sungkyunkwan University
We discuss many available settings for NTRU related problems. Most of the NTRU variants rely on the difficulty of the following problem: For a suitably chosen ring $\mathcal R$ equipped with norm and for $f, g\in \mathcal R $ with small norm, define $h=\bar{g}/\bar{f} \in \bar{\mathcal R}$ where $\bar{R}$ is a finite quotient ring induced from $R$ and $\bar{f}, \bar{g}$ are images of $f, g$ in $\bar{R}$. The problem is that, for such given $h\in\bar{R}$ and unknown $f, g\in R$, recover $f, g\in R$ or $f_1, g_1\in R$ with small norm satisfying $h=\bar{g_1}/\bar{f_1} \in \bar{\mathcal R}$. We explain some relations between various lattice reductions of several NTRU variants.
2010 Mathematics Subject Classification: 11C99, 11Y16
Key Words and Phrases: NTRU, lattice, LLL algorithm
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Private database queries using fully homomorphic encryption (Hyung Tae Lee)
- 이형태(Nanyang Technological Univ.)
Hyung Tae Lee, Nanyang Technological University
Private database query (PDQ) processing has received much attention from the fields of both cryptography and databases. While previous approaches to design PDQ protocols exploit several cryptographic tools concurrently, the appearance of fully homomorphic encryption (FHE) schemes enables us to design PDQ protocols without the aid of additional tools.
In this talk, we present a set of basic PDQ protocols for conjunctive, disjunctive, and threshold queries with equality conditions, which protect the constants in the query statement, together with the client's data stored at the server. Furthermore, we consider an extension that additionally hides query types.
This talk is mainly based on the following two recent results:
\begin{itemize}[leftmargin=5mm]
\item[1.] M. Kim, H.T. Lee, S. Ling, and H. Wang, On the Efficiency of FHE-Based Private Queries, IEEE Transactions on Dependable and Secure Computing, to appear, 2016.
\item[2.] M. Kim, H.T. Lee, S. Ling, S.Q. Ren, B.H.M. Tan, and H. Wang, Better Security for Queries on Encrypted Databases, preprint, 2016. Available at http://ia.cr/2016/470.
\end{itemize}
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: private database queries, fully homomorphic encryption, encrypted database
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Recent progress on discrete logarithm problem over finite fields (Taechan Kim)
- 김태찬(NTT Secure Platform Laboratories, Japan)
Taechan Kim, NTT Secure Platform Laboratories, Japan
In this talk, we present a new algorithm of number field sieve variants proposed by Kim and Barbulescu, called extended tower number field sieve (exTNFS).
It improves the complexity of discrete logarithm problem over finite field of medium sized characteristics. As a result, the key size of pairing-based crypto systems should be increased.
If time permits, we also introduce a variant recently proposed by Kim and Jeong that generalizes Kim-Barbulescu's exTNFS.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: number field sieve, discrete logarithm problem
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Cryptanalysis of the multilinear maps over the integers (Hansol Ryu )
- 류한솔(국가보안기술연구소)
Hansol Ryu, National Security Research Institute
Multilinear maps serve as a basis for a wide range of cryptographic applications. Until now, there are three types of multilinear maps: the first is constructed using ideal lattices, the second is defined over the integers, and the last one uses standard lattices.
In this talk, we present cryptanalysis of multilinear maps over the integers suggested by Coron, Lepoint, and Tibouchi at Crypto13 and Crypto15. Our attacks share the essence and utilize low-level encodings of zero. This leads to find all the secret parameters of multilinear maps in polynomial time of the security parameter.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: multilinear map
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Naive implementation of elliptic curve cryptography (Hwajeong Seo)
- 서화정(A Star)
Hwajeong Seo, A Star
Cryptography technology is developed by cryptographer to protect the private information from potential threats and the technology is realized by cryptography engineer in hardware/software implementations. Particularly, the cryptography is based on mathematics and the implementation is computer architecture. In order to narrow down the wide gap between theory and practice, cryptography engineer should convert the mathematical equations into computer instructions since this task cannot be done automatically due to high complexity of optimal conversion. In this talk, we explore the basic steps to implement the cryptography over computers. Our target cryptography is the most well-known Montgomery curve (Curve25519) based public key cryptography and our target platform is the most common 32-bit personal computer. This talk includes all required primitive operations for elliptic curve cryptography on the computers, which will be good reference materials for both cryptographer and cryptography engineer.
2010 Mathematics Subject Classification: 11G05
Key Words and Phrases: elliptic curve cryptography, software implementation
- ⋅ 22nd-D-16:05 − 16:45 [Invited Talk] How cryptography has been advanced in Korea - Know the past well, then understand the present (Kwangjo Kim)
- 김광조(한국과학기술원)
Kwangjo Kim, KAIST
About 3 decades ago in Korea, the strong demand for building the secure communication systems was raised by the government due to the military and political conflicts between two Koreas.
The first priority at that time was to provide the top level confidentiality service in our own way
since cryptographic technologies are very sensitive and export-controlled from the advanced countries.
With the limited references on cryptography, a Korean team had to research and develop our own style secure devices used for radio communication for islands, data communication for diplomatic channels. etc.
In this talk, the speaker will introduce how the cryptographic research was advanced and an interesting history on developing Korean block ciphers
compared with international progress mainly discussed at Crypto conferences.
This will help the audience to understand the current researches in Korea.
2010 Mathematics Subject Classification: 53A25
Key Words and Phrases: Secure Device, Block Cipher,
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] Coming quantum-resistant cryptography (Sang Geun Hahn)
- 한상근(한국과학기술원)
Sang Geun Hahn, KAIST
This talk will overview the current status of the coming NIST proposal for quantum resistant cryptography and response of Korean cryptographers.
2010 Mathematics Subject Classification: 11, 14
Key Words and Phrases: post quantm cyptography
- ⋅ 22nd-E-17:50 − 18:30 [Invited Talk] Code-based cryptography using rank metric codes (Jon-Lark Kim)
- 김종락(서강대)
Jon-Lark Kim, Sogang University
Code-based cryptography is considered as one of the quantum-resistant cryptosystems. Recently rank metric codes were used to shorten the key size of code-based cryptography. In this talk, we give an introduction to code-based (public key) cryptography and describe the current status of rank metric code-based cryptography.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: code-based cryptography, error-correcting codes, rank metric codes, McEliece cryptography
- Mathematical Biology
- ⋅ 22nd-A-09:00 − 09:40 [Invited Talk, The winner of 2015 Sangsan Prize for Young Mathematicians] Molecular mechanisms leading to the Kuramoto model of coupled oscillators (Jae Kyoung Kim)
- 김재경(한국과학기술원)
Jae Kyoung Kim, KAIST
The Kuramoto model has been widely used to describe the synchronizations of a large set of
coupled oscillators. In particular, the Kuramto model successfully captures the key features of
synchronization of 20,000 coupled cellular rhythms in the circadian clocks of our brain.
However, due to the abstractness of the Kuramoto model, specific molecular mechanisms
underlying the sinusoidal coupling terms have not been identified. In this talk, I will discuss that
the combination of intracellular transcriptional repression mechanisms and intercellular coupling
mechanisms in the mammalian circadian clocks can lead to such sinusoidal coupling.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: coupled oscillator, biological rhythms, Kuramoto model, circadian rhythms, mathematical modeling, global stability analysis, weekly coupled oscillator theory
- ⋅ 22nd-A-09:50 − 10:10 The role of microenvironment in regulation of tumor growth (Yangjin Kim)
- 김양진(건국대)
Yangjin Kim, Konkuk University
Malignant gliomas are the most common type of brain cancer, which arise from glial cells, and in their most aggressive form are called GBMs. GBMs are highly invasive and difficult to treat because cells migrate into surrounding healthy brain tissue rapidly, and thus these tumors are difficult to completely remove surgically. GIMs, which can comprise up to one third of the total tumor mass (Markovic et al., 2009), are present in both intact glioma tissue and necrotic areas. They apparently originate from both resident brain macrophages (microglia) and newly recruited monocyte-derived macrophages from the circulation. Activated GIMs exhibit several phenotypes: one called M1 for classically activated, tumor suppressive, and another called M2 for alternatively activated, tumor promoting, and immunosuppressive (Mantovani et al., 2013). Within a tumor the balance between these phenotypes is typically shifted to the M2 form (Pollard, 2009). Numerous factors secreted by glioma cells can influence GIM recruitment and phenotypic switching, including growth factors, chemokines, cytokines and matrix proteins (Coniglio et al., 2012; Wang et al., 2012). In this work, we focus on mutual interaction between a glioma and M1/M2 microglia mediated by CSF-1, TGFbeta, and EGF. Up-regulated TGFbeta leads to up-regulation of Smad within the tumor cells and secretion of MMPs, leading to proteolysis for EMT process and cell infiltration. The mathematical model consists of densities of glioma cells, M1 type cells, M2 type cells, and concentrations of CSF-1, EGF, TGFbeta, Extracellular matrix, and MMPs. We developed the model to investigate the mutual interactions between tumor cells in the upper chamber and microglia in the lower chamber. In the experiments, Boyden invasion assay was used to show that this mutual interaction induces glioma infiltration in vitro and in vivo. We show that our simulation results are in good agreement with the experimental data and we generate several hypotheses that should be tested in future experiments in vivo.
For the second part of the talk, we investigate the role of microenvironment in glioma cell infiltration through the dense network of normal cells in the brain. We focus on the ERK-RKIP-ZEB-E-cadherin signaling networks (system of ODEs) embedded in the PDE model that control the EMT, therefore, increasing further growth and tumorigenicity.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: cancer research glioma microenvironment
- ⋅ 22nd-A-10:10 − 10:30 Instabilities of a rotating helical rod (Yongsam Kim, Yun-Young Park, Sookkyung Lim)
- 김영삼 *(중앙대), 박윤영(중앙대), 임숙경(Univ. of Cincinnati)
Yongsam Kim*, Chung-Ang University, Yun-Young Park, Chung-Ang University, Sookkyung Lim, University of Cincinati
Bacteria such as Vibrio alginolyticus swim through the fluid by utilizing the rotational motion of their helical flagellum driven by a rotary motor. The flagellar motor is embedded in the cell body and turn either clockwise (CW) or counterclockwise (CCW), which may lead to straight forward or backward swimming or reorientation of the cell. We investigate the dynamics of the helical flagellum by adopting the Kirchhoff rod theory in which the flagellum is described as a space curve associated with orthonormal triads which measure the degree of bend and twist of the rod. The hydrodynamic interaction with the flagellum is involved using the regularized Stokes formulation. We focus on two different types of instabilities; (1) whirling instability of a rotating helical filament in the absence of a hook and (2) buckling instability of a flagellum in the presence of a compliant hook that links the flagellar filament to the rotary motor. We show that the helical filament without a hook displays three regimes of dynamical motions; a stable twirling, unstable whirling, and stable overwhirling motions depending on the physical parameters such as rotational frequency and elastic properties of the flagellum. The helical filament with a hook experiences buckling instability when the motor switches the direction of rotation and the elastic properties of the hook changes. We vary the physical parameters such as the bend modulus of the hook and the length of the hook to investigate their effect on the buckling angle which subsequently affects the reorientation of the cell body.
2010 Mathematics Subject Classification: 65-04, 65M06, 76D05, 76M20
Key Words and Phrases: fluid-structure interaction, twirling, whirling, overwhirling, regularized fundamental solutions, Kirchhoff rod theory, hook, buckling
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Modelling Ebola virus dynamics: implications for therapy (Shingo Iwami)
- Shingo Iwami, Kyushu University
Ebola virus (EBOV) causes a severe, often fatal Ebola virus disease (EVD), for which no approved antivirals exist. Recently, some promising anti-EBOV drugs, which are experimentally potent in animal models, have been developed. However, because the quantitative dynamics of EBOV replication in humans is uncertain, it remains unclear how much antiviral suppression of viral replication affects EVD outcome in patients. Here, we developed a novel mathematical model to quantitatively analyse human viral load data obtained during the 2000/01 Uganda EBOV outbreak and evaluated the effects of different antivirals. We found that nucleoside analogue- and siRNA-based therapies are effective if a therapy with $a >50\%$ inhibition rate is initiated within a few days post-symptom-onset. In contrast, antibody-based therapy requires not only a higher inhibition rate but also an earlier administration, especially for otherwise fatal cases. Our results demonstrate that an appropriate choice of EBOV-specific drugs is required for effective EVD treatment.
2010 Mathematics Subject Classification: 34A34
Key Words and Phrases: virus dynamics, data analysis
- ⋅ 22nd-B-11:35 − 11:55 Stochastic modeling and computation for reaction networks (Chang Hyeong Lee)
- 이창형(울산과학기술원)
Chang Hyeong Lee, UNIST
In this talk, we present stochastic modeling and computational methods for the time-evolution of reaction networks where various reactions occur between several species. We show simulation results of some motivating examples including biochemical systems and epidemic models.
2010 Mathematics Subject Classification: 92b05
Key Words and Phrases: reaction network, stochastic modeling
- ⋅ 22nd-B-11:55 − 12:15 Model predictive control of an HBV model (Hee-Dae Kwon)
- 권희대(인하대)
Hee-Dae Kwon, Inha University
In this talk, we consider a guideline for efficient drug treatment strategies for hepatitis B
virus (HBV) infection. We introduce and analyze a mathematical model that describes the
HBV infection during antiviral therapy. The reproduction number R0 is determined. The local/
global stability of virus-free steady state is investigated. We formulate a control problem
which minimizes the viral load as well as treatment costs. In order to reflect the status of
patients not only at the initial time but also at the follow-up visits, we consider the model
predictive control based on ensemble Kalman filter and differential evolution. The ensemble
Kalman filter is employed to estimate full information of the state from incomplete observation
data. We derive piecewise constant drug schedule applying techniques of differential evolution
algorithm. Numerical simulations are performed using various weights in the objective
functional to suggest optimal treatment strategies in different situations.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: feedback control, HBV, model predictive control, Ensemble Kalman filter
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front (Masaharu Taniguchi)
- Masaharu Taniguchi, Okayama University
We study traveling fronts to the unbalanced Allen--Cahn equation
or cooperation-diffusion systems in the N-dimensional Euclidean space
for $N$ that is larger or equals 3. We consider $(N-2)$-dimensional
smooth surfaces as boundaries of strictly convex compact sets in
in the $(N-1)$-dimensional Euclidean space, and define an equivalence
relation between them. We prove that there exists a traveling front
associated with such a given surface and show its stability.
The associated traveling fronts coincide up to phase transition
if and only if the given surfaces satisfy the equivalence relation.
2010 Mathematics Subject Classification: 35C07, 35B20, 35K57
Key Words and Phrases: traveling front, Allen–Cahn equation, non-symmetric
- ⋅ 22nd-C-14:20 − 14:40 Lotka-Volterra type predator-prey equations with a constant term (Yong Jung Kim)
- 김용정(한국과학기술원)
Yong Jung Kim, KAIST
Population models such as the logistic and Lotka-Volterra type competition and prey-predator equations are approximations of population dynamics. Most of the models, if not all of them, contain linear or higher order terms, but not a constant one. We will consider three examples of population dynamics that contain a constant term. We will see that the addition of a constant effect introduces new interesting dynamics to population models.
2010 Mathematics Subject Classification: 92C17, 35K51, 35K59
Key Words and Phrases: Allee effect, population dynamics
- ⋅ 22nd-C-14:40 − 15:00 Mathematical modeling of host-virus interactions with heterogeneous cell types and their spatial aspects (Hesung Now, Seongwon Lee, Hyung Ju Hwang, Joo-Yeon Yoo)
- 노해성(포항공대), 이성원*(국가수리과학연구소), 황형주(포항공대), 유주연(포항공대)
Hesung Now, POSTECH, Seongwon Lee*, NIMS, Hyung Ju Hwang, POSTECH, Joo-Yeon Yoo, POSTECH
Viruses directly infect the living cells for their replication and propagation. Because the propagation of viruses disturbs the homeostasis of the host, the host has developed the defense mechanism for avoiding the viral infection. In the meanwhile, viruses also have been evolved to avoid the host defense mechanism for their survival. During the long evolution processes with these struggles, the complicated biological interactions have been established between the host and viruses. Understanding of these host-virus interactions is required for the development of the therapeutic strategies for the virus-induced diseases.
Until now, researches for understanding host-virus interactions have been mainly performed at the cellular and organism level. Although these kind of approaches revealed important aspects of the interactions, the heterogeneity and spatial aspects of viruses and host cells have been ignored. To understand the effect of heterogeneity and spatial aspects of host cells on the viral infection, we develop an experimental model system and its corresponding mathematical model. With this approach, we hope to find the novel biological factors determining host-virus interactions. We extend the ODE model proposed by Hur et al.~(2013) to the PDE model with more dependent variables, to deal with heterogeneous cell types and spatial aspects.
2010 Mathematics Subject Classification: 92C45
Key Words and Phrases: host-virus interactions, mathematical modeling
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Mathematical modeling of metapopulation dynamics: Revisiting and expanding its meaning (Hiromi Seno)
- Hiromi Seno, Tohoku University
We revisit the metapopulation dynamics model of typical Levins type, and reconsider its mathematical modeling. We give an idea to reconstruct it, making use of the difference in time scale between the state transition and the dispersal of individuals within the patchy habitat. Our rational modeling leads to some unknown mathematical structures for it. As for the well-known ``Levins model'' with two states for the patch of a habitat composed of a number patches available for the reproduction, `empty' (i.e., with no resident) and `occupied' (i.e., with some resident), we reconstruct the two-state metapopulation dynamics model in a general form with our idea of the mathematical modeling, and show that the typical Levins type of metapopulation dynamics model appears only for a specific case with some additional assumptions for mathematical simplification. Further, it is shown that the bistable situation is much likely to occur in our model, which cannot appear in the ``Levins model''. We discuss moreover the three-state metapopulation dynamics model. In our discussion, we focus on the rationality of mass-action terms for the patch state transition in the Levins model.
2010 Mathematics Subject Classification: 97M60, 93A30, 92D25, 92D40, 92B99
Key Words and Phrases: metapopulation dynamics, mathematical modelling, quasi-stationary state approximation
- ⋅ 22nd-D-16:05 − 16:25 The role of virtual dispersal in two-patch SIR dynamics and optimal strategies (Sunmi Lee)
- 이선미(경희대)
Sunmi Lee, Kyung Hee University
A two-patch epidemic model is considered to assess the impact of disease transmission dynamics in heterogeneous environments. The two-patch system models the movement of individuals between and within patches/environments using a patch residence-times matrix P with entries that budget within and between host patch relative residence times across two regions. Recently, a two-group epidemic framework via virtual dispersal has been developed in which the risk of infection is a function of the residence time and local environmental risk. This approach generalizes the traditional two-group epidemic models with heterogeneous mixing. In this work, we employ this approach to a general two-patch SIR model in order to investigate the effect of state dependent dispersal behavior on the disease dynamics.
Three single-outbreak scenarios are considered i) polar scenario ii) symmetric scenario and iii) high mobility scenario. Next, optimal control theory is used to identify and evaluate patch-specific control measures aimed at reducing disease prevalence at a minimal cost. Optimal policies are computed under different residence-matrix configurations mentioned above as well as transmissibility scenarios characterized by the magnitude of the basic reproduction number.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: a two-patch SIR model, virtual dispersal, optimal strategies
- ⋅ 22nd-D-16:25 − 16:45 Numerical simulations of larval dispersal during upwelling season over Oregon coast (Sangil Kim, Jack Barth)
- 김상일*(한국외대), Jack Barth(Oregon State Univ.)
Sangil Kim*, Hankuk University of Foreign Studies, Jack Barth, Oregon State University
Larval dispersal is explored in numerical, primitive-equation model simulations of ocean circulation along the Oregon coast, a region of strong wind-driven currents and variable topography. The model configuration uses realistic topography, and open boundary conditions. The forcing is COAMPS wind of year 2001 during the summer upwelling season. The numerical coastal circulation grid has 3-km resolution on average. A large number of Lagrangian particles as models of planktonic larvae are released daily at different depths for 90 days (from Day 120 to 210) during the upwelling season near the sea surface at every grid point with depth shallower than the 200-m isobath (offshore approximately 80 km). The horizontal circulation at each single depth is only considered for the float trajectories and the competency time window is assumed to be 20 to 40 days after the floats are released. To analyze the simulation results for larval dispersal, using techniques such as regional transition and connectivity matrices, floats are clustered by 3 different regions such as inshore (0-50-m isobath), midshelf (50-120-m isobath) and offshore (>120-m isobath). Larval settlement is defined in separate analyses as all waters shallower than either 200 or 50 m in depth. We will connectivity matrices and statistical calculations such as time-mean trajectories, time to exit the continental shelf region (depths less than the 200-m isobath), and the fraction retained or transported to different portions of the shelf.
2010 Mathematics Subject Classification: 92B99
Key Words and Phrases: Larval dispersal, numerical primitive-equation model, Lagrangian particles, Oregon coast
- ⋅ 22nd-E-17:00 − 17:40 [Invited Talk] The legacy of Kermack and McKendrick again (Hisashi Inaba)
- Hisashi Inaba, The University of Tokyo
In a seminal series of papers published from 1927 to 1937, Kermack and McKendrick developed SIR-type endemic models with local time structure, which can take into account the demography of the host population, the waning immunity, the variable susceptibility and infectivity, and the reinfection of recovered individuals. The ideas of the variable susceptibility and immunity for the recovered individuals has become increasingly important in understanding emerging and reemerging infectious diseases, since the reinfection makes the control of infectious diseases more difficult, and the waning immunity is widely observed if there is no (natural or artificial) boosting. If any enhancement of epidemiological reproductivity by reinfection exists, we also expect that endemic steady states backwardly bifurcate when the basic reproduction number $R_0$ crosses unity, which implies that attaining a subcritical level of $R_0$ is not necessarily a complete policy for disease prevention. The main aim of my talk is to demonstrate the possible usefulness of the ``forgotten'' Kermack--McKendrick model to understand endemic phenomena for infectious diseases.
2010 Mathematics Subject Classification: 92D30
Key Words and Phrases: endemic model, infectious diseases, the basic reproduction number, structured population
- ⋅ 22nd-E-17:50 − 18:10 Global stability analysis for epidemic models with diffusion terms and space-depend\-ent coefficients (Toshikazu Kuniya, Jinliang Wang)
- Toshikazu Kuniya*, Kobe University, Jinliang Wang, Heilongjiang University
In this study, we consider an SIR epidemic model and an HIV infection model with diffusion terms and space-dependent coefficients. For each of them, we consider the case where only one population can diffuse. We investigate the global asymptotic stability of each of their equilibria by constructing suitable Lyapunov functions and using the Green's first identity. In the construction of such Lyapunov functions, to obtain a hint of the form of the suitable Lyapunov functions, we discretize the PDE models into the corresponding ODE models and construct Lyapunov functions for them.
2010 Mathematics Subject Classification: 35Q92, 92D30
Key Words and Phrases: SIR epidemic model, HIV infection, diffusion, global stability, Lyapunov function
- ⋅ 22nd-E-18:10 − 18:30 Epidemic models with waning immunity (Yukihiko Nakata)
- Yukihiko Nakata, Shimane University
In this talk we would like to introduce a mathematical model formulated by a system of delay differential equations in order to give a possible explanation of periodic outbreak of mycoplasma pneumoniae observed in Japan. Stability analysis and numerical studies for periodic solutions will be presented together with biological interpretations. We then study a general SIS type epidemic model formulated by a scalar integral equation, motivated by a recent open problem regarding global stability of an equilibrium. Comparison of two models will illustrate importance of immunity period and its distribution in the periodic oscillation of infectious diseases. The talk is based on collaboration with R. Omori and G. Rost.
2010 Mathematics Subject Classification: 34K05, 92D30
Key Words and Phrases: epidemic model, delay equation, age structure, stability
- ⋅ 23rd-F-09:00 − 09:40 [Invited Talk] The effect of stochasticity in adaptive dynamics (Joe Yuichiro Wakano, Yoh Iwasa)
- Joe Yuichiro Wakano*, Meiji University, Yoh Iwasa, Kyushu University
Adaptive dynamics formalism demonstrates that, in a constant environment, a continuous trait may first converge to a singular point followed by spontaneous transition from a unimodal trait distribution into a bimodal one, which is called ``evolutionary branching''. Most previous analyses of evolutionary branching were done in an infinitely large population. Here we study the effect of stochasticity caused by the finiteness of the population size on evolutionary branching. By analyzing the dynamics of trait variance, we obtain the condition for evolutionary branching as the one under which the trait variance explodes. Genetic drift reduces the trait variance, and it causes stochastic fluctuation of the trait variance. In a very small population, evolutionary branching does not occur. In larger populations, evolutionary branching may occur, but it occurs in two different manners: in the deterministic branching, branching occurs quickly when the population reaches the singular point, whilst in the stochastic branching, the population stays at the singularity for a while until it branches out. The condition for these cases and the mean branching-out time are calculated in terms of population size, mutational effects, and selection intensity, and are confirmed by direct computer simulations of the individual-based model.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: evolutionary branching, SDE
- ⋅ 23rd-F-09:50 − 10:10 Towards construction of dialogical control (Ryo Kobayashi)
- Ryo Kobayashi, Hiroshima University
Conventional control theory has developed highly sophisticated framework by separating the system and the environment, in which the interactions between the system and the environment are treated as a ``disturbance". The theory has been made big successes in the areas where it is applicable. Typical examples are machines (including robots) working in the factories where environment is completely known.
However, mobile robots are not the case. They encounter unknown environments as they move into new sites. In such situations, the interaction between the system and the environment can no more be regarded as disturbance. That means the system cannot be closed (like the conventional theory assumed), and we definitely need a new framework of control. The most promising approach seems to learn from the animals, because even the lower animals can make locomotion easily in the complex environment. Thus, collaboration between mathematical biology and robotics can be just the potential way for achieving our goal. I will introduce our challenge to construct a novel control principle for mobile robots.
2010 Mathematics Subject Classification: mobile robot, control
Key Words and Phrases: 93A99
- ⋅ 23rd-F-10:10 − 10:30 A challenging interdisciplinary approach to elucidate a mystery of remodeling process in nuclear architecture: Theory and Experiment (Sungrim Seirin Lee, Hiroshi Ochiai)
- Sungrim Seirin Lee*, Department of Mathematical and Life Sciences, Hiroshima University, Hiroshi Ochiai, PRESTO, Japan Science and Technology Agency (JST)
Nuclear architecture, which plays an important role in organizing the function of the nucleus, is composed of heterochromatin and euchromatin. Conventional nuclear architecture is found when the distribution of heterochromatin is enriched in the periphery of the nucleus. Conventional architecture is the primary structure in the majority of eukaryotic cells, and the rod cells of diurnal mammals contain this structure. In contrast, inverted nuclear architecture occurs when the heterochromatin is distributed in the center of the nucleus; this occurs in the rod cells of nocturnal mammals. Surprisingly, the inverted architecture found in the rod cells of the adult mouse is formed through reorganization of the conventional architecture during terminal differentiation. Although an experimental approach has shown the relationship between these two types of nuclear architecture at the molecular level, the mechanisms mediating the long-range reorganization processes remain unknown. Here, we suggest a new mathematical approach to understanding the dynamics of nuclear architecture, by which we found that the deformation of nucleus can play a critical role in the process of chromatin remodeling. With the interdisciplinary work, we succeeded in developing an in vitro experiment and found that the dynamical deformation of nucleus promotes the clustering of chromocenters. With the basis of theoretical observation, we prove that the deformation of nucleus is sufficient condition to induce the remodeling of chromatin architecture. This interdisciplinary work has been started from the theoretical hypothesis and we suggest a new framework of interdisciplinary research in life sciences.
2010 Mathematics Subject Classification: 92B99
Key Words and Phrases: mathematical modeling, phase-field method, chromatin dynamics
- ⋅ 23rd-G-10:45 − 11:05 Mathematical modelling of pertussis resurgence in the UK (Yoon Hong Choi)
- 최윤홍(Public Health England)
Yoon Hong Choi, Public Health England
In 2012, the UK experienced a resurgence of pertussis and an increase in infant deaths. This occurred eight years after acellular pertussis (aP) vaccine replaced whole cell (wP) vaccine and despite continued high coverage for the primary series and pre-school aP booster. We developed a mathematical model to describe pertussis transmission dynamics in England and Wales since the 1950s and used it to investigate the cause of the resurgence and the potential impact of additional vaccination strategies. Long-term simulation results indicated that the likely cause of the resurgence was the replacement of wP by less efficacious aP vaccine and that an elevated level of pertussis would continue. Our findings contributed to the recent recommendation by the World Health Organisation that countries currently using wP vaccine for primary immunisation should stay with it. Improved pertussis vaccines that provide better protection against infection are needed.
2010 Mathematics Subject Classification: 93A30
Key Words and Phrases: mathematical modelling
- ⋅ 23rd-G-11:05 − 11:25 Deterministic and stochastic modeling of the transmission dynamics of dengue fever in Korea with seasonality (Hyojung Lee, Chang Hyeong Lee)
- 이효정*(울산과학기술원), 이창형(울산과학기술원)
Hyojung Lee*, UNIST, Chang Hyeong Lee, UNIST
Dengue fever is a mosquito-borne viral disease and currently endemic in many tropical and sub-tropical regions in the world. Dengue virus has serotypes 1-4 and is transmitted by the bite of an infected Aedes mosquito. Infected people with one serotype confer permanent immunity to it. People re-infected by the other strain increase the risk of developing more severe disease from known as dengue hemorrhagic fever (DHF) and dengue shock syndrome (DSS). We consider a mathematical model with two strains for primary and secondary infection. Especially in Korea, all of infected people who were returned from the visit to endemic areas have been reported. As the number of the international traveler increases, the risk of dengue outbreaks has been elevated. So we incorporate the term of imported cases into the model. Moreover, climate changes by global warming are strongly associated with significant increases in dengue incidence. To explore seasonal population dynamics of mosquitoes, we also incorporate the climate-dependent entomological parameters in the model. In this talk, first, we investigate the relation between the dengue outbreak and seasonality based on the data of RCP scenarios obtained by the Korea Meterological Administration (KMA). Second, we also aim to investigate the impact of infected international travelers on the transmission dynamics of infected individuals. Finally, we compare quantitatively dynamics of dengue outbreaks between stochastic model and deterministic model for the cases at the initial of the epidemic.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: Dengue fever, Epidemic model, Stochastic model, Climate change
- ⋅ 23rd-G-11:35 − 11:55 A dynamic compartmental model for the 2015 MERS outbreak in Korea (Jonggul Lee, Gerardo Chowell, Eunok Jung)
- 이종걸*(건국대), Gerardo Chowell(Georgia State Univ.), 정은옥(건국대)
Jonggul Lee*, Konkuk University, Gerardo Chowell, Georgia State University, Eunok Jung, Konkuk University
The 2015 Middle East respiratory syndrome (MERS) outbreak in the Republic of Korea has provided an opportunity to improve our understanding of the spread of MERS linked to healthcare settings. Here we designed a dynamic transmission model to analyze the MERS outbreak in the Republic of Korea based on confirmed cases reported during the period May 20–July 4, 2015. Our model explicitly incorporates superspreading events and time-dependent transmission and isolation rates. Our model was able to provide a good fit to the trajectory of the outbreak and was useful to analyze the role of hypothetical control scenarios. Specifically, we assessed the impact of the timing of control measures, especially as- sociated with a reduction of the transmission rate and diagnostic delays on outbreak size and duration. Early interventions within 1 week after the epidemic onset, for instance, including the initial government announcement to the public about the list of hospitals exposed to MERS coronavirus (MERS-CoV), showa promising means to reduce the size $(>71\%)$ and duration $(>35\%)$ of the MERS epidemic. Finally, we also present results of an uncertainty analysis focused on the role of superspreading events.
2010 Mathematics Subject Classification: 92D30
Key Words and Phrases: MERS, superspreader, nosocomial infections, mathematical modeling, infectious diseases
- ⋅ 23rd-G-11:55 − 12:15 Assessment of optimal strategies in a two-patch dengue transmission model with seasonality (Jung Eun Kim, Hyojung Lee, Sunmi Lee, Chang Hyeong Lee)
- 김정은*(울산과학기술원), 이효정(울산과학기술원), 이선미(경희대), 이창형(울산과학기술원)
Jung Eun Kim*, UNIST, Hyojung Lee, UNIST, Sunmi Lee, Kyung Hee University, Chang Hyeong Lee, UNIST
Emerging and re-emerging dengue fever has posed serious problems to public health officials in many tropical and subtropical countries. The continuous traveling in seasonally varying areas makes more difficult to control the spread of dengue fever. In this work, we consider a two-patch dengue model which can capture the movement of host individuals between and within patches using a residence-time matrix. We investigate the effect of human movement and seasonality on the two-patch dengue transmission dynamics. The role of seasonality and residence-time configurations has been highlighted in terms of the seasonal reproduction number and the cumulative incidence. Moreover, optimal control theory has been employed to identify and evaluate patch-specific control measures aimed at reducing dengue prevalence in the presence of seasonality. Our findings highlight that optimal patch-specific control strategies are sensitive to seasonality and residence-time scenarios.
2010 Mathematics Subject Classification: 92Bxx
Key Words and Phrases: Epidemic modeling, Optimal control theory
- Applied Mathematics
- ⋅ 22nd-A-09:00 − 09:20 Generation and propagation of surface and internal long waves by atmospheric pressure disturbances (Young-Kwang Choi, Tae-Chang Jo)
- 최영광*(인하대), 조태창(인하대)
Young-Kwang Choi*, Inha University, Tae-Chang Jo, Inha University
MCC (Miyata, Choi, and Camassa) type strongly nonlinear dispersive long wave model in a two-layer fluid system with non-rigid top boundary is considered. Dynamics of the surface and internal long waves generated by atmospheric pressure disturbances are numerically studied. The model is also compared with single layer linear and nonliear shallow water equations and two-layer shallow water equations. The Proudman resonance interacting with surface waves is investigated. Similar resonance phenomena with internal waves have been also observed from MCC type model with much slower speeds of the atmospheric pressure disturbances. A mechanism to generate surface waves sequentially, dynamics of waves with non-uniform bottom topography, and wave collision are also discussed.
2010 Mathematics Subject Classification: 74J30
Key Words and Phrases: internal waves, MCC type model, Proudman resonance
- ⋅ 22nd-A-09:20 − 09:40 CVOD based model reduction for the Rosenau-RLW equation (Guangri Piao)
- 박광일(Yanbian Univ.)
Guangri Piao, Yanbian University
Using POD to get the dimension reduction models of differential equations, we always need to consider the ensemble of snapshots. The snapshots arise from approximate solutions of the PDEs, which correspond to several sets of parameter values appearing in the problem specification and/or values evaluated at several time instants during the evolution process. Sometimes the ensemble of snapshots is very large, and it is ifficult to apply POD to the ensemble of snapshots directly. So, in this paper, we first use CVT to compress the snapshots set, and then get the reduced-order bases by POD. Numerical experiments show that the combination of CVT and POD (CVOD) methods are more advantageous than the usual combination of even selection and POD (ESPOD) method in the model reduction of the Rosenau-RLW equation.
2010 Mathematics Subject Classification: 78M34
Key Words and Phrases: model reduction for the Rosenau-RLW equation
- ⋅ 22nd-A-09:50 − 10:10 Emergent dynamics of Cucker-Smale flocking ensembles (Seung-Yeal Ha, Dongnam Ko, Xiongtao Zhang, Yinglong Zhang)
- 하승열(서울대), 고동남*(서울대), Xiongtao Zhang(서울대), Yinglong Zhang(서울대)
Seung-Yeal Ha, Seoul National University, Dongnam Ko*, Seoul National University, Xiongtao Zhang, Seoul National University, Yinglong Zhang, Seoul National University
Merging and separation of flocking groups are often observed in our natural complex systems. Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. We will see that the emerging of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that mea- sure the local spatial and velocity fluctuations. Under these methods, we can get additional informations of asymptotic behavior, such as emerging speed and non-flocking conditions.
2010 Mathematics Subject Classification: 34D50
Key Words and Phrases: Cucker-Smale, flocking, local flocking, coupling strength, interactions
- ⋅ 22nd-A-10:10 − 10:30 Scalability of frames generated by dynamical operators (Roza Aceska, Yeonhyang Kim)
- Roza Aceska(Ball State Univ), 김연향*(Central Michigan Univ.)
Roza Aceska, Ball State University, Yeonhyang Kim*, Central Michigan University
Let $A$ be an operator in a finite dimensional Hilbert space $H$, and let $G\subset H$ be a finite set of vectors. It is known that, under certain conditions on $A$ and $G$, the set of iterations $FG(A) = \{A^jg\,|\, g\in G,~ 0\le j\le L(g)\}$ is a frame for $H$. We explore the relations between $A, G$ and the respective number of iterations $L$ that makes the system $FG(A)$ a scalable frame.
2010 Mathematics Subject Classification: 42C15
Key Words and Phrases: frames, tight frames, scalable frames, dynamical operators
- ⋅ 22nd-B-10:45 − 11:25 [Invited Talk] Dual-stiffness spring-mass fabric model and simulation of parachute inflation (Xiaolin Li, Zheng Gao, Xiaolei Chen)
- Xiaolin Li*, Stony Brook University, Zheng Gao, Stony Brook University, Xiaolei Chen, Stony Brook University
A mesoscale spring model based on Rayleigh-Ritz analysis is used to mimic the fabric surface as a wrinkale sheet in parachute simulation. Such elastic structure is coupled with fluid solver through the immersed boundary and impulse method. We will discuss several challenging problems in this multi-physics system including turbulence modeling, fabric collision, parachutist coupling, and computational parallelization. We will show numerical proof of convergence, verification and validation of numerical components, and programming design for simulations of different air-delivery assemblies.
2010 Mathematics Subject Classification: 65Zxx, 74Bxx, 76Nxx
Key Words and Phrases: front tracking, dual-stiffness spring model, impulse method, parachute inflation
- ⋅ 22nd-B-11:35 − 12:15 [Invited Talk] Competition and survival in cyclic games (Younghae Do, Junpyo Park, Bongsoo Jang, Ying-Cheng Lai)
- 도영해*(경북대), 박준표(울산과학기술원), 장봉수(울산과학기술원), Ying-Cheng Lai(Arizona State Univ.)
Younghae Do*, Kyungpook National University, Junpyo Park, UNIST, Bongsoo Jang, UNIST, Ying-Cheng Lai, Arizona State University
Understanding mechanisms for species diversity is indispensable in ecosystems and can be elucidated by adopting evolutionary games. Nonhierarchical cyclic competitions among species are represented by `rock-paper-scissors' and `rock-paper-scissors-lizard-spock' games, which typically provide outstanding paradigms of biodiversity of cyclic competing three or five species, respectively. Competitions within and between species are fundamental roles to hinder or induce the biodiversity of species. In this talk, we present new results.
2010 Mathematics Subject Classification: 92Dxx,37L60
Key Words and Phrases: population dynamics, game theory, RPS
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Anomalous phenomena in granular particle systems (Qiang Zhang)
- Qiang Zhang, City University of Hong Kong
We apply methods developed in dynamic system to study the behavior of granular particles in an extremely simple setting of inelastic, spherical particles falling under gravity and colliding with walls of a symmetric funnel. We show that, even in such simple setting, several surprising phenomena can occur:
(1) One might naively expect that, on average, particles would fall through funnels with steeper sides more quickly, exert a smaller total impulse on the funnel walls, and lose less energy. However, we show that there are special ranges of angles of the funnel walls for which exactly the opposite occurs. Typically, the particle will experience a sequence of collisions that is highly sensitive to the location at which it enters the funnel and nearby particle trajectories become widely dispersed. However, in the special angular ranges this is not the case and the particle can experience sequences of collisions that have a highly coherent structure.
(2) We show that such anomalous phenomena occur in both frictionless and frictional particle systems and the frictional force dramatically enhances the anomalous phenomena. This is due to the stability of the highly coherent structure in these systems.
(3) At certain funnel wall angles and certain injection frequency of the particles, the systems alternates in time between two different states, namely, the phenomenon of intermittency can occur in such simple systems.
Based on methods developed in dynamic systems, we provide a theoretical analysis that can predict and explain the surprising behavior observed.
2010 Mathematics Subject Classification: 70Fxx
Key Words and Phrases: granular particles, anomalous behavior, intermittency
- ⋅ 22nd-C-14:20 − 15:00 [Invited Talk] Kolmogorov flows: a classical topic with new discoveries (Sun-Chul Kim, Hisashi Okamoto)
- 김선철*(중앙대), Hisashi Okamoto(RIMS, Kyoto Univ., Japan)
Sun-Chul Kim*, Chung-Ang University, Hisashi Okamoto, Research Institute for Mathematical Sciences, Kyoto University, Japan
In 1958, A. N. Kolmogorov suggested a simple type of forced Navier-Stokes solutions to study many interesting aspects of Navier-Stokes flows with a close connection to turbulence. After then, a lot of works have been done in regard to the stability, bifurcation and multiplicity of solutions by many researchers. Recently, the authors discovered some new interesting and important asymptotic properties of Kolmogorov flows which seem to be a clue to understand the mechanics of two-dimensional fluid flows including 2d turbulence. Numerical examples and proper asymptotic analysis are presented to explain the obtained result.
2010 Mathematics Subject Classification: 76D
Key Words and Phrases: Kolmogorov flows, Navier-Stokes equations, unimodality, turbulence
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] Coordinate descent and incremental method for regularized minimization (Sangwoon Yun)
- 윤상운(성균관대)
Sangwoon Yun, Sungkyunkwan University
We consider the regularized minimization problem whose objective function is the sum of a smooth function and a convex function. The special cases of regularized minimization problems are bounded constrained minimization problem, linearly constrained minimization problem, L1-regularized least squares problem, L1-regularized logistic regression problem, and total variation regularized convex minimization problem. The coordinate descent method is introduced to solve the regularized minimization problem when the (possibly nonsmooth) convex function has a particular structure such as separability. In machine learning, the smooth function has often the form of the sum of several functions. In this case, the incremental method is considered to solve the problem.
2010 Mathematics Subject Classification: 49M27, 90C25
Key Words and Phrases: regularized minimization, coordinate descent method, incremental method
- ⋅ 22nd-D-16:05 − 16:25 Incremental gradient method for Karcher mean on symmetric cones (Sangho Kum, Sangwoon Yun)
- 금상호*(충북대), 윤상운(성균관대)
Sangho Kum*, Chungbuk National University, Sangwoon Yun, Sungkyunkwan University
In this talk, we deal with the minimization problem for computing Karcher mean on a symmetric cone. The objective of this minimization problem consists of the sum of squares of the Riemannian distances with many given points in a symmetric cone. Moreover, the problem can be reduced to a bound constrained minimization problem. These motivate us to adapt an incremental gradient method. So we propose an incremental gradient method and establish its global convergence properties exploiting the Lipschitz continuity of the gradient of the Riemannian distance function.
2010 Mathematics Subject Classification: 65K05, 90C25
Key Words and Phrases: Karcher mean, incremental gradient method, symmetric cone
- ⋅ 22nd-D-16:25 − 16:45 On sequential optimality conditions for convex optimization problems (Jae Hyoung Lee, Gue Myung Lee)
- 이재형 (부경대), 이규명*(부경대)
Jae Hyoung Lee, Pukyong National University, Gue Myung Lee*, Pukyong National University
In this talk, we give two kinds of sequential optimality conditions for convex optimization problems, which are expressed sequences of epsilon subgradients of involved (proper and lower semi-continuous) convex functions and which hold without any constraint qualification. Moreover, we give examples illustrating our sequential optimality conditions.
2010 Mathematics Subject Classification: 90C26
Key Words and Phrases: convex optimiation problems, sequential optimality conditions, convex functions
- ⋅ 22nd-E-17:00 − 17:20 On some fuzzy differential equations (Minghao Chen)
- 진명호(Harbin Institute of Technology)
Minghao Chen, Harbin Institute of Technology
At present, the researches on solutions to fuzzy differential equations are mainly in the following three ways: H-derivatives and Bede's generalized derivatives which are generalized from H-derivatives; Zadeh's Extension Principle; differential inclusions. And the theories of fuzzy differential equations are different under these three approaches. In this paper some of the works of the fuzzy differential equations will be introduced.
2010 Mathematics Subject Classification: 35R13
Key Words and Phrases: fuzzy differential equations
- ⋅ 22nd-E-17:20 − 17:40 Convergence analysis of boosted proximal point algorithm for DC programming (Liguo Jiao, Zhe Hong, Do Sang Kim)
- Liguo Jiao*(부경대), Zhe Hong(부경대), 김도상(부경대)
Liguo Jiao *, Pukyong National University, Zhe Hong, Pukyong National University, Do Sang Kim, Pukyong National University
We consider the minimization problems of the form $P(\varphi, g, h)$: $\min\{f(x) = \varphi(x) + g(x) - h(x): x \in \mathbb R^n\}$, where $\varphi$ is a differentiable function and $g$, $h$ are convex functions, and introduce iterative methods to finding a critical point of $f$ when $f$ is differentiable. We show that the point computed by proximal point algorithm at each iteration can be used to determine a descent direction for the objective function at this point. This algorithm can be considered as a combination of proximal point algorithm together with a linesearch step that uses this descent direction. We also study convergence results of these algorithms and the inertial proximal methods proposed by P. E. Maing\'e and A. Moudafi (SIAM J Optim 19, 397--413, 2008) under the main assumption that the objective function satisfies the Kurdika-{\L}ojasiewicz property.
2010 Mathematics Subject Classification: 49J52, 65K10
Key Words and Phrases: DC programming, Kurdyka-{\L}ojasiewicz inequality, proximal mapping, critical points
- ⋅ 22nd-E-17:50 − 18:10 Numerical methods for solving initial value problems of some kinds of nonlinear impulsive fractional differential equations (Yuanfeng Jin)
- 김원봉(Yanbian Univ.)
Yuanfeng Jin, Yanbian University
This article is concerned with the numerical solutions for initial value problems of nonlinear impulsive fractional differential equations which are actively studied recently. In this paper we construct numerical schemes for solving initial value problems of I-type impulsive fractional differential equation and II-type impulsive fractional differential equation and estimate their convergence and stability.
2010 Mathematics Subject Classification: 34A08
Key Words and Phrases: nonlinear impulsive fractional differential equations
- ⋅ 22nd-E-18:10 − 18:30 On the existence of Hermitian positive definite solution of $X^{p}-A^{*}e^{X}A=I_{n}$ (Chacha stephen Chacha, Hyun-Min Kim)
- Chacha stephen Chacha*(부산대), 김현민(부산대)
Chacha stephen Chacha*, Pusan National University, Hyun-Min Kim, Pusan National University
In this work, we explored Hermitian positive definite solution(HPDS) of a new nonlinear matrix equation $ X^{p}-A^{*}e^{X}A=I_{n}$, where $A $ is a stable matrix, which means that $\rho(A)<1,$ $p \in {\mathbb Z}_{+}\backslash \{1\},$ $X$ an unknown Hermitian positive definite matrix and $I_{n}$ an $n\times n $ identity matrix. We derived the necessary and sufficient conditions for existence of HPDS. Furthermore, the convergence analysis of the matrix sequence to HPDS, the existence of unique HPD solution and an iterative algorithm were discussed. Finally, numerical tests and results of the problem are reported to verify the effectiveness of the suggested iterative approach. It was depicted that our nonlinear matrix equation above, have a unique HPD solution for all Markov matrices $(\rho(A )=1)$ and matrices($A_{i}$) satisfying $\rho(A_{i})<1.$
2010 Mathematics Subject Classification: 65F60
Key Words and Phrases: Hermitian positive definite solution, stable matrix, nonlinear matrix equation, iterative algorithm
- ⋅ 23rd-F-09:20 − 09:40 Iteration with stepsize parameter and condition numbers for one nonlinear matrix equation (Syed Muhammad Raza Shah Naqvi, Jie Meng, Hyun-Min Kim)
- Syed Muhammad Raza Shah Naqvi*(부산대), Jie Meng(부산대), 김현민(부산대)
Syed Muhammad Raza Shah Naqvi*, Pusan National University, Jie Meng, Pusan National University, Hyun-Min Kim, Pusan National University
We consider the nonlinear matrix equation $X^p+A^TXA=Q$, where $p$ is a positive integer, $A$ is an arbitrary $n\times n$ matrix, and $Q$ is a Hermitian positive definite matrix. A fixed-point iteration with stepsize parameter for obtaining the Hermitian positive definite solution of the matrix equation is proposed. The explicit expressions of the normwise, mixed and componentwise condition numbers and the backward error are derived. Several numerical examples are presented to show the efficiency of the proposed iterative method with proper stepsize parameter and the sharpness of the three condition numbers.
2010 Mathematics Subject Classification: 15A24, 65F10, 65H10
Key Words and Phrases: matrix equation, Hermitian positive definite, fixed-point iteration, condition number, mixed and componentwise, backward error
- ⋅ 23rd-F-09:50 − 10:10 Portfolio optimization problem under a multiscale Heston's stochastic volatility model (Jai Heui Kim, Sotheara Veng)
- 김재희(부산대), Sotheara Veng*(부산대)
Jai Heui Kim, Pusan National University, Sotheara Veng*, Pusan National University
We study the portfolio optimization problem under a multi-scale stochastic volatility model in which a fast mean reverting volatility factor is incorporated to the Heston's stochastic volatility model. Using an asymptotic analysis method, we are able to derive explicitly approximations to the value function and the optimal strategy for hyperbolic absolute risk aversion (HARA) utility functions. When the structure of correlation is of particular form, we give a proof of the accuracy of the approximation to the value function in the case of power utility functions. We also introduce a practical strategy that does not depend on the incorporated factor which is unobservable in the market.
2010 Mathematics Subject Classification: 90C39, 90C59, 91G10, 35Q93
Key Words and Phrases: portfolio optimization, Heston's model, stochastic volatility, asymptotic analysis, ergodic process
- ⋅ 23rd-F-10:10 − 10:30 Normwise, mixed and componentwise condition numbers of a matrix polynomial equation arising in stochastic models (Jie Meng, Sang-Hyup Seo, Hyun-Min Kim)
- 맹걸*(부산대), 서상협(부산대), 김현민(부산대)
Jie Meng*, Pusan National University, Sang-Hyup Seo, Pusan National University, Hyun-Min Kim, Pusan National University
We consider a matrix polynomial equation (MPE) $A_nX^n+A_{n-1}X^{n-1}+\cdots +A_0=0$, where $A_n, A_{n-1},\ldots, A_0 \in \mathbb{R}^{m\times m}$ are the coefficient matrices, and $X\in \mathbb{R}^{m\times m}$ is the unknown matrix. One sufficient condition for the existence of elementwise minimal nonnegative solution is derived. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are presented. Some numerical examples are given to show the sharpness of the three condition numbers.
2010 Mathematics Subject Classification: 15A24, 65F10, 65H10
Key Words and Phrases: matrix equation, Hermitian positive definite, fixed-point iteration, coupled fixed-point theorem
- Mathematical Education
- ⋅ 22nd-C-13:30 − 14:10 [Invited Talk] Reasoning, proof, and justification: modifying textbook tasks to integrate reasoning into the non-geometry curriculum (Denisse R. Thompson)
- Denisse R. Thompson, University of South Florida
Reasoning and proof are important mathematical processes in classrooms around the world. Yet, students often struggle with reasoning and proof and teachers wonder how to integrate these processes into their classroom instruction on a regular basis. A framework from research on textbook curriculum is used to illustrate types of exercises that can focus on proof-related reasoning. Typical textbook examples from algebra, precalculus, and statistics are shared, with suggestions for modifications to enhance reasoning and justification.
2010 Mathematics Subject Classification: 97
Key Words and Phrases: modifying textbook tasks, reasoning in the curriculum
- ⋅ 22nd-C-14:20 − 14:40 The spirit of the age of mathematics and sculpture (Young Hee Kye)
- 계영희(고신대)
Young Hee Kye, Kosin University
In this study, We consider the co-relationship about this contemporary sculpture and topology. So we can enjoy of many works such Rodin, Brancusi, Modigliani, Giacometti, Henri Moore, and Dali, and then we can find the characteristic of modern times' sculpture are simplification and abstraction such as topology of mathematics.
2010 Mathematics Subject Classification: 97A02
Key Words and Phrases: topology, contemporary scuplture
- ⋅ 22nd-C-14:40 − 15:00 Study on KSA-KAIST continuity education system for mathematically gifted students (Hun Kim)
- 김훈(한국과학영재학교)
Hun Kim, Korea Science Academy of KAIST
Korea Science Academy of KAIST is the first science-gifted institute in Korea which offers a differentiated curriculum. One of the most important education system of KSA is KSA-KAIST continuity education.
In this talk, we will observe this continuity education and suggest more efficient system through investigating mathematically gifted students who graduated from KSA and KAIST.
2010 Mathematics Subject Classification: 97B40
Key Words and Phrases: continuity education, gifted student
- ⋅ 22nd-D-15:15 − 15:55 [Invited Talk] History of early mathematics curricula in Korea (Young Wook Kim)
- 김영욱(고려대)
Young Wook Kim, Korea University
We report on some of the historical documents on the preparation for mathematics curriculum in the early period of Korea. One of these documents sheds light on the work of the Ministry of Education on the establishment of the first curriculum in the mid 1950's. Also some documents from late 1960's to early 1970's show some details in the process of construction of the third mathematics curriculum.
2010 Mathematics Subject Classification: 01A60, 97-03, 97A30
Key Words and Phrases: mathematics curriculum, history in establishment of curriculum
- ⋅ 22nd-D-16:05 − 16:25 Mathematics suitable for future students (Young Wook Kim, Hye Sook Park, Sung-Eun Koh, Sangwook Ree, Young Rock Kim, Dosang Joe, Jeongwook Chang)
- 김영욱(고려대), 박혜숙*(서원대), 고성은(건국대), 이상욱(수원대), 김영록(한국외대), 조도상(국가수리과학연구소), 장정욱(단국대)
Young Wook Kim, Korea University, Hye Sook Park*, Seowon University, Sung-Eun Koh, Konkuk University, Sangwook Ree, The University of Suwon, Young Rock Kim, Hankuk University of Foreign Studies, Dosang Joe, NIMS, Jeongwook Chang, Dankook University
The rapidly changing contemporary society shows enormous amount of reshaping in the abilities needed for the future generation. Recently many country concentrates on the researches to keep up with the changes needed for the education of the future generation. We conclude that now is the time when there were paradigm shifts in science and technology. Therefore we try to suggest on the contents for mathematics learning which is necessary in the preparation for the future society.
At first, we do the following;
(1) Review and analysis of reports on future studies and future mathematics
(2) First establishment of competency for future in the mathematical viewpoint
(3) Analysis of classifications and suggestions for future jobs
(4) Collection of contents and directions for mathematics for future.
(This is the part of the an intermediate report of the project performed by Kim, Young-Wook etc.).
2010 Mathematics Subject Classification: 97B10
Key Words and Phrases: future generation competency, future jobs, mathematics learning for future generation
- ⋅ 22nd-D-16:25 − 16:45 Counting is an important ingredient of mathematics education (Youngmee Koh, Sangwook Ree)
- 고영미(수원대), 이상욱*(수원대)
Youngmee Koh, The University of Suwon, Sangwook Ree*, The University of Suwon
Mathematics is a kind of language, and even a tool of cognition for human beings.
Mathematics has been used to communicate and to develop the civilizations
through the history. So mathematics is one of the most important subjects for human
to teach and learn. Especially, developed countries believe that mathematics
will play very important roles in the developments of future industries and so
future society. In this talk, we clarify that combinatorics which is mainly represented
by counting is an important ingredient of future mathematics education.
To do so, we investigate the characteristics of combinatorics from the educational
and cognitive perspectives.
2010 Mathematics Subject Classification: mathematics education, cognition, counting
Key Words and Phrases: 05-01, 97A40, 97B20, 97C30, 97C70, 97K20
- Poster Session
- ⋅ 22nd-D-16:15 − 16:45 Optimal tool axis control for 5-axis CNC machining using tractrix (Chang Yong Han)
- 한창용(경희대)
Chang Yong Han, Kyung Hee Univeristy
We consider the tool orientation problem of a 5-axis CNC machine that cuts along a curved toolpath on a smooth workpiece surface in which the tool axis maintains a fixed angle with respect to the surface normal. We control the variation of the remaining azimuthal angle of the tool axis about the surface normal so that the instantaneous speed of the tool axis motion is minimal, which is a judicious choice for the efficient and robust machine performance. The ensuing curve traced by the tool axis turns out to be a generalized tractrix on the unit sphere determined by the curve traced by the surface normal vector.
2010 Mathematics Subject Classification: 00A69
Key Words and Phrases: 5-axis CNC machining, tool orientation, tractrix
- ⋅ 22nd-D-16:15 − 16:45 Existence of 1-sum 3-flows of Cayley graphs (Seong Woo Hur, Hyunsu Kim, Boram Park)
- 허승우(경기과학고), 김현수(경기과학고), 박보람*(아주대)
Seong Woo Hur, Gyeonggi Science High School, Hyunsu Kim, Gyeonggi Science High School, Boram Park*, Ajou University
For a graph $G$, for positive integers $l$ and $k$, an $l$-sum $k$-flow is an assignment of values from $\{\pm 1,\ldots, \pm (k-1)\}$ to the edges of $G$ such that for every vertex $v$ of $G$, the sum of values of all edges incident with $v$ equals $l$. It has been an interesting problem to determine the existence of an $l$-sum $k$-flow of a regular graph. Recently, it was shown that every regular graph of an even order admits a 1-sum 5-flow, and in the same paper, it was asked if every $4k$-regular connected graph with an even order admits a 1-sum 4-flow or not. In this paper, we answer the question positively in the class of Cayley graphs over a finite Abelian group, by showing that every connected Cayley graph with an even order allows a 1-sum 3-flow. We also explore more observations on 1-sum 2- and 3-flows.
2010 Mathematics Subject Classification: 05C21, 05C22
Key Words and Phrases: 1-sum 3-flow, Cayey graph, regular graph
- ⋅ 22nd-D-16:15 − 16:45 Factorization theory in general Hurwitz rings (Dong Kyu Kim, Jung Wook Lim)
- 김동규*(경북대), 임정욱(경북대)
Dong Kyu Kim*, Kyungpook National University, Jung Wook Lim, Kyungpook National University
We define the domains with respect to factorization and define the general composite Hurwitz rings. We introduce some properties of Hurwitz rings and we find the eqivalent conditions for these domains to be general composite Hurwitz rings.
2010 Mathematics Subject Classification: 13G99
Key Words and Phrases: factorization, Hurwitz ring
- ⋅ 22nd-D-16:15 − 16:45 Boundedness of the Segal-Bargmann transform on fractional Fock-Sobolev spaces (Hong Rae Cho, Hyunil Choi, Han-Wool Lee)
- 조홍래(부산대), 최현일*(부산대), 이한울(부산대)
Hong Rae Cho, Pusan National University, Hyunil Choi*, Pusan National University, Han-Wool Lee, Pusan National University
Let $s\in\mathbb{R}$ and $2\leq p\leq\infty$.
We prove that the Segal-Bargmann transform $\mathcal B$ is a bounded operator
from fractional Hermite-Sobolev spaces
$W^{s,p}_H(\mathbb{R}^n)$ to fractional Fock-Sobolev spaces $F^{s,p}_\mathscr{R}$.
2010 Mathematics Subject Classification: Primary 26A33, 30H20 Secondary 42B35
Key Words and Phrases: fractional Fock-Sobolev space, fractional radial derivative, fractional Hermite operator, fractional Hermite-Sobolev space
- ⋅ 22nd-D-16:15 − 16:45 An application of the fixed point theorem to nonlinear fractional differential equations (Jinsil Lee, Yong-Hoon Lee)
- 이진실*(부산대), 이용훈(부산대)
Jinsil Lee*, Pusan National University, Yong-Hoon Lee, Pusan National University
In this paper, we introduce the existence of positive solutions for nonlinear fractional differential equation with a singular weight.\\
Consider the following fractional nonlinear differential equation
\begin{equation*}\tag*{($E)$}
\begin{cases}
D^{\alpha}_{0+}u(t)+h(t)f(u(t))= 0,\quad t\in (0,1),\\
u(0)= 0 = u(1),
\end{cases}
\end{equation*}
where {$D^{\alpha}_{0+}$} is the Riemann-Liouville fractional derivative of order $\alpha$,
$\alpha$ is a real number in $(1,2]$, {$h\in L^{1}_{loc}(0,1$}) satisfies the condition
$(H)\int_{0}^{1}s^{\alpha-1}(1-s)^{\alpha-1}h(s)ds<+\infty$ and {$f\in C([0,\infty), [0,\infty)$}).
Based on above assumptions, we firstly set up the solution operator to apply fixed point theorem to our equation $(E).$
As an application, we examine the existence of solution of $(E)$ when $f_0=\lim_{u\to 0}\frac{f(u)}{u}$, $f_\infty=\lim_{u\to \infty}\frac{f(u)}{u}$ are 0 or finite or $\infty$ respectively. In addition, we show the nonexistence and existence of the solution for our equation under special case.
2010 Mathematics Subject Classification: 34B16
Key Words and Phrases: fractional differential equation, positive solution, existence of solution, singular weight
- ⋅ 22nd-D-16:15 − 16:45 Elliptic gradient constraint problem (Souksomvang Phoui, Hi Jun Choe)
- Souksomvang Phoui*(연세대), 최희준(연세대)
Souksomvang Phoui*, Yonsei University, Hi Jun Choe, Yonsei University
We study the existence and regularity of gradient constraint problem. It arises in elastoplasticity and finance.
First, we consider linear double obstacle problem which comes from viscosity solution to Hamilton-Jacobi equation and find the solution has $C^{1,\alpha}$, regularity by estimating Campanato type integral oscillation. Then, by perturbation method and fixed point theorem in $C^{1,\alpha}$, space, we prove the existence of $C^{1,\alpha}$, solution.
2010 Mathematics Subject Classification: 35Jxx
Key Words and Phrases: gradient constraint, viscosity solution, Hamilton-Jacobi equation, obstacle problem, $C^{1,\alpha}$ regularity
- ⋅ 22nd-D-16:15 − 16:45 Blow-up time for a quasilinear parabolic equation with inner source and nonlinear boundary condition (Rui Yang, Zhong Bo Fang)
- Rui Yang*(부산대), Zhong Bo Fang(Ocean Univ. of China)
Rui Yang*, Pusan National University, Zhong Bo Fang, Ocean University of China
In this paper, a quasilinear parabolic equation with inner source and nonlinear boundary
condition is studied. Based on the technique of differential inequality, a lower bound for the blow-up time if blow-up does occur is established. Moreover, a specific class of problems to guarantee occurrence of blow-up and derive an upper bound for the blow-up time are also considered.
2010 Mathematics Subject Classification: 35K65, 35B30, 35B40
Key Words and Phrases: quasilinear parabolic, blow-up time, lower bound, upper bound
- ⋅ 22nd-D-16:15 − 16:45 Hypersurface in quasi K\"ahler manifold (Jihong Bae, Wonmin Shin)
- 배지홍*(성균관대), 신원민(성균관대)
Jihong Bae*, Sungkyunkwan University, Wonmin Shin, Sungkyunkwan University
We discuss an oriented hypersurface of a quasi K\"ahler manifold and give a necessary and sufficient condition for such a hypersurface to be a quasi contact metric manifold with respect to the naturally induced almost contact metric structure.
2010 Mathematics Subject Classification: 53C25, 53D10
Key Words and Phrases: contact metric manifold, quasi contact metric manifold
- ⋅ 22nd-D-16:15 − 16:45 On tricolorability of $1$-tangles (Yongju Bae, Hun Lee)
- 배용주(경북대), 이헌*(경북대)
Yongju Bae, Kyungpook Nationcal University, Hun Lee*, Kyungpook Nationcal University
An $n$-tangle diagram in knot or link projection is a region in the projection plane surrounded by a circle such that there is the disjoint union of $n$ arcs into the circle.
In this talk, We will define colorability of an $n$-tangle diagram and study how effects colorability on $1$-tangle.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: tangle, colorability
- ⋅ 22nd-D-16:15 − 16:45 On the invertibility of the closure of $(n,n)$-tangles (Yongju Bae, Yongjae Park)
- 배용주(경북대), 박용재*(경북대)
Yongju Bae, Kyungpook National University, Yongjae Park*, Kyungpook National University
An $(n,n)$-tangle is an embedding of $n$ arcs and $m$ circles into a $3$-ball with $2n$ fixed points which are boundaries of $n$ arcs.
For given $(n,n)$-tangles $T$ and $T^{'}$, we will define the product $T\bullet T^{'}$ which is also an $(n,n)$-tangle. In this talk, we will study conditions of $T$ satisfying that there exists an $(n,n)$-tangle $T^{'}$ such that the closure of $T\bullet T^{'}$ is the trivial link.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: tangle, invertibility
- ⋅ 22nd-D-16:15 − 16:45 Approximating an irrational power of a matrix (Yeonji Kim, Hyun-Min Kim, Jong-Hyeon Seo)
- 김연지*(부산대), 김현민(부산대), 서종현(부산대)
Yeonji Kim*, Pusan National University, Hyun-Min Kim, Pusan National University, Jong-Hyeon Seo, Pusan National University
We present a new algorithm for approximating arbitrary irrational powers of a complex square matrix using binary number representation and the matrix square root algorithms. There are several methods for computing the matrix square root. We derive the algorithm for some special matrices by using different matrix square root methods. We also derive a version of the extended algorithm that combine the Schur decomposition. Our numerical experiments show similar accuracy as compared to existing methods.
2010 Mathematics Subject Classification: 65F30
Key Words and Phrases: an irrational power of a matrix, matrix square root, binary number representation system, Schur decomposition
- ⋅ 22nd-D-16:15 − 16:45 Multiscale method for flluid flow in porous media (Muhammad Arshad, Eun-Jae Park, Dong-wook Shin)
- Muhammad Arshad*(연세대), 박은재(연세대), 신동욱(연세대)
Muhammad Arshad*, Yonsei University, Eun-Jae Park, Yonsei University, Dong-wook Shin, Yonsei University
We consider the discretization of second order nonlinear elliptic problem by multiscale mortar mixed finite element method. Many physical system of interest leads to nonlinear partial differential equation with heterogeneity, e.g. fluid flow in porous media. For the problems with heterogeneous coefficient fluctuating on fine scale, the direct discretization require full fine scale resolution of variation in coefficient on the whole domain which results in a large system of equations whose solution would be complicated computationally. Main purpose of the multiscale methods is to alleviate computational load.
Our method, based on domain decomposition and mortar finite element methods. The domain is decomposed in to small subdomains (coarse element).
The mortar finite element space is introduced to impose flux continuity across interface and the problems are discretized on fine scale on each subdomain. The local problem on each subdomain is solved by standard mixed finite element method with appropriate dirichlet boundary conditions. The efficiency of the method comes from divide and conquer strategy. We derive a priori error estimate for both flux and pressure and show the optimal
order convergence on the fine scale by a proper choice of mortar space and polynomial degree
of approximation. Numerical experiments are performed in confirmation of theory.
2010 Mathematics Subject Classification: 65N30, 65N55
Key Words and Phrases: mixed finite element, multiscale, mortar finite element, error estimates, domain decomposition