KMS Meeting - Programs and Abstracts
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- Plenary Lecture
- ⋅ 15:20 − 16:10 Chair: Jun-Muk Hwang(KIAS)
- ⋅ 5th 15:20 − 16:10 $\mathbb Q$-Gorenstein deformation theory and its applications (Yongnam Lee)
- 이용남(카이스트)
Yongnam Lee, KAIST
The notion of singularity of class T, which is defined as a quotient surface singularity admitting a $\mathbb Q$-Gorenstein smoothing, was introduced by Koll\'ar and Shepherd-Barron [2]. They also gave an explicit description of the singularity of class T. Previous study for singularities for class T and $\mathbb Q$-Gorenstein deformation theory use the property that an index one cover is \'etale on the Gorenstein locus, which does not hold in characteristic $p > 0$ when the Gorenstein index is divisible by $p$. By developing the theory of $\mathbb Q$-Gorenstein deformation functor, we can generalize their results to surfaces in positive characteristics. In [5] we prove the existence of versal $\mathbb Q$-Gorenstein $\Lambda$-deformation for some typical $\mathbb Q$-Gorenstein $k$-schemes for a fixed complete Noetherian ring $\Lambda$ with an algebraically closed residue field $k$.
After the paper [3], which constructs a simply connected Campedelli surface (a minimal projective surface of general type with $p_g=q=0$, $K^2=2$) by using a $\mathbb Q$-Gorenstein smoothing and Milnor fiber of a smoothing (or rational blow-down surgery), several interesting examples of surfaces of general type with $p_g=q=0$ were constructed via $\mathbb Q$-Gorenstein smoothings. Now, these $\mathbb Q$-Gorenstein smoothing methods are extended to some other type of surfaces and to surfaces in positive characteristics. To find a new family of simply connected surfaces of general type with $p_g=q=0$ is one of the fundamental problems in the study of algebraic surfaces. Surfaces with $p_g =q= 0$ are interesting in view of Castelnuovo's criterion: An irrational surface with $q = 0$ must have $P_2\ge 1$. This class of surfaces has been studied extensively by algebraic geometers and topologists for a long time. In particular, simply connected surfaces of general type with $p_g=0$ are little known. Before the paper [3] the only previously known simply connected, minimal, complex surface of general type with $p_g=0$ was Barlow surface [1]. Barlow surface has $K^2 = 1$. When a surface is defined over a field of positive characteristic, the existence of algebraically simply connected minimal surface of general type with $p_g = 0$ is known only for some special characteristics. The paper [4] constructs such a surface of general type defined over an algebraically closed field of any characteristic applying the construction given in [3]. Construction in [3] is as follows: First, we consider a special pencil of cubics in $\mathbb P^2$ and blow up many times to get a projective surface $M$, which contains a disjoint union of five linear chains of smooth rational curves representing the resolution graphs of singularities of class $T$. Then, we contract these linear chains of rational curves from the surface $M$ to produce a projective surface $X$ with five singularities of class $T$ and with $K^2_X = 2$. We can prove the existence of a global $\mathbb Q$-Gorenstein smoothing of the singular surface $X$, in which a general fiber $X_t$ of the $\mathbb Q$-Gorenstein smoothing is a simply connected minimal surface of general type with $p_g = 0$ and $K^2 = 2$. This method of construction works to other types of rational elliptic surfaces, which are used to construct a simply connected minimal surface of general type with $p_g = 0$ and with $K^2$ = 1, 3, or 4. The paper [4] shows that the construction of singular surfaces also works in positive characteristic, but several key parts in the proof to show the existence of a global $\mathbb Q$-Gorenstein smoothing should be modified. We construct a deformation of a normal projective surface $X$ with toric singularities of class $T$ assuming some extra conditions. As a consequence, we have a so-called $\mathbb Q$-Gorenstein smoothing not only over the base field $k$ but also over a complete discrete valuation ring with the residue field $k$. By the smoothing over the discrete valuation ring and the Grothendieck specialization theorem, the algebraic simply connectedness of the smooth fiber is reduced to that of a smooth fiber of a $\mathbb Q$-Gorenstein smoothing of a reduction of $X$ of our construction to the complex number field.
\begin{thebibliography}{9}
\bibitem{1} R. Barlow, {\it A simply connected surface of general type with $p_g =0$}, Invent. Math. {\bf 79} (1984), 293--301.
\bibitem{2} J. Koll\'ar and N. I. Shepherd-Barron, {\it Threefolds and deformations of surface singularities}, Invent. Math. {\bf 91} (1988), 299--338.
\bibitem{3} Y. Lee and J. Park, {\it A simply connected surface of general type with $p_g=0$ and $K^2=2$}, Invent. Math. {\bf 170} (2007), 483--505.
\bibitem{4} Y. Lee and N. Nakayama, {\it Simply connected surfaces of general type in positive characteristic via deformation theory}, to appear in PLMS.
\bibitem{5} Y. Lee and N. Nakayama, {\it $\mathbb Q$-Gorenstein deformation theory over a complete Noetherian local ring}, In preparation.
\end{thebibliography}
2010 Mathematics Subject Classification: Primary 14J29; Secondary 14B07, 14J17
Key Words and Phrases: surface of general type, $\mathbb Q$-Gorenstein smoothing, singularity of class $T$
- Special Invited Lecture
- ⋅ 16:20 − 17:10 [ Chair: Jong Hae Keum(KIAS)
- ⋅ 5th 16:20 − 17:10 Simply connected symplectic 4-manifolds with $b_{2}^{+}=1$ and $c_{1}^{2}=2$ - Before and After 2004 (Jongil Park)
- 박종일(서울대)
Jongil Park, Seoul National University
One of the fundamental problems in the topology of $4$-manifolds is to classify all smooth (symplectic, complex) $4$-manifolds homeomorphic to the same underlying
topological $4$-manifold. Even though gauge theory has been very successful
in addressing this question, the complete answer is still far from reach. In particular, for smooth structures on simply connected closed smooth $4$-manifolds, most known results have been obtained
for the case of $4$-manifolds with either $b_2^+ > 1$ and odd or $b_2^+ = 1$ and $c_1^2 \leq 0$.
In the case of $b_2^+ = 1$ and $c_1^2 > 0$, the only known result is the theorem of D. Kotschick that the Barlow surface (\cite{b}) is not diffeomorphic to ${\mathbf CP}^2 \sharp 8{\overline{{\mathbf CP}}^2}$(\cite{k}).
Since I discovered a new simply connected symplectic $4$-manifold with $b_2^+ =1$ and $c_1^2=2$ (\cite{p}) in 2004 by using a rational blow-down surgery, many new simply connected $4$-manifolds with small Euler characteristic have been constructed (\cite{fs, lp, pps, pss, ss, ssw}) and now it is one of the most active research areas in $4$-manifolds to find a new family of smooth (symplectic, complex) $4$-manifolds with $b_2^+ =1$ and $c_1^2 > 0$.
The aim of this talk is to review briefly how to construct such a symplectic $4$-manifold with $b_2^+ =1$ and $c_1^2=2$ . And then I'll explain how this construction affects the study of simply connected $4$-manifolds with $b_2^+=1$ (equivalently, $p_g=0$ in complex category) in three levels - smooth category, symplectic category and complex category.
\begin{thebibliography}{BD}
\bibitem{b} R. Barlow, {\it A simply connected surface of general type with
$p_g =0$}, Invent. Math. {\bf 79} (1984), 293--301.
\bibitem{fs} R. Fintushel and R. Stern, {\it Double node neighborhoods and families of simply
connected 4-manifolds with $b^+=1$}, J. Amer. Math. Soc. {\bf 19} (2006), 171--180.
\bibitem{k} D. Kotschick, {\it On manifolds homeomorphic to
${\mathbf CP}^2 \sharp 8{\overline{{\mathbf CP}}^2}$},
Invent. Math. {\bf 95} (1989), 591--600.
\bibitem{lp} Y. Lee and J. Park, {\it A surface of general type with $p_g=0$ and $K^2=2$},
Invent. Math. {\bf 170} (2007), 483--505.
\bibitem{p} J. Park, {\it Simply connected symplectic 4-manifolds with
$b_2^+=1$ and $c_1^2=2$}, Invent. Math. {\bf 159} (2005),
657--667.
\bibitem{pps} H. Park, J. Park, and D. Shin, {\it A simply connected surface of general type
with $p_g=0$ and $K^2=3, 4$}, Geom. Topol. {\bf 13} (2009), 743--767; 1483--1494.
\bibitem{pss} J. Park, A. Stipsicz, and Z. Szab\`{o}, {\it Exotic smooth structures on
${\mathbf CP}^2 \sharp 5{\overline{{\mathbf CP}}^2}$}, Math. Res. Letters {\bf 12} (2005), 701--712.
\bibitem{ss} A. Stipsicz and Z. Szab\`{o}, {\it An exotic smooth structure on ${\mathbf CP}^2
\sharp 6{\overline{{\mathbf CP}}^2}$}, Geom. Topol. {\bf 9} (2005), 813--832.
\bibitem{ssw} A. Stipsicz, Z. Szab\`{o}, and J. Wahl, {\it Rational blowdowns and smoothings
of surface singularities}, J. Topology {\bf 1} (2008), 477--517.
\end{thebibliography}
2010 Mathematics Subject Classification: 53D05, 14J26, 57R55, 57R57
Key Words and Phrases: exotic smooth structure, rational blow-down surgery, symplectic 4-manifolds, complex surface of general type
- Invited Lectures
- ⋅ 10:40 − 11:20 [ Chair: Jongil Park(Seoul National University)
- ⋅ 5th 10:40 − 11:20 Finite group action, Floer theory and orbifolds (Cheol Hyun Cho)
- 조철현(서울대)
Cheol Hyun Cho, Seoul National University
This will be an introductory talk regarding Lagrangian Floer theory on orbifolds, its homological mirror symmetry, and related phenomenons. In particular, the role of group actions, and orbifold holomorphic curve will be explained, as well as open crepant resolution conjecture.
2010 Mathematics Subject Classification: 53DXX
Key Words and Phrases: Floer homology, orbifolds, group action
- ⋅ 10:40 − 11:20 [Mathematics for Information Sciences] Chair: Seog-Jin Kim(Konkuk University)
- ⋅ 5th 10:40 − 11:20 Even cycle decomposition of graphs with no odd $K_4$ Minor (Sang-il Oum, Tony Huynh and Maryam Verdian-Rizi)
- 엄상일*(카이스트), Tony Huynh(카이스트), Maryam Verdian-Rizi(카이스트)
Sang-il Oum*, KAIST, Tony Huynh, KAIST, Maryam Verdian-Rizi, KAIST
An even cycle decomposition of a graph $G$ is a partition of $E(G)$ into cycles of even length. Evidently, every Eulerian bipartite graph has an even cycle decomposition. Seymour [Circuits in planar graphs. J. Combin. Theory Ser. B 31 (1981), no. 3, 327--338] proved that every $2$-connected loopless Eulerian planar graph with an even number of edges also admits an even cycle decomposition. Later, Zhang [On even circuit decompositions of {E}ulerian graphs. J. Graph Theory 18 (1994), no. 1, 51--57] generalized this to graphs with no $K_5$-minor. In this paper we propose a conjecture involving signed graphs which contains all of these results. Our main result is a weakened form of this conjecture. Namely, we prove that every $2$-connected loopless Eulerian odd-$K_4$-minor free signed graph with an even number of odd edges has an even cycle decomposition.
2010 Mathematics Subject Classification: 05C45
Key Words and Phrases: Eulerian, even cycle decomposition
- ⋅ 10:40 − 11:20 [Cryptography] Chair: Hyang-Sook Lee(Ewha Womans University)
- ⋅ 5th 10:40 − 11:20 Side channel attacks and their applications (Young-Ho Park)
- 박영호(세종사이버대)
Young-Ho Park, Sejong Cyber University
In cryptography, a side channel attack is any attack based on side channel information gained from the physical implementation of a cryptosystem. It takes advantage of implementation-specific characteristics to recover the secret parameters involved in the computation. It is therefore much less general but often much more powerful than mathematical cryptanalysis. In this talk, we are going to survey side channel attacks and their applications.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: side channel attack, cryptanalysis, elliptic curve cryptosystem, information security
- ⋅ 11:30 − 12:10 [Topology] Chair: In Kang Kim(KIAS)
- ⋅ 5th 11:30 − 12:10 Deformation spaces of Kleinian groups and beyond (Ken'ichi Ohshika)
- Ken'ichi Ohshika, Osaka University \& KIAS
The main object in Kleinian group theory is now studying topological structure of deformation spaces of Kleinian groups. Each deformation space is embedded in a character variety, and it is also an important problem to determine how it is embedded and what kind of representations are nearby. In this talk, I shall discuss recent progresses in this field including my own work on boundaries of deformation spaces.
2010 Mathematics Subject Classification: 57M50, 30F40
Key Words and Phrases: Kleinian group, hyperbolic 3-manifold, deformation space
- ⋅ 11:30 − 12:10 [Mathematical Education] Chair: Joonkook Shin(Chungnam National University)
- ⋅ 5th 11:30 − 12:10 Harmonic mean (Hong-Jong Kim)
- 김홍종(서울대)
Hong-Jong Kim, Seoul National University
In everyday life, we see, hear, and feel harmony. I will explain some of them.
2010 Mathematics Subject Classification: 97B20
Key Words and Phrases: harmony, mean, perspective
- ⋅ 13:40 − 14:20 [Algebra] Chair: Bokhee Im(Chonnam National University)
- ⋅ 5th 13:40 − 14:20 Mock modular forms of small weights (Soon-Yi Kang)
- 강순이(강원대)
Soon-Yi Kang, Kangwon National University
One can construct a harmonic weak Maass form using weak Maass-Poincar\'{e} series, but for weights between 0 and 2, care needs to be taken as the Poincar\'{e} series does not converge. Many interesting examples of harmonic weak Maass forms or mock modular forms are found in these weights though, such as Ramanujan's mock theta functions, traces of singular moduli, Niebur-Poincar\'{e} series, generating functions of partitions or compositions. In this talk, we present these mock modular forms and various properties of them.
2010 Mathematics Subject Classification: 05A17, 11F03, 11F30, 33D15
Key Words and Phrases: mock modular forms, harmonic weak Maass forms, mock theta functions, traces of singular moduli, partitions, compositions
- ⋅ 13:40 − 14:20 [ Chair: Chan Woo Yang(Korea University)
- ⋅ 5th 13:40 − 14:20 Radial Fourier multipliers in high dimensions (Yaryong Heo)
- 허야용(고려대)
Yaryong Heo, Korea University
This is a joint work with F. Nazarov and A. Seeger. A simple characterization of convolution operators bounded on $L^p$ is known only in two cases: $p=1$ and $p=2$. We will deal with convolution operators with radial kernels acting on functions defined in $R^d$. And we will try to give a simple characterizations of radial functions that are multipliers in $L^p$, $p \neq 2$.
2010 Mathematics Subject Classification: 42B15
Key Words and Phrases: convolution operators, multipliers
- ⋅ 13:40 − 14:20 [Geometry] Chair: Seongtag Kim(Inha University)
- ⋅ 5th 13:40 − 14:20 Higher dimensional versions of the Enneper surface, catenoid and helicoid (Jaigyoung Choe, Jens Hoppe)
- 최재경*(고등과학원), Jens Hoppe(서강대)
Jaigyoung Choe*, KIAS, Jens Hoppe, Sogang University
The Enneper surface, catenoid and helicoid are the simplest complete minimal surfaces in $\mathbb R^3$. They can be constructed by applying the Weierstrass representation formula. But there is no such formula for higher dimensional minimal surfaces. However, using some geometric properties, we construct the higher dimensional generalizations of these surfaces in $\mathbb R^n$.
2010 Mathematics Subject Classification: 53A10
Key Words and Phrases: minimal surface
- ⋅ 13:40 − 14:20 [Applied Mathematics] Chair: Kwang-Yeon Kim(Kangwon National University)
- ⋅ 5th 13:40 − 14:20 Mathematical modeling of ionic fluids in ion channel: energetic variational approach (YunKyong Hyon, Bob Eisenberg and Chun Liu)
- 현윤경*(국가수리과학연구소), Bob Eisenberg(Rush Medical Center), Chun Liu(Penn\-sylvania State Univ.)
YunKyong Hyon*, National Institute for Mathematical Sciences, Bob Eisenberg, Rush Medical Center, Chun Liu, Pennsylvania State University
We discuss a mathematical model for the transport of ions through ion channels. The resulting system of partial differential equations is derived in the frame of the energetic variational approaches, taking into account the coupling between electrostatics, diffusion and protein (ion channel) structure. We also incorporate the geometric constraints of the ion channel through a potential energy controlling the local maximum volume inside the ion channel. In the mathematical modeling, a diffusive interface (labeling) description is also employed to describe the geometric configuration of the channels. The energy functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system, we consider channel recordings of voltage clamp to measure the current flowing through the ion channel. The results of one-dimensional numerical simulations are presented to demonstrate some signature effects of the channel.
2010 Mathematics Subject Classification: 76A05, 76M99, 65C30
Key Words and Phrases: mathematical modeling, energetic variational approach, Poisson-Nernst-Planck equation, diffuse interface method, finite size effect, ion dynamics
- ⋅ 14:30 − 15:10 [Analysis] Chair: Seung-Hyeok Kye(Seoul National University)
- ⋅ 5th 14:30 − 15:10 Equivariant degree formula and gauge theory (Namkwon Kim)
- 김남권(조선대)
Namkwon Kim, Chosun University
We consider some gauge theory here and present existence results for PDEs in those gauge theory. The relevant PDE problem here is basically nonlinear and nonmonotone (That is, certain crossing happens). The equivariant degree formula gives nice framework for these problems and will be presented.
2010 Mathematics Subject Classification: 35J, 81T
Key Words and Phrases: elliptic PDE, equivariant degree formula, Chern-Simons gauge theory
- ⋅ 12:00 − 12:40 [Probability and Statistics] Chair: Hyun Jae Yoo(Hankyong University)
- ⋅ 6th 12:00 − 12:40 Potential theory of subordinate Brownian motions with Gaussian components (Zoran Vondra\v{c}ek, Panki Kim and Renming Song)
- Zoran Vondra\v{c}ek*(Univ. of Zagreb), 김판기(서울대), Renming Song(Univ. of Illinois)
Zoran Vondra\v{c}ek*, University of Zagreb, Panki Kim, Seoul National University, Renming Song, University of Illinois
In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a L\'evy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in $C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded $C^{1,1}$ open set $D$ and identify the Martin boundary of $D$ with respect to the subordinate Brownian motion with the Euclidean boundary.
2010 Mathematics Subject Classification: Primary 60J45; Secondary 31B25, 47G20, 60J75, 31B05
Key Words and Phrases: boundary Harnack principle, subordinate Brownian motion, harmonic function, Green function, Martin boundary, L\'evy system, exit distribution
- Algebra
- ⋅ 10:40 − 11:40 Chair: Pyung-Lyun Kan(Chungnam National University)
- ⋅ 5th 10:40 − 11:00 Inversion problems of $k$-involutory matrices over the complex field (Hana Kim, Gi-Sang Cheon)
- 김하나*(성균관대), 천기상(성균관대)
Hana Kim*, Sungkyunkwan University, Gi-Sang Cheon, Sungkyunkwan University
A matrix satisfying $A^k=I$, the identity matrix, is said to be a $k$-involutory matrix. If $k=2$, then $A$ is an usual involution or an involutory matrix. Matrices with this property occur in the science of cryptography. In this talk, we are particularly interested in $k$-involutory matrices in the Riordan group over the complex field. We first discuss the inverse of a $k$-involutory Riordan matrix by means of diagonally striped matrices. For $m=0,1,2,\ldots$, an $m$-diagonally striped matrix is defined as a matrix consisting of diagonals of nonzero (and possibly some zero) entries such that there are $m$ zero diagonals between the adjacent diagonals. This leads to the concept of an $m$-pseudo involution. We clarify a relationship between $m$-pseudo involutions, $2m$-involutory matrices, and diagonally striped matrices. In addition, we prove two conjectures concerning the zero-nonzero pattern of the inverse of a lower triangular, diagonally striped matrix with three nonzero diagonals. Finally, we suggest that the inversion problems of $k$-involutory matrices of the Riordan group may be applied to those of the general linear group GL$(n,\mathbb{C})$.
2010 Mathematics Subject Classification: 15A09
Key Words and Phrases: involutory matrix, diagonally striped matrix, Riordan matrix
- ⋅ 5th 11:00 − 11:20 Redundant blow-ups and Cox rings of rational surfaces (Jinhyung Park, DongSeon Hwang)
- 박진형*(카이스트), 황동선(아주대)
Jinhyung Park*, KAIST, DongSeon Hwang, Ajou University
The classification problem of rational surfaces with finitely generated Cox rings, equivalently Mori dream rational surfaces, have been received a lot of attention recently. We introduce the notion of redundant blow-ups based on Sakai's work in order to understand the classification problem. We first show that `almost every' redundant blow-up preserves finite generation of Cox ring, and `almost every' Mori dream rational surface with non-negative anticanonical Iitaka dimension can be obtained by redundant blow-ups from `certain' minimal classes. This method is also applied to construct series of new examples of Mori dream rational surfaces with given invariants.
2010 Mathematics Subject Classification: 14J26
Key Words and Phrases: Cox ring, rational surface, Zariski decomposition
- ⋅ 5th 11:20 − 11:40 Variety of minimal rational tangents on double covers of projective space (Hosung Kim, Jun-Muk Hwang)
- 김호성*(고등과학원), 황준묵(고등과학원)
Hosung Kim*, KIAS, Jun-Muk Hwang, KIAS
Let $\phi: X \to \mathbb P^n$ be a double cover branched along a smooth hypersurface of degree $2m, 2 \leq m \leq n-1$. We study the varieties of minimal rational tangents $\mathcal C_x \subset \mathbb T_x(X)$ at a general point $x$ of $X$. We describe the homogeneous ideal of $\mathcal C_x$ and show that the projective isomorphism type of $\mathcal C_x$ varies in a maximal way as $x$ varies over general points of $X$. Our description of the ideal of $\mathcal C_x$ implies a certain rigidity property of the covering morphism $\phi$. As an application of this rigidity, we show that any finite morphism between such double covers with $m=n-1$ must be biregular. We also prove that Liouville-type extension property holds with respect to minimal rational curves on $X$.
2010 Mathematics Subject Classification: 14J45
Key Words and Phrases: Fano manifolds, variety of minimal rational tangents
- ⋅ 14:30 − 15:00 Chair: Bokhee Im(Chonnam National University)
- ⋅ 5th 14:30 − 15:00 On schurity of $p$-schemes of order $p^3$ (Kijung Kim)
- 김기정(부산대)
Kijung Kim, Pusan National University
In this talk, we give a criterion to find schurian schemes within a class of $p$-schemes of order $p^3$. Moreover, we construct non-schurian $p$-schemes of order $p^3$.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: association scheme, $p$-scheme, schurian
- ⋅ 09:30 − 11:00 Chair: Gi-Sang Cheon(SungKyunKwan University)
- ⋅ 6th 09:30 − 10:00 Chain conditions in the $D+E[\Gamma^*]$ constructions (Jung Wook Lim, Dong Yeol Oh)
- 임정욱*(서강대), 오동렬(한밭대)
Jung Wook Lim*, Sogang University, Dong Yeol Oh, Hanbat National University
Let $D \subseteq E$ denote an extension of commutative rings with identity, $I$ be a nonzero proper ideal of $D$, $\Gamma$ mean a nonzero torsion-free additive grading monoid with $\Gamma \cap -\Gamma =\{0\}$ and $\Gamma^*=\Gamma \setminus \{0\}$. Let $E[\Gamma]$ be the semigroup ring of $\Gamma$ over $E$, $D + E[\Gamma^*]=\{f \in E[\Gamma] \mid f(0) \in D\}$ and $D + I[\Gamma^*] = \{f \in D[\Gamma] \mid$ the coefficients of nonconstant terms of $f$ belong to $I\}$. In this talk, we give necessary and sufficient conditions for the rings (resp., domains) $D + E[\Gamma^*]$ and $D + I[\Gamma^*]$ to be Noetherian (resp., to satisfy the ascending chain condition on principal ideals).
2010 Mathematics Subject Classification: 13A02, 13A15, 13E05, 13E99
Key Words and Phrases: Noetherian ring, ascending chain condition on principal ideals, $D+E[\Gamma^*]$
- ⋅ 6th 10:00 − 10:30 Gorenstein sequences of small socle degree (Jeaman Ahn)
- 안재만(공주대)
Jeaman Ahn, Kongju National University
The h-vectors of Artinian graded rings are characterized by a theorem of Macaulay, but it is an open question to characterize the h-vectors of Gorenstein Artinian graded algebras. This has been the object of intense study as evidenced by the several papers. The purpose of the talk is to introduce recent results and open problems on Gorenstein sequences of small socle degree. In particular, we give a complete characterization for codimension $11$ and socle degree 4. These results allow us to improve interesting results of Boij-Zanello for socle degree 4 or 5.
2010 Mathematics Subject Classification: Primary 13D40; Secondary 13H10, 14C20
Key Words and Phrases: Hilbert function, Gorenstein algebra, minimal free resolution, graded Betti number
- ⋅ 6th 10:30 − 11:00 Linear operators that preserve arbitrary two term ranks of matrices (Seok-Zun Song)
- 송석준(제주대)
Seok-Zun Song, Jeju National University
The term rank of a matrix $A$ is the least number of lines (rows or columns) needed to include all the nonzero entries in $A$. In this talk, we show a characterization of linear operators that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear operator $T$ on a matrix space over antinegative semirings preserves term rank if and only if $T$ preserves arbitrary two term ranks $k$ and $h$ if and only if $T$ strongly preserves any one term rank $k$.
2010 Mathematics Subject Classification: 15A86
Key Words and Phrases: term rank, linear operator
- ⋅ 11:10 − 11:40 Chair: Sijong Kwak(KAIST)
- ⋅ 6th 11:10 − 11:40 On hypersurfaces containing projective varieties (Eui Sung Park)
- 박의성(고려대)
Eui Sung Park, Korea University
Classical Castelnuovo's Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension $c$ is at most $c(c+1)/2$ and the equality is attained if and only if the variety is of minimal degree. Also a generalization of Castelnuovo's Lemma by G. Fano implies that the next case occurs if and only if the variety is a del Pezzo variety. For curve case, these results are extended to equations of arbitrary degree respectively by J. Harris and S. L'vovsky. In this talk, I will speak about some extension of these results to arbitrary dimensional varieties and to the next cases.
2010 Mathematics Subject Classification: 14N05
Key Words and Phrases: Hilbert function
- Analysis
- ⋅ 09:00 − 10:20 Chair: Sunggeum Hong(Chosun University)
- ⋅ 5th 09:00 − 09:20 Unitary orbit technique in matrix inequalities (Eun-Young Lee)
- 이은영(경북대)
Eun-Young Lee, Kyungpook National University
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen-Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.
2010 Mathematics Subject Classification: Primary 15A60, 47A30, 47A60
Key Words and Phrases: operator inequalities, positive linear map, trace, unitary orbit, convex function
- ⋅ 5th 09:20 − 09:50 Moments of discrete measures with dense jumps induced by $\beta$-expansions (DoYong Kwon)
- 권도용(전남대)
DoYong Kwon, Chonnam National University
Let $\beta>1$. Through an appeal to $\beta$-expansions we define a strictly increasing and left-continuous function $\mu_\beta$ on $[0,1]$. Then $\mu_\beta$ turns out to be a pure jump distribution. In other words, its consequential Lebesgue-Stieltjes measure is discrete, i.e., a summation of point masses. The present talk studies the moment of this discrete measure. We express the moment in terms of Bernoulli numbers and polylogarithms, and investigate its asymptotics as well.
2010 Mathematics Subject Classification: 26A30, 42C05, 11A63
Key Words and Phrases: moment, $\beta$-expansion, Sturmian word
- ⋅ 5th 09:50 − 10:20 The It\^{o} integral with respect to analogue of Wiener process (Kun Sik Ryu)
- 류근식(한남대)
Kun Sik Ryu, Hannam University
In this note, we define the It\^{o} integral with respect to analogue of Wiener process and investigate its various properties and examples.
2010 Mathematics Subject Classification: 28C20, 65G20
Key Words and Phrases: analogue of Wiener measure space, It\^{o} integral, It\^{o} formula, martingale
- ⋅ 10:40 − 12:00 Chair: Byung-Geun Oh(Hanyang University)
- ⋅ 5th 10:40 − 11:00 Some properties of hyperholomorphic functions and hyper-conjugate harmonic functions in Clifford analysis (Su Jin Lim, Kwang Ho Shon)
- 임수진*(부산대), 손광호(부산대)
Su Jin Lim*, Pusan National University, Kwang Ho Shon, Pusan National University
Let ${\mathcal A_n}$ be the Clifford algebra constructed over the field $\mathbb{C}$ of complex numbers with orthonormal basis $\{e_1, e_2, \ldots , e_n\}$. A basis for ${\mathcal A_n}$ satisfy the following : $e_j^2=-e_0 (j=1,2,\ldots,n)$ $e_ie_j+e_je_i=0 (i\neq j, i,j=1,2,\ldots,n)$ where $e_0$ is the identity element. We research properties of hyperholomorphic functions in Clifford analysis and investigate a generalized Cauchy Riemann system.
2010 Mathematics Subject Classification: 30G35, 32A10, 32A30
Key Words and Phrases: hyperholomorphic, Clifford analysis, quaternion, octonion
- ⋅ 5th 11:00 − 11:30 Geometric result for the elliptic problem with nonlinearity crossing three eigenvalues (Q-Heung Choi, Tacksun Jung)
- 최규흥*(인하대), 정택선(군산대)
Q-Heung Choi*, Inha University, Tacksun Jung, Kunsan National University
We investigate the number of the solutions for the elliptic boundary value problem. We obtain a theorem which shows the existence of six weak solutions for the elliptic problem with jumping nonlinearity crossing three eigenvalues. We get this result by using the geometric mapping defined on the finite dimensional subspace. We use a contraction mapping principle to reduce the problem on the infinite dimensional space to that on the finite dimensional subspace. We construct a three dimensional subspace with three axis spanned by three eigenvalues and a mapping from the finite dimensional subspace to the one dimensional subspace.
2010 Mathematics Subject Classification: 34C15, 34C25, 35Q72
Key Words and Phrases: elliptic boundary value problem, jumping nonlinearity, geometric mapping, contraction mapping principle
- ⋅ 5th 11:30 − 12:00 A bifurcation problem for the biharmonic operator (Tacksun Jung, Q-Heung Choi)
- 정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University
We investigate the number of the solutions for the biharmonic boundary value problem with variable coefficient nonlinear term. We get a theorem which shows the existence of $m$ weak solutions for the biharmonic problem with variable coefficient. We obtain this result by using the critical point theory induced from the invariant function and invariant linear subspace.
2010 Mathematics Subject Classification: 35J30, 35J40
Key Words and Phrases: biharmonic boundary value problem, critical point theory, invariant function, invariant subspace, $(P.S.)_{c}$ condition, eigenvalue problem
- ⋅ 09:30 − 11:20 Chair: Deok Hoon Boo(Chungnam National University)
- ⋅ 6th 09:30 − 10:00 Averaging operators over nondegenerate quadratic surfaces in finite fields (Doowon Koh)
- 고두원(충북대)
Doowon Koh, Chungbuk National University
In this talk, we study mapping properties of the averaging operator related to quadratic surfaces in the finite field setting. Sharp averaging estimates were well known in odd dimensions, but not in even dimensions. We investigate the sharp strong-type estimates of averaging operators over nondegenerate quadratic varieties in even dimensions. Critical endpoint estimates are successfully carried out to improve upon the previously known results. Our results are sharp in the sense that a further improvement could not be expected if the quadratic variety contains a huge dimensional subspace and the dimension is even.
2010 Mathematics Subject Classification: 42B05
Key Words and Phrases: averaging operators, finite fields
- ⋅ 6th 10:00 − 10:20 Restriction estimates for the space curves with respect to fractal measures (Seheon Ham, Sanghyuk Lee)
- 함세헌*(서울대), 이상혁(서울대)
Seheon Ham*, Seoul National University, Sanghyuk Lee, Seoul National University
There are well-known estimates for the Fourier transform restricted to the space curves. To obtain some sharp estimates, one may use an multilinear interpolation theorem a little complicated to prove. In this talk, we introduce another argument to avoid the interpolation theorem using a decomposition of the domain of the curves into dyadic intervals. Form this argument, it is easy to get an $L^p$ -- $L^q$ estimate of the extension operator of the Fourier transform except the edge line of the necessary condition area. Moreover, we estimate it with respect to some fractal measures.
2010 Mathematics Subject Classification: 42B10
Key Words and Phrases: Fourier restriction theorem
- ⋅ 6th 10:20 − 10:50 Weighted Fourier algebras on compact quantum groups and complexification (Hun Hee Lee)
- 이훈희(충북대)
Hun Hee Lee, Chungbuk National University
In this talk we will focus on the Fourier algebra on compact quantum groups and their weighted versions. As in the case of compact groups we will try to investigate the connection with weighted Fourier algebras and complexification of (quantum) groups. Examples we will be examining include Woronowicz's SUq(2), free orthogonal quantum groups, free unitary quantum groups.
2010 Mathematics Subject Classification: 43A30, 47L87
Key Words and Phrases: Fourier algebra, compact quantum group, SUq(2)
- ⋅ 6th 10:50 − 11:20 Boundary integral operator for the fractional Laplace equation in a bounded Lipschitz domain (Tongkeun Chang)
- 장통근(연세대)
Tongkeun Chang, Yonsei University
We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.
2010 Mathematics Subject Classification: 45EXX
Key Words and Phrases: fractional Laplace, Lipschitz bounded domain, boundary value problem
- Geometry
- ⋅ 09:00 − 10:30 Chair: Jongsu Kim(Sogang University)
- ⋅ 5th 09:00 − 09:30 Category of Lie algebroids (Hoil Kim)
- 김호일(경북대)
Hoil Kim, Kyungpook National University
We describe the category of Lie algebroids with the relation of Lie groupoids. This category contains the category of generalized complex geometry which combines complex geometry and symplectic geometry. We also explain K group, Chern map, Chow map and categorical cycles. In this context we propose the generalized mirror (duality) behaviour.
2010 Mathematics Subject Classification: 14Fxx
Key Words and Phrases: category, Lie algebroids, Lie groupoids, generalized geometry, complex geometry, symplectic geometry, K group, Chern map, Chow map, cycles, mirror (duality)
- ⋅ 5th 09:30 − 10:00 Triangulations of the disk with positive curvature (Maryam Verdian-Rizi)
- Maryam Verdian-Rizi, KAIST
This talks regards the structure of triangulations of the disk with vertices of positive curvature. We show that such triangulations can be obtained from the triangular grid by cutting or cutting and gluing, depending on the sum of the curvature on the boundary of the disk.
2010 Mathematics Subject Classification: 52CXX
Key Words and Phrases: triangulation, grid construction
- ⋅ 5th 10:00 − 10:30 Higher genus Riemann type minimal surfaces (Filippo Morabito)
- Filippo Morabito, Korea University
We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in Euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic examples of infinite genus. Joint work with Martin Traizet.
2010 Mathematics Subject Classification: 53, 58
Key Words and Phrases: Riemann type minimal surface, Euclidean three space
- ⋅ 10:40 − 12:00 Chair: Jaigyoung Choe(KIAS)
- ⋅ 5th 10:40 − 11:00 Helices in a 4-dimensional Euclidean space (Jin Ho Choi, Young Ho Kim)
- 최진호*(울산대), 김영호(경북대)
Jin Ho Choi*, University of Ulsan, Young Ho Kim, Kyungpook National University
In this talk, we introduce some integral curves associated by a Frenet curve of order $d$ in a $n$-dimensional Euclidean space. Using these curves, we give some new characterizations of helices and general helices in a 4-dimensional Euclidean space.
2010 Mathematics Subject Classification: 53A04
Key Words and Phrases: helices, general helices, classical differential geometry
- ⋅ 5th 11:00 − 11:30 Ellipsoids and elliptic hyperboloids in the Euclidean space (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\mathbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume of regions between parallel hyperplanes. We also give a few characterizations of elliptic paraboloids in the $(n+1)$-dimensional Euclidean space ${\mathbb E}^{n+1}$.
2010 Mathematics Subject Classification: 53A07
Key Words and Phrases: ellipsoid, elliptic hyperboloid, $(n+1)$-dimensional volume, $n$-dimensional surface area
- ⋅ 5th 11:30 − 12:00 Gronwall conjecture (Seungho Wang)
- 왕승호(고등과학원)
Seungho Wang, KIAS
In an article published in 1912, Gronwall remarked that a planar 3-web which admits more than one distinct linearizations is locally equivalent to an algebraic web. This remains a conjecture after a number of relevant results since then. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point.
2010 Mathematics Subject Classification: 53A60
Key Words and Phrases: planar 3-web, path geometry, linearization
- ⋅ 14:30 − 15:00 Chair: Yoon-Tae Jung(Chosun University)
- ⋅ 5th 14:30 − 15:00 Helical immersions and chord properties (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
First, we study curves in a Euclidean space of arbitrary dimension such that the chord joining any two points on the curve meets it at the same angle. \smallskip Next, we study hypersurfaces in a Euclidean space of arbitrary dimension such that the chord joining any two points on the hypersurface meets it at the same angle. \smallskip Finally, we study submanifolds in a Euclidean space of arbitrary dimension such that the chord joining any two points on the submanifold meets it at the same angle.
2010 Mathematics Subject Classification: 53A10, 53B25
Key Words and Phrases: hypersurface, submanifold, $W$-curve, chord property, helical immersion, geodesic normal section
- ⋅ 09:20 − 11:00 Chair: Sun-Suk Jin(Gongju National University of Education)
- ⋅ 6th 09:20 − 09:40 Applications of the curvature identity on 4-dimensional Riemannian manifolds (Yunhee Euh, JeongHyeong Park and Kouei Sekigawa)
- 어윤희*(성균관대), 박정형(성균관대), Kouei Sekigawa(Niigata Univ.)
Yunhee Euh*, Sungkyunkwan University, JeongHyeong Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University
We recall a curvature identity for 4-dimensional compact Riemannian manifolds as derived from the generalized Gauss-Bonnet formula. We extended this curvature identity to non-compact 4-dimensional Riemannian manifolds. We shall discuss critical metrics for the squared $L^2$-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature by making use of a curvature identity on 4-dimensional Riemannian manifolds.
2010 Mathematics Subject Classification: 53B20, 53C20
Key Words and Phrases: critical metric, 4-dimensional curvature identity
- ⋅ 6th 09:40 − 10:10 Connections on the bundle of linear frames (Joon-Sik Park)
- 박준식(부산외대)
Joon-Sik Park, Busan University of Foreign Studies
Let $L(M)$ be the bundle of all linear frames over $M$, $u$ an arbitrarily given point of $L(M)$, and $\nabla: {\mathfrak X}(M) \times {\mathfrak X}(M)\rightarrow {\mathfrak X}(M)$ a linear connection on $M$. Then the following facts are well known: in terms of local coordinates of $u \in L(M)$ and Christoffel's symbols defined by $\nabla$, \par (i) the horizontal subspace and the connection form at the point $u$ may be written, and \par (ii) conversely the covariant differentiation on $M$ which is defined by the parallelity on $L(M)$, determined from (i), is identified with $\nabla$. \par These are very fundamental on the study of the theory of connections. In this note we show that the local expressions of the horizontal subspace and the connection form at the point $u$ do not depend on the choice of a local coordinate system around the point $u \in L(M)$, which is rarely seen.
2010 Mathematics Subject Classification: 53C07, 53A15
Key Words and Phrases: bundle of linear frames, connection, covariant derivative
- ⋅ 6th 10:10 − 10:30 On critical almost Hermitian structures (Jungchan Lee, Jeong Hyeong Park and Kouei Sekigawa)
- 이정찬*(성균관대), 박정형(성균관대), Kouei Sekigawa(Niigata Univ.)
Jungchan Lee*, Sungkyunkwan University, Jeong Hyeong Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University
Let $M$ be an even dimensional compact smooth manifold admitting an almost complex structure. Let ${(\lambda, \mu)} \in \mathbb{R}^2 -(0,0)$. We discuss the critical points of the functional $\mathcal{F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* )dv_g$ on the space of all almost Hermitian structures $\mathcal{AH}(M)$ on $M$ and its subspace ${\mathcal{AH}_{c}(M) }$ with a certain positive constant $c$, where $\tau$ and $\tau^*$ are the scalar curvature and the $*$-scalar curvature of $(J, g)$, respectively. We also investigate the curvature properties of a two-parameter family of Hermitian structures on a product manifold $\overline{M}$ of Sasakian manifolds and give a necessary and sufficient condition for a Hermitian structure belonging to the family to be Einstein and provide concrete examples. In addition, we discuss the first variational problem of the functional $\mathcal {F}_{\lambda, \mu}$ on the spaces $\mathcal{AH}(\overline{M})$ and $\mathcal{AH}_c(\overline{M})$. Further, we provide some examples illustrating our arguments.
2010 Mathematics Subject Classification: 53C15, 53C25
Key Words and Phrases: critical almost Hermitian structure, Einstein-Hilbert functional, Sasakian manifold
- ⋅ 6th 10:30 − 11:00 Calibrations and symplectic knot spaces (Jae-Hyouk Lee)
- 이재혁(이화여대)
Jae-Hyouk Lee, Ewha Womans University
Calibrations are closed forms satisfying special volume comparison which induces minimality on the corresponding calibrated submanifolds. We consider a calibration on a symplectic manifold $M$ given by wedge of sympelctic forms on it and describe geometry of the manifold $M$ along the calibration. Then we construct a symplectic knot space on $M$ and explain the correspondence between the geometry of $M$ and the symplectic geometry of the symplectic Knot spaces.
2010 Mathematics Subject Classification: 53C38, 53D30
Key Words and Phrases: calibration, symplectic knot space
- ⋅ 11:10 − 12:50 Chair: Joon-Sik Park(Pusan University of Foreign Studies)
- ⋅ 6th 11:10 − 11:30 Hopf hypersurfaces in complex two-plane Grassmannians with $\mathfrak D$-parallel shape operator (Eunmi Pak, Hyunjin Lee and Young Jin Suh)
- 박은미*(경북대), 이현진(경북대), 서영진(경북대)
Eunmi Pak*, Kyungpook National University, Hyunjin Lee, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this paper we consider a generalized parallelism of shape operator for a real hypersurface $M$ in complex two-plane Grassmannian, namely, $\mathfrak D$-parallel shape operator of $M$. Using this notion, we prove that there does not exist a real hypersurface in complex two-plane Grassmannian with $\mathfrak D$-parallel shape operator.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: complex two-plane Grassmannians, Hopf hypersurface, $\mathfrak D$-parallel shape operator
- ⋅ 6th 11:30 − 12:00 Real hypersurfaces in complex two-plane Grassmannians with $\mathfrak D^{\bot}$-parallel shape operator (Imsoon Jeong, Hyunjin Lee and Young Jin Suh)
- 정임순*(경북대), 이현진(경북대), 서영진(경북대)
Imsoon Jeong*, Kyungpook National University, Hyunjin Lee, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this talk, we give a non-existence theorem for real hypersurfaces in complex two-plane Grassmannians $G_{2}(\mathbb C^{m+2})$ with $\mathfrak D^{\bot}$-parallel shape operator, where the distribution $\mathfrak D^{\bot}$ is spanned by structure vector fields $\xi_{\nu}$, $\nu=1,2,3$.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, parallel shape operator
- ⋅ 6th 12:00 − 12:30 Long-time convergence of harmonic map heat flows of a closed Riemann surface to a Riemannian manifold (Kwangho Choi)
- 최광호(서울대)
Kwangho Choi, Seoul National University
We study the long-time convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinite-time limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no bubbling occurs at infinite time.
2010 Mathematics Subject Classification: 53C44, 58E20
Key Words and Phrases: harmonic map heat flow
- ⋅ 6th 12:30 − 12:50 Levi-Civita and generalized Tanaka-Webster covariant derivatives for real hypersurfaces in complex two-plane Grassmannians (Hyunjin Lee, Imsoon Jeong and Young Jin Suh)
- 이현진*(경북대), 정임순(경북대), 서영진(경북대)
Hyunjin Lee*, Kyungpook National University, Imsoon Jeong, Kyungpook National University, Young Jin Suh, Kyungpook National University
It is known that submanifolds in Kaehler manifolds have many kinds of connections. Among them, we consider two connections, that is, Levi-Civita and Tanaka-Webster connections for real hypersurfaces in complex two-plane Grassmannians. When they are equal to each other, we give some characterizations in complex two-plane Grassmannians.
2010 Mathematics Subject Classification: 54C40
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurfaces, $\mathfrak D^{\bot}$-invariant hypersurface, Levi-Civita connection, generalized Tanaka-Webster connection
- Topology I
- ⋅ 09:00 − 10:30 Chair: Suyoung Choi(Ajou University)
- ⋅ 5th 09:00 − 09:30 An obstruction to embedding right-angled Artin groups in mapping class groups (Sang-hyun Kim, Thomas Koberda)
- 김상현*(카이스트), Thomas Koberda(Yale Univ.)
Sang-hyun Kim*, KAIST, Thomas Koberda, Yale University
For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\Gamma$ to embed into the mapping class group of a surface $S$, we show that the chromatic number of $\Gamma$ cannot exceed the chromatic number of the clique graph of the curve graph $\mathcal{C}(S)$. Thus, the chromatic number of $\Gamma$ is a global obstruction to embedding the right-angled Artin group $A(\Gamma)$ into the mapping class group $\mathrm{Mod}(S)$.
2010 Mathematics Subject Classification: 20F36
Key Words and Phrases: right-angled Artin groups
- ⋅ 5th 09:30 − 10:00 Automorphisms of surface braid groups (Byunghee An)
- 안병희(고등과학원)
Byunghee An, KIAS
Let $\Sigma$ be an orientable surface without boundary. Then the braid group $\mathbf{B}_n(\Sigma)$ is defined by the fundamental group of configuration space on $\Sigma$. The automorphism groups Aut$(\mathbf{B}_n(\Sigma))$ are known only for a disc or closed surface $\Sigma$. In this talk, we compute Aut$(\mathbf{B}_n(\Sigma))$ for all orientable surface $\Sigma$.
2010 Mathematics Subject Classification: 20F36, 20E36
Key Words and Phrases: braid group, automorphism
- ⋅ 5th 10:00 − 10:30 Volume invariant and maximal representations of discrete subgroups of Lie groups (Sungwoon Kim, Inkang Kim)
- 김성운*(고등과학원), 김인강(고등과학원)
Sungwoon Kim*, KIAS, Inkang Kim, KIAS
We introduce a volume invariant of representations of lattices in semisimple Lie groups which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that this volume invariant exactly characterizes discrete, faithful representations.
2010 Mathematics Subject Classification: 22E46, 57R20, 53C35
Key Words and Phrases: volume invariant, representation variety, maximal representation
- ⋅ 13:40 − 15:00 Chair: Cheol-Hyun Cho(Seoul National University)
- ⋅ 5th 13:40 − 14:10 Hyperbolic structure of $C^1$-generic diffeomorphisms (Keonhee Lee)
- 이건희(충남대)
Keonhee Lee, Chungnam National University
A basic problem in differentiable dynamical systems is to understand how a robust dynamic property on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. In this talk we discuss some recent works on the robust dynamics of $C^1$ generic diffeomorphisms; we show that for $C^1$ generic diffeomorphisms on a closed $C^{\infty}$-manifold $M$, every continuum-wise expansive homoclinic class is hyperbolic (joint work with T. Das and M. Lee) and every Lipschitz shadowable chain component is hyperbolic (joint work with A. Fakhari and K. Tajbakhsh). Finally we introduce two open problems in this direction.
2010 Mathematics Subject Classification: 34C20, 34C50
Key Words and Phrases: homoclinic class, continuum-wise expansive, hyperbolic, generic, chain component
- ⋅ 5th 14:10 − 14:30 Examples of matrix factorizations from SYZ (Hansol Hong, Cheol-Hyun Cho and Sangwook Lee)
- 홍한솔*(서울대), 조철현(서울대), 이상욱(서울대)
Hansol Hong*, Seoul National University, Cheol-Hyun Cho, Seoul National University, Sangwook Lee, Seoul National University
We find matrix factorization corresponding to an anti-diagonal in ${\mathbb C}P^1 \times {\mathbb C}P^1$, and circle fibers in weighted projective lines using the idea of Chan and Leung of Strominger-Yau-Zaslow transformations. For the tear drop orbifolds, we apply this idea to find matrix factorizations for two types of potential, the usual Hori-Vafa potential or the bulk deformed (orbi)-potential. We also show that the direct sum of anti-diagonal with its shift, is equivalent to the direct sum of central torus fibers with holonomy $(1,-1)$ and $(-1,1)$ in the Fukaya category of ${\mathbb C}P^1 \times {\mathbb C}P^1$, which was predicted by Kapustin and Li from B-model calculations.
2010 Mathematics Subject Classification: 53D37, 53D40, 57R18
Key Words and Phrases: matrix factorization, SYZ transformation, mirror symmetry, Lagrangian Floer theory
- ⋅ 5th 14:30 − 15:00 Topological fields and topological organism (Jae-Ryong Kim)
- 김재룡(국민대)
Jae-Ryong Kim, Kookmin University
Topology may described a pattern of existence of elements of a given set $X$. The family $\tau(X)$ of all topologies given on a set $X$ form a complete lattice. We will give some topologies on this lattice $\tau(X)$ using a topology on $X$ and regard $\tau(X)$ a topological space. A topology $\tau $ on $X$ can be regarded a map from $X$ to $\tau(X)$ naturally. Such a map will be called it topology field. If $X$ is a topological flow group with acting group $T$, then naturally we can get a another topological flow $\tau(X)$ with same acting group $T$. If the topological flow $X$ is minimal, we can prove $\tau(X)$ is also minimal. The disjoint unions of the topological spaces can describe some topological systems (topological organisms). Here we will give a definition of topological organism. Our purpose of this study is to describe some properties concerning patterns of relationship between topology fields and topological organisms.
2010 Mathematics Subject Classification: 54A10, 57S99, 93A30
Key Words and Phrases: topological field, topological organism
- ⋅ 09:30 − 11:00 Chair: Sang-hyun Kim(KAIST)
- ⋅ 6th 09:30 − 10:00 Torus cobordism of lens spaces (Soumen Sarkar, Dong Youp Suh)
- Soumen Sarkar*(카이스트), 서동엽(카이스트)
Soumen Sarkar*, KAIST, Dong Youp Suh, KAIST
We give a new definition of lens spaces. We discuss the equivariant oriented cobordism of lens spaces with respect to the natural compact torus actions. The main tool is the quasitoric theory.
2010 Mathematics Subject Classification: 55N22, 57R90
Key Words and Phrases: group actions, lens spaces, cobordism
- ⋅ 6th 10:00 − 10:30 Chern-Simons functional under gauge transformations (Joohee Kim, Yanghyun Byun)
- 김주희*(한양대), 변양현(한양대)
Joohee Kim*, Hanyang University, Yanghyun Byun, Hanyang University
We consider a general principal $G$-bundle $P$ over an oriented connected closed 3-manifold, where $G$ is a compact connected Lie group, and a Chern-Simons functional $CS:\mathcal{A}(P)\rightarrow \mathbb{R}$ defined by choosing a reference connection, where $\mathcal{A}(P)$ denotes the space of connections on $P$. After briefly reviewing some general behavior of this functional under guage transformations we concentrate on the case when $P$ is a flat bundle. It will be shown that in this case the degree $\deg \varphi$ of a gauge transformation $\varphi:P\rightarrow P$ can be defined as an integer so that $CS(\varphi^*A)-CS(A)=\deg \varphi$ holds for any $A\in\mathcal{A}(P)$.
2010 Mathematics Subject Classification: 55R10
Key Words and Phrases: Chern-Simons functional, gauge transformation, flat bundle
- ⋅ 6th 10:30 − 11:00 Subquandles of a finite Takasaki quandle (Seongjeong Kim, Yongju Bae)
- 김성정*(경북대), 배용주(경북대)
Seongjeong Kim*, Kyungpook National University, Yongju Bae, Kyungpook National University
Takasaki quandle is the qundle defined by the binary operation $a*b=2b-a$ on an abelian group $G$. Takasaki quandle depends on the algebraic properties of the given abelian group. In this talk, we will study the relationship between subquandles of a finite Takasaki quandle and subgroups of the underlying abelian group.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: quandle, subquandle, Takasaki quandle
- ⋅ 11:10 − 12:40 Chair: Jong Bum Lee(Sogang University)
- ⋅ 6th 11:10 − 11:40 The algebraic structure of Alexander quandles via group automorphisms (Seonmi Choi, Yongju Bae)
- 최선미*(경북대), 배용주(경북대)
Seonmi Choi*, Kyungpook National University, Yongju Bae, Kyungpook National University
An Alexander quandle $M_t$ is an abelian group with a quandle operation $a\ast b = ta+(1-t)b$ where $t$ is a group automorphism of the underlying abelian group. In this talk, we will study the algebraic structure of an Alexander quandle $M_t$ by using the property of the group automorphism $t$. In particular, we will study the relationship between Alexander quandles $M_t$ and $M_{1-t}$, in the case that $1-t$ is a group automorphism.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: quandle, Alexander quandle, group automorphism
- ⋅ 6th 11:40 − 12:00 Stick number of Montesinos knots and links (Sung Jong No, Seung Sang Oh, Kyung Pyo Hong and Hwa Jeong Lee)
- 노성종*(고려대), 오승상(고려대), 홍경표(고려대), 이화정(중앙대)
Sung Jong No*, Korea University, Seung Sang Oh, Korea University, Kyung Pyo Hong, Korea University, Hwa Jeong Lee, Chung-Ang University
The stick number $s(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$. In this talk, we find an upper bound on the stick number of a Montesinos knot or link $K$, except some special cases, in terms of the minimal crossing number $c(K)$ which is $s(K) \leq c(K) +2$. And we also introduce an upper bound of lattice stick number of knots.
2010 Mathematics Subject Classification: 57M27
Key Words and Phrases: montesinos, stick knot, lattice stick
- ⋅ 6th 12:00 − 12:20 A multi-variable polynomial invariant for virtual links (Kyoung Il Park)
- 박경일(한국과학영재학교)
Kyoung Il Park, Korea Science Academy of KAIST
Y. H. Im, K. Lee and S. Y. Lee are defined some index polynomial invariants for virtual links. We construct a multi-variable polynomial invariant which is a generalization of the index polynomial invariant for virtual links. We give several properties of this polynomial and examples. Moreover, we show that this polynomial can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.
2010 Mathematics Subject Classification: 57Mxx
Key Words and Phrases: virtual link, index polynomial, invertibility
- ⋅ 6th 12:20 − 12:40 On embeddings of orbifolds (Hyung-seok Shin, Cheol-Hyun Cho and Hansol Hong)
- 신형석*(서울대), 조철현(서울대), 홍한솔(서울대)
Hyung-seok Shin*, Seoul National University, Cheol-Hyun Cho, Seoul National University, Hansol Hong, Seoul National University
In this talk, we study the relation between orbifold embeddings and equivariant immersion maps of manifolds on which Lie groups act.
2010 Mathematics Subject Classification: 57R18
Key Words and Phrases: orbifold
- Topology II
- ⋅ 09:30 − 11:00 Chair: Ki-Heon Yun(SungShin Women's University)
- ⋅ 6th 09:30 − 10:00 A toric-like foliation on complex manifolds (Jerome Tambour)
- Jerome Tambour, KAIST
Following the works of Lopez de Medrano, Verjovsky and Meersseman, Bosio described in 2001 a construction of non kahler compact complex manifolds. Those manifolds, known as LVMB manifolds, are parametrized by simplicial complete fans and, when the fan is rational, one can also define a foliation on a LVMB manifold N whose leaves are closed and diffeomorphic to compact tori. Moreover, the leaves space can naturally be identified with a compact toric variety. The Betti numbers of the cohomology of this torus variety are closely related to the combinatorics of the fan describing N. In the non rational case, one can still define a foliation on a LVMB manifold N but the leaves are not closed anymore. The leaves space being non Hausdorff, it is more convenient to study instead the basic cohomology of the foliation. In the case where the fan is shellable, computations have been made by Battaglia and Zaffran. In this talk, using Ishida's complexes (developed to compute the cohomology of toric varieties), we will present the computation in the general case. As a consequence of this, we obtain the Dehn-Sommerville equations for starshaped spheres (i.e., spheres which are the underlying complex of a complete simplicial fan).
2010 Mathematics Subject Classification: 57R19
Key Words and Phrases: complex manifolds, foliation, Betti numbers
- ⋅ 6th 10:00 − 10:30 Surfaces in four-manifolds with non-simply connected complements and exotic embeddings (Hee Jung Kim, Daniel Ruberman)
- 김희정*(포항공대), Daniel Ruberman(Brandeis Univ.)
Hee Jung Kim*, POSTECH, Daniel Ruberman, Brandeis University
In this talk, we will study surfaces embedded in simply-connected 4-manifolds whose complements are not simply-connected. In particular, we will construct, for any finitely presented group $G$ satisfying some a simple condition, a simply-connected symplectic 4-manifold containing a symplectic surface whose complement has fundamental group $G$. Also, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth s-cobordism and hence are topologically equivalent for good groups.
2010 Mathematics Subject Classification: 57R57, 57N13
Key Words and Phrases: knot, embedded surface, twist-rim surgery, symplectic manifold
- ⋅ 6th 10:30 − 11:00 Crystallographic groups of Sol (Jong Bum Lee)
- 이종범(서강대)
Jong Bum Lee, Sogang University
We classify all the closed $3$-dimensional orbifolds with $\mathrm{Sol}$-geometry. These are aspherical orbifolds and so their fundamental groups, called SC-groups, determine the orbifolds completely. Thus we will classify all the SC-groups, together with all the SB-groups, up to isomorphism.
2010 Mathematics Subject Classification: 57S25, 22E25
Key Words and Phrases: infra-solvmanifolds, SC-groups, $\mathrm{Sol}$-geometry
- ⋅ 11:10 − 12:30 Chair: Keonhee Lee(Chungnam National University)
- ⋅ 6th 11:10 − 11:30 Intrinsically knotted graph with 22 edges (Hyoung Jun Kim, Hwa Jeong Lee, Min Jung Lee and Seung Sang Oh)
- 김형준*(고려대), 이화정(중앙대), 이민정(고려대), 오승상(고려대)
Hyoung Jun Kim*, Korea University, Hwa Jeong Lee, Chung-ang University, Min Jung Lee, Korea University, Seung Sang Oh, Korea University
We proved that only 14 graphs are intrinsically knotted graph with 21 edges. Furthermore, we will show the process of the finding intrinsically knotted graphs with 22 edges.
2010 Mathematics Subject Classification: 68R10
Key Words and Phrases: intrinsic, knotted, graph
- ⋅ 6th 11:30 − 12:00 Schubert calculus for weighted Grassmannians and weighted Schur functions (Tomoo Matsumura, Hiraku Abe)
- Tomoo Matsumura*, KAIST, Hiraku Abe, Tokyo Metropolitan University
The weighted Grassmannian is introduced by Corti-Reid and it is an orbifold generalization of the classical Grassmannian. In the joint work with Hiraku Abe, we have introduced the weighted Schubert classes in the rational (equivariant) cohomology of the weighted Grassmannian and shown that they form a basis. We have computed the (equivariant) structure constants in an explicit formula which generalizes Knutson-Tao's puzzle formula. Moreover, we have shown a certain equivariant positivity for the constants, analogous to Graham's equivariant positivity. As the usual (equivariant) Schubert classes in the classical Grassmannian are represented by (factorial) Schur functions, we found a weighted version of them and have shown the correspondence to the (equivariant) weighted Schubert classes.
2010 Mathematics Subject Classification: Primary 14N15; Secondary 55N91, 57R18
Key Words and Phrases: Schubert calculus, Grassmannian, orbifolds, Schur functions
- ⋅ 6th 12:00 − 12:30 On primitive stable representations of free Kleinian groups (Woojin Jeon, Inkang Kim and Ken'ichi Ohshika)
- 전우진*(고등과학원), 김인강(고등과학원), Ken'ichi Ohshika(Osaka Univ.)
Woojin Jeon*, KIAS, Inkang Kim, KIAS, Ken'ichi Ohshika, Osaka University
A. Lubotzky and Y. Minsky suggested a dynamical decomposition of the PSL$(2,C)$-character variety of a free group $F$ with respect to the action of Out$(F)$. One part of this decomposition consists of primitive stable characters. We discuss a geometric criterion for a representation to be primitive stable. This is joint work with I. Kim, K. Ohshika and C. Lecuire.
2010 Mathematics Subject Classification: 51M10, 57S25
Key Words and Phrases: primitive stability, Cannon-Thurston map
- Probability and Statistics
- ⋅ 09:30 − 10:00 Chair: Panki Kim(Seoul National University)
- ⋅ 6th 09:30 − 10:00 Limit distributions of open quantum random walks (Hyun Jae Yoo, Norio Konno)
- 유현재*(한경대), Norio Konno(Yokohama National Univ.)
Hyun Jae Yoo*, Hankyong National University, Norio Konno, Yokohama National University
We consider the limit distributions of open quantum random walks on the one-dimen\-sional lattice space. By introducing the concept of dual process, it is possible to represent the walks in the Fourier transform space. In contrast to the usual quantum walks, it turns out that the limit distributions of open quantum random walks are Gaussian. We will consider several concrete examples.
2010 Mathematics Subject Classification: 60F05, 81P16, 81P45
Key Words and Phrases: open quantum random walks, dual process, central limit theorem
- ⋅ 10:00 − 10:50 Chair: Soo Hak Sung(Pai Chai University)
- ⋅ 6th 10:00 − 10:30 How small are the increments of partial sums of a discrete random field? (Yong-Kab Choi)
- 최용갑(경상대)
Yong-Kab Choi, Gyeongsang National University
In this paper we establish inferior limit results and path properties for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) discrete random field with multidimensional indices.
2010 Mathematics Subject Classification: 60F15, 60G17, 60G60
Key Words and Phrases: linearly quadrant dependence, random field, stationary random variable
- ⋅ 6th 10:30 − 10:50 A note on the almost sure central limit theorem for self-normalized partial sums of mixing sequences (Kyo-Shin Hwang)
- 황교신(경상대)
Kyo-Shin Hwang, Gyeongsang National University
In this paper we study the almost sure central limit theorem, contained the general weight sequences, for weakly dependent random variables. We here give an almost sure central limit theorem for self-normalized partial sums of $\phi$-mixing random variables which is in domain of attraction of the normal law with mean zero, possibly infinite variance and mixing coefficient rates $\phi(n)$ satisfying $\sum_{n\ge 1}\phi^{1/2}(n)<\infty$ under a fairly general growth condition on $d_k=k^{-1}\exp(\ln^\alpha k), 0\le \alpha<1/2$.\\
2010 Mathematics Subject Classification: 60FXX
Key Words and Phrases: almost sure central limit theorem, $\phi$-mixing, domain of attraction of the normal law, self-normalized partial sum, strictly stationary, weight sequence
- ⋅ 11:00 − 11:50 Chair: Jiyeon Lee(Yeungnam University)
- ⋅ 6th 11:00 − 11:30 Green function of subordinate Brownian motion: stable and beyond (Panki Kim)
- 김판기(서울대)
Panki Kim, Seoul National University
A subordinate Brownian motion is a L\'evy process which can obtained by replacing the time of Brownian motion by an independent increasing L\'evy process. In this talk, we consider a large class of subordinate Brownian motions without diffusion term. We discuss an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in bounded $C^{1,1}$ open set.
2010 Mathematics Subject Classification: 60J45
Key Words and Phrases: geometric stable process, Green function, harmonic function, subordinate Brownian motion, Harnack inequality, boundary Harnack inequality
- ⋅ 6th 11:30 − 11:50 Precise asymptotic approximations for the kernels corresponding to the Levy processes (Minsuk Yang, Sihun Jo)
- 양민석*(연세대), 조시훈(연세대)
Minsuk Yang*, Yonsei University, Sihun Jo, Yonsei University
Using basic complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the Levy processes whose characteristic functions are radial and satisfy some regularity conditions.
2010 Mathematics Subject Classification: 60G51
Key Words and Phrases: Levy process, Mellin transform
- Applied Mathematics
- ⋅ 14:30 − 15:00 Chair: Jae Heon Yun(Chungbuk National University)
- ⋅ 5th 14:30 − 15:00 Error estimates for the fully discrete Gauge-Uzawa method to solve Boussinesq equations (Jae-Hong Pyo)
- 표재홍(강원대)
Jae-Hong Pyo, Kangwon National University
The stabilized Gauge-Uzawa method [SGUM], which is a 2nd order projection type algorithm to solve the time depend Navier-Stokes equations, has been newly constructed and studied in some papers. The goal of this paper is to extend the SGUM to the evolution Boussinesq equations and to prove stability and to estimate errors on the fully discrete finite element space. We present numerical simulations at the end of paper.
2010 Mathematics Subject Classification: 65M12
Key Words and Phrases: Boussinesq equations, projection method, Gauge-Uzawa method
- ⋅ 09:30 − 10:40 Chair: Do Sang Kim(Pukyong National University)
- ⋅ 6th 09:30 − 09:50 Fitting bivariate cumulative returns with temporal Student-t copula (Jung Kyung Lee, Hi Jun Choe, Cheong Hee Ahn)
- 이중경*(연세대), 최희준(연세대), 안청희(아주대)
Jung Kyung Lee*, Yonsei University, Hi Jun Choe, Yonsei University, Cheong Hee Ahn, Ajou University
Financial forecasting changes in the stock market index has been an important subject that has attracted researchers' attention from a variety of fields for many years. We especially have many questions of a possible fitting of probability distributions to financial data. We perform statistical method of fitting joint cumulative returns between the KOSPI(Korea Composite Stock Price Index), and the Dow Jones Industrial Average(U.S.) from the data contained daily, weekly and monthly stock returns data for the period of January 1, 2000 to August 31, 2010. Modifying the method of IFM(inference functions for margins), we practice two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be Gaussian,generalized Student-T for each time, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-fit test. The purpose of this paper is to prove whether our fitting results for cumulative returns which contained daily, weekly and monthly for ten-year period are highly reliable or not through empirical analysis. Finally, we shows that Korea's KOSPI stock index follows U.S.'s Dow growth trend. The correlation coefficient between the two countries(KOSPI vs Dow) is $0.73$, which is very large.
2010 Mathematics Subject Classification: 62P05
Key Words and Phrases: copula
- ⋅ 6th 09:50 − 10:20 Approximate solutions of multiobjective optimization problems (Thai Doan Chuong, Do Sang Kim)
- Thai Doan Chuong*(Saigon Univ.), 김도상(부경대)
Thai Doan Chuong*, Saigon University, Do Sang Kim, Pukyong National University
In this talk, we establish Fritz-John type necessary conditions and sufficient conditions for approximate Pareto solutions of such a problem. As a by-product, Fritz-John type necessary conditions for (weakly) Pareto solutions of the considered problem are obtained by passing to the limit from the corresponding ones of the approximate Pareto solutions. In addition, we state a dual problem formulated in an approximate form to the reference problem and explore duality relations between them.
2010 Mathematics Subject Classification: 90C29, 90C46
Key Words and Phrases: approximate solutions, optimality condition, duality, fuzzy form, limiting/Mordukhovich subdifferential
- ⋅ 6th 10:20 − 10:40 Primal-dual interior-point algorithms for semidefinite optimization based on a new kernel function (Jin-Hee Jin, Gyeong-Mi Cho, You-Young Cho and Yong-Hoon Lee)
- 진진희*(부산대), 조경미(동서대), 조유영(부산대), 이용훈(부산대)
Jin-Hee Jin*, Pusan National University, Gyeong-Mi Cho, Dongseo University, You-Young Cho, Pusan National University, Yong-Hoon Lee, Pusan National University
In this talk, we consider the standard form of semidefinite optimization problem as follows; \begin{equation}\label{(1)} \min \{C \bullet X : A_{i} \bullet X = b_{i}, ~i=1,2, \ldots, m,~ X \succeq 0\}, \end{equation} where each $A_{i}\in \mathbf{S}^{n}$, $b \in \mathbf{R}^m$, $C \in \mathbf{S}^n$, and its dual problem \begin{equation}\label{(2)} \max \{b^Ty : \sum_{i=1} ^{m} y_{i} A_{i}+S=C,~S \succeq 0\}, \end{equation} where $y \in \mathbf{R}^m$. We propose the primal-dual interior-point algorithms based on a new kernel function. We analyze the iteration complexity of the algorithm and get the best known complexity, that is, $\mathcal{O}(\sqrt{n}\, \log\frac{n}{\epsilon})$ and $\mathcal{O}(\sqrt{n}\log n\, \log\frac{n}{\epsilon})$ iteration bound for small- and large-update methods, respectively.
2010 Mathematics Subject Classification: 90C22
Key Words and Phrases: interior-point method, kernel function, complexity, semidefinite programming
- ⋅ 11:10 − 12:10 Chair: Yong Hoon Lee(Pusan National University)
- ⋅ 6th 11:10 − 11:30 Interior-point methods for $P_*(\kappa)$-linear complementarity problem based on a new class of kernel functions (You-Young Cho, Gyeong-Mi Cho and Yong-Hoon Lee)
- 조유영*(부산대), 조경미(동서대), 이용훈(부산대)
You-Young Cho*, Pusan National University, Gyeong-Mi Cho, Dongseo University, Yong-Hoon Lee, Pusan National University
In this talk, we introduce new primal-dual interior-point methods for $P_*(\kappa)$-linear complementarity problem based on a new class of eligible kernel functions. Interior-point algorithm is one of the most efficient numerical methods for optimization problems. We show that the algorithms have $\mathcal{O}\bigl((1+2\kappa)\sqrt{n}\log n\log\frac{n}{\epsilon}\bigr)$ and $\mathcal{O}\bigl((1+2\kappa)\sqrt{n}\log\frac{n}{\epsilon}\bigr)$ iteration bounds for large- and small-update methods, respectively. These are currently the best known complexity results for such methods.
2010 Mathematics Subject Classification: 90C33
Key Words and Phrases: interior-point method, kernel function, complexity, linear complementarity problem
- ⋅ 6th 11:30 − 11:50 New finite difference method for American put option (Beom Jin Kim, Hi Jun Choe)
- 김범진*(연세대), 최희준(연세대)
Beom Jin Kim*, Yonsei University, Hi Jun Choe, Yonsei University
We present a simple numerical method to find the optimal exercise boundary in an American put option. Our method solves an equation of a fixed boundary problem derived from a free boundary problem. We exploit an intermediate function that has Lipschitz character near the optimal exercise boundary and produces a convergent algorithm to locate the optimal exercise boundary. We also present several numerical results which illustrate comparison to other methods.
2010 Mathematics Subject Classification: 91G60
Key Words and Phrases: American put option
- ⋅ 6th 11:50 − 12:10 A note on the analysis of a perpetual American option under the CEV (Ji Hun Yoon, Jeong-Hoon Kim and Sun-Yong Choi)
- 윤지훈*(연세대), 김정훈(연세대), 최선용(연세대)
Ji Hun Yoon*, Yonsei University, Jeong-Hoon Kim, Yonsei University, Sun-Yong Choi, Yonsei University
We consider a perpetual American option under both the constant elasticity of variance model and a generalized model where the constant elasticity is perturbed by a small fast mean-reverting Ornstein-Uhlenbeck process. We analyze changes of option price and optimal exercise price under the unperturbed and perturbed CEV diffusions with respect to the parameters of the constant elasticity and the market price of elasticity risk. The revealed results may provide a useful information, particularly, on real option problems.
2010 Mathematics Subject Classification: 60Hxx
Key Words and Phrases: CEV, American option, perturbation method
- Mathematical Education
- ⋅ 10:00 − 10:30 Chair: Hyang-Keun Shin(Sunchon National University)
- ⋅ 5th 10:00 − 10:30 Early history of linear algebra (Sang-Gu Lee, Jae Hwa Lee and Yoon Mee Ham)
- 이상구*(성균관대), 이재화(성균관대), 함윤미(경기대)
Sang-Gu Lee*, Sungkyunkwan University, Jae Hwa Lee, Sungkyunkwan University, Yoon Mee Ham, Kyonggi University
Until the 1950s, linear algebra was considered only as one of abstract and advanced mathematics subject among in graduate mathematics courses, mainly dealing with module in algebra. Since the 1960s, it has been a main subject in undergraduate mathematics education because matrices has been used all over. In Korea, it was considered as a course only for mathematics major students until 1980s. However, now it is a subject for all undergraduate students including natural science, engineering, social science since 1990s. In this talk, we deal with the birth and development of early linear algebra from a historical perspective in order to explain why linear algebra became so popular in college mathematics education. Contributions of Chinese and H. Grassmann will be extensively examined.
2010 Mathematics Subject Classification: 01A55, 01A73, 01A90
Key Words and Phrases: Chinese, the nine chapters on the mathematical art, Seki Kowa, Leibniz, H. Grassmann
- ⋅ 10:40 − 11:10 Chair: Hyang-Keun Shin(Sunchon National University)
- ⋅ 5th 10:40 − 11:10 Current problems of mathematics education in South Korea (Young Han Choe)
- 최영한(한국수학교육학회)
Young Han Choe, Korean Soc. of Math. Edu.
In S. Korea, mathematics has been a major blame for huge private expenditures on so-called "private education", which includes 1 : 1 tutoring, group tutoring, home-delivery of studies, on-line classes, and various classes in "private academies". The private expenditures on out-of-school education often exceed public spending on schools. Many researches show that mathematics learning in South Korea is provided mostly through private education. In 1997 (again in 2007 and in 2011), S. Korean government reformed mathematics curriculum to put a boost on individual students' interests, affections and other attributes toward school mathematics. However, most teachers just repeat the same contents and problems drilling students to memorize all the techniques for solving simple and routine problems, instead of touching extra contents to enhance each student's understanding in mathematics and to attract students with interest in mathematics according to their ability. This results in nothing but students with less interest in mathematics. On the contrary to high scores in mathematical performance of international comparative studies, Korean students expressed an extremely low level of interest in mathematics compared with their counterparts of other countries. Korean students came in the second to the last in students' interest level of mathematics among OECD countries. Korean students had also difficulties in explaining their thoughts logically in answers to open realistic problems and in justifying the answers. Low level of interest in mathematics is not only a problem for an individual student but also a dilemma of the whole nation.
2010 Mathematics Subject Classification: 97D30
Key Words and Phrases: interest in mathematics, private education
- ⋅ 13:40 − 15:10 Chair: Ik Pyo Kim(Daegu University)
- ⋅ 5th 13:40 − 14:10 The inquiry of attitude of math students toward group discussions and presentations (Seong-A Kim)
- 김성아(동국대)
Seong-A Kim, Dongguk University
We designed a class room experiment employing small group discussions in a class of senior students who are preservice math teacher. Through this research experiment, we expected students' deep understanding of the contents covered in the course as well as the power of thinking and communication in mathematics. Students in the class were divided into 5 small groups in the beginning of the course. Each group was asked to present their finding after group discussion on the given topic and then each student wrote a short article on his/her conclusions about the topic individually. Sometimes students summarized their findings into a short piece of writing to lecturer by group. We found that students' attitude toward group discussions changed to be more positive and they even came to enjoy those. They also realized that they could learn how to think and communicate through discussions and finally could improve their writing in mathematics.
2010 Mathematics Subject Classification: 97D40
Key Words and Phrases: group discussion, presentation, mathematical communication, preservice math teacher
- ⋅ 5th 14:10 − 14:40 Creative hands-on activities on sphere designs (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
We design various creative patterns on the hands-on composition of sphere.
2010 Mathematics Subject Classification: 97U30
Key Words and Phrases: sphere pattern designs
- ⋅ 5th 14:40 − 15:10 Learning motivation and self-efficacy by feedback in engineering math education (Young Sik Kim)
- 김영식(홍익대)
Young Sik Kim, Hong Ik University
We investigate the learning motivation and self-efficacy by feedback in engineering math education.
2010 Mathematics Subject Classification: 97U99
Key Words and Phrases: feedback, education
- Mathematics for Information Sciences
- ⋅ 09:00 − 10:20 Chair: Sang-il Oum(KAIST)
- ⋅ 5th 09:00 − 09:30 On Cameron--Erd\H{o}s problem of Sidon sets (Sang June Lee, Yoshiharu Kohayakawa, Vojtech Rodl and Wojciech Samotij)
- 이상준*(고등과학원), Yoshiharu Kohayakawa(Univ. of Sao Paulo), Vojtech Rodl (Emory Univ.), Wojciech Samotij(Univ. of Cambridge)
Sang June Lee*, KIAS, Yoshiharu Kohayakawa, University of Sao Paulo, Vojtech Rodl, Emory University, Wojciech Samotij, University of Cambridge
A set~$A$ of positive integers is called a \textit{Sidon set} if all the sums~$a_1+a_2$, with~$a_1\leq a_2$ and~$a_1$,~$a_2\in A$, are distinct. In this talk we consider \emph{Cameron--Erd\H{o}s problem} which was suggested in 1990. The problem is to estimate the number of Sidon sets contained in $[n]:=\{1,2,\dots, n\}$. Results of Chowla, Erd\H{o}s, Singer, and Tur\'an from the 1940s imply that the maximum size of Sidon sets in $[n]$ is $\sqrt{n}(1+o(1))$. From this result, one can trivially obtain that the number of Sidon sets contained in $[n]$ is between $2^{(1+o(1))\sqrt{n}}$ and $2^{c\sqrt{n}\log n}$ for some absolute constant $c$. However, these bounds have not been notably improved for about 20 years. We obtain a new upper bound $2^{c\sqrt{n}}$ which is sharp up to a constant factor in the exponent when compared to the previous lower bound $2^{(1+o(1))\sqrt{n}}$. For the proof, we define a graph from our setting such that, roughly speaking, a Sidon set in $[n]$ corresponds to an independent set of the graph. In addition, the graph satisfies some dense condition. We show that in a graph satisfying the dense condition, the number of independent sets of given size $t$ is much smaller than the trivial bound ${n\choose t}$. By applying it repeatedly, we have the new upper bound on the number of Sidon sets contained in $[n]$.
2010 Mathematics Subject Classification: 05A16
Key Words and Phrases: Sidon set, Cameron--Erd\H{o}s
- ⋅ 5th 09:30 − 10:00 Cycle-saturated graphs with minimum number of edges (Younjin Kim, Zoltan Furedi)
- 김연진*(카이스트), Zoltan Furedi(Renyi Institute of Mathematics of the Hungarian Academy of Sciences)
Younjin Kim*, KAIST, Zoltan Furedi, Renyi Institute of Mathematics of the Hungarian Academy of Sciences
A graph $G$ is called $H$-saturated if it does not contain any copy of $H$, but for any edge $e$ in the complement of $G$ the graph $G + e$ contains some $H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by $sat(n,H)$. We prove $sat(n, C_k) = n + n/k + O((n/k^2) + k^2)$ holds for all $n \ge k \ge 3$, where $C_k$ is a cycle with length $k$.
2010 Mathematics Subject Classification: 05C38, 05C35
Key Words and Phrases: graphs, cycles, extremal graphs, minimal saturated graphs
- ⋅ 5th 10:00 − 10:20 Graphs of small rank-width are pivot-minors of graphs of small tree-width (O-joung Kwon, Sang-il Oum)
- 권오정*(카이스트), 엄상일(카이스트)
O-joung Kwon*, KAIST, Sang-il Oum, KAIST
We prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$. We also prove that graphs of rank-width at most $1$, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-width at most $1$ are exactly vertexminors of paths.
2010 Mathematics Subject Classification: 05C83
Key Words and Phrases: graph, rank-width, tree-width, pivot-minor
- ⋅ 11:30 − 12:00 Chair: Seunghyun Seo(Kangwon National University)
- ⋅ 5th 11:30 − 12:00 Injectively $\Delta$-choosable graphs (Seog-Jin Kim, Won-Jin Park)
- 김석진*(건국대), 박원진(서울대)
Seog-Jin Kim*, Konkuk University, Won-Jin Park, Seoul National University
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor receive distinct colors. A graph $G$ is said to be injectively $k$-choosable if any list $L(v)$ of size at least $k$ for every vertex $v$ allows an injective coloring $\phi(v)$ such that $\phi(v) \in L(v)$ for every $v \in V(G)$. The least $k$ for which $G$ is injectively $k$-choosable is the injective choosability number of $G$, denoted by $\chi_i^{l} (G)$. In this paper, we obtain new sufficient conditions to be $\chi_i^l (G) = \Delta(G)$.
2010 Mathematics Subject Classification: 05Cxx
Key Words and Phrases: injective coloring, list coloring, maximum average degree
- ⋅ 13:40 − 15:10 Chair: Hyun Kwang Kim(POSTECH)
- ⋅ 5th 13:40 − 14:10 On the weight enumerators of the projections of the 2-adic Golay code of length 24 to $\mathbb{Z}_{2^e}$ (Sunghyu Han)
- 한성휴(한국기술교육대)
Sunghyu Han, Korea University of Technology and Education
Dougherty et al. calculated some of the weight enumerators of the projections of the 2-adic Golay code of length 24 to $\mathbb{Z}_{2^e}$ [Lifted codes and their weight enumerators, Discrete Mathematics, 305 (2005) 123--135]. In this paper, we calculate the missing values so that we complete the calculation of the weight enumerators of the projections of the 2-adic Golay code of length 24 to $\mathbb{Z}_{2^e}$ for all $e \geq 1$.
2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: Golay codes, lifted codes, p-adic codes, weight enumerators
- ⋅ 5th 14:10 − 14:40 Construction of self-dual codes with an automorphism of order $p$ (Hyun Jin Kim, Heisook Lee, June Bok Lee and Yoonjin Lee)
- 김현진*(이화여대 수리과학연구소), 이혜숙(이화여대), 이준복(연세대), 이윤진(이화여대)
Hyun Jin Kim*, Institute of Mathematical Sciences, Ewha Womans University, Heisook Lee, Ewha Womans University, June Bok Lee, Yonsei University, Yoonjin Lee, Ewha Womans University
We develop a construction method for finding self-dual codes with an automorphism of order $p$ with $c$ independent $p$-cycles. In more detail, we construct a self-dual code with an automorphism of type $p-(c,f+2)$ and length $n+2$ from a self-dual code with an automorphism of type $p-(c,f)$ and length $n$, where an automorphism of type $p-(c, f)$ is that of order $p$ with $c$ independent cycles and $f$ fixed points; we note that the extremality is preserved.
2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: self-dual, extremal, automorphism, code
- ⋅ 5th 14:40 − 15:10 Algorithm for finding modules in complex networks (EunYoun Kim)
- 김은연(국가수리과학연구소)
EunYoun Kim, National Institute for Mathematical Sciences
The identification of modules in complex networks is important for the understanding of systems. Here, we propose an ensemble clustering method incorporating vertex groupings in various sizes and the sequential removal of weak ties between vertieces which are rarely grouped together. This method successfully detects modules in various networks, such as hierarchical random networks and the American college football network, with known modular structures. Some of the results are compared with those obtained by modularity optimization and K-means clustering.
2010 Mathematics Subject Classification: 94C15
Key Words and Phrases: network, module
- Cryptography
- ⋅ 09:00 − 10:30 Chair: Kyung-Ah Shim(National Institute for Mathematical Sciences)
- ⋅ 5th 09:00 − 09:20 Efficient $r$-th root extraction over finite field (Gook Hwa Cho, Namhun Koo and Soonhak Kwon)
- 조국화*(성균관대), 구남훈(성균관대), 권순학(성균관대)
Gook Hwa Cho*, Sungkyunkwan University, Namhun Koo, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University
Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root algorithm based on linear recurrence relation which is a generalization of Adleman-Manders-Miller algorithm. More precisely, for given $r$-th residue $a\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $Tr(\alpha)^r=a$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial over $\mathbb F_q$.
2010 Mathematics Subject Classification: 11Y16
Key Words and Phrases: finite Field, $r$-th root, linear recurrence relation, Adleman-Manders-Miller algorithm
- ⋅ 5th 09:20 − 09:40 A generalization of Atkin's square root algorithm (Namhun Koo, Gook Hwa Cho and Soonhak Kwon)
- 구남훈*(성균관대), 조국화(성균관대), 권순학(성균관대)
Namhun Koo*, Sungkyunkwan University, Gook Hwa Cho, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University
In this paper, we propose a square root algorithm for finite field which generalizes Atkins's square root finding algorithm for finite field $\mathbb{F}_q$ of $q$ elements where $q\equiv 5\pmod{8}$ and Kong et al.'s algorithm for the case $q\equiv 9\pmod{16}$. Proposed algorithm requires one exponentiation for specific case and then faster than Tonelli-Shanks algorithm. We implement our algorithm and Tonelli-Shanks algorithm using Sage and PARI/GP and compare their performance.
2010 Mathematics Subject Classification: 11Y40
Key Words and Phrases: square root finding algorithm, finite field, Atkin's algorithm, Kong's algorithm, Tonelli-Shanks algorithm
- ⋅ 5th 09:40 − 10:00 Improved trade-off algorithms through more pre-computation (Kyung Chul Jeong)
- 정경철(ETRI 부설연구소)
Kyung Chul Jeong, The Attached Institute of ETRI
For some cryptanalytic algorithms which require pre-computation, there is the trade-off relation between the performance and the required resources. For this kind of algorithms, to achieve better trade-off relation is main concern. We can improve these algorithms through the investigation into the random function behaviors. As a result, in pre-computation Pollard discrete logarithm algorithm, time complexity can be reduced after more pre-computation.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: random function, discrete logarithm problem, Pollard rho
- ⋅ 5th 10:00 − 10:30 Cryptanalysis of a public key cryptosystem based on the matrix combinatorial problem (Moon Sung Lee)
- 이문성(서울대)
Moon Sung Lee, Seoul National University
In this talk, we present a cryptanalysis of a public key cryptosystem based on the matrix combinatorial problem proposed by Wang and Hu in [INFORMATICA Vol. 21 (2010), no. 4, 611--626]. Using lattice-based methods finding small integer solutions of modular linear equations, we recover the secret key of the cryptosystem for a certain range of parameters including the suggested parameters.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: public key cryptosystem, matrix combinatorial problem, modular linear equation, lattice
- ⋅ 11:30 − 12:00 Chair: Aaram Yun(UNIST)
- ⋅ 5th 11:30 − 12:00 The $l$-th power DHP and the $l$-th root DHP (I-Yeol Kim, Dongyoung Roh and Sang Geun Hahn)
- 김이열*(국가수리과학연구소), 노동영(ETRI 부설연구소), 한상근(카이스트)
I-Yeol Kim*, National Institute for Mathematical Sciences, Dongyoung Roh, Attached Institute of Electronics and Telecommunications Research Institute, Sang Geun Hahn, KAIST
There are many variants of the computational Diffie-Hellman problem that are necessary to provide security of many cryptographic schemes. Two of them are the square Diffie-Hellman problem and the square root Diffie-Hellman problem. Recently it is proved that these two problems are polynomial-time equivalent under a certain condition. In this talk, we generalize this result. We introduce the $l$-th power Diffie-Hellman problem and the $l$-th root Diffie-Hellman problem and show that these two problems are polynomial-time equivalent for $l=O$ $(\log p)$ under a certain condition, where $p$ is the order of the underlying group.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: $l$-th power DHP, $l$-th root DHP
- ⋅ 13:40 − 15:00 Chair: Yongjin Yeom(Kookmin University)
- ⋅ 5th 13:40 − 14:10 New preimage attacks on Hash modes of AES-256 (Deukjo Hong, Dong-Chan Kim and Daesung Kwon)
- 홍득조*(ETRI 부설연구소), 김동찬(ETRI 부설연구소), 권대성(ETRI 부설연구소)
Deukjo Hong*, The Attached Institute of ETRI, Dong-Chan Kim, The Attached Institute of ETRI, Daesung Kwon, The Attached Institute of ETRI
We study the slow diffusion of the AES key schedule for 256-bit keys and find weakness which can be used in the preimage attack on its Davis-Meyer mode. Our preimage attack works for 8 rounds of AES-256 with the computational complexity of $2^{124.9}$, while the best previous attack works for 7 rounds of AES-256. It is also extended to the preimage attack on some well-known double-block-length hash modes assuming the underlying block cipher is 8-round AES-256, whose computational complexity is $2^{252.9}$.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: block cipher, AES, preimage attack
- ⋅ 5th 14:10 − 14:30 Practical homomorphic encyrption (Jinsu Kim, Moonsung Lee and Jung Hee Cheon)
- 김진수*(서울대), 이문성(서울대), 천정희(서울대)
Jinsu Kim*, Seoul National University, Moonsung Lee, Seoul National University, Jung Hee Cheon, Seoul National University
최근 fully homomorphic encryption에 대한 연구가 활발히 진행되고 있다. 클라우드 컴퓨팅 환경이 현실화되면서 암호문 간의 연산을 지원하는 암호에 대한 연구가 활발히 진행되고 있다. 2009년 Gentry가 제시한 ideal lattice 기반에서부터 2010년 Dijk 등이 제시한 정수 기반 fully homomorphic encryption에 이르기까지 많은 연구가 진행되었다. 이들 중 가장 간단한 형태인 정수 기반 fully homomorphic encrpytion은 공개키 사이즈가 너무 크다는 단점을 가지고 있다. 그에 따라 2011년 CRYPTO와 2012년 Eurocrypt 에는 공개키 사이즈를 줄이는 방법에 대한 연구가 발표되었다. 또한, 이 스킴들은 모두 1-bit의 암호화에 대하여 논의하고 있는데, 1-bit를 암호화하면 $\lambda^5$ bit 사이즈의 암호문이 생성되므로 현실적으로 사용하기에는 비효율적이다. 본 연구에서는 보다 현실적이고 실용적인 적용을 위하여 1-bit에 대한 암호화가 아닌 $\lambda^3$ bit를 암호화하는 기법을 제시하고자 한다. CRT기법을 이용하여 평문의 사이즈를 기존 대비 $\lambda^3$배 까지 늘렸다. 기존의 fully homomorphic encryption은 computational approximate gcd문제에 기반하였는데, 제시된 암호스킴은 새로운 암호학적 가정인 decisional approximate gcd문제에 기반하였다.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: homomorphic, pratical, CRT, approximate gcd
- ⋅ 5th 14:30 − 15:00 LEA: A software-oriented lightweight block cipher (Dong-Chan Kim, Deukjo Hong, Jung-Keun Lee and Daesung Kwon)
- 김동찬*(ETRI 부설연구소), 홍득조(ETRI 부설연구소), 이정근(ETRI 부설연구소), 권대성(ETRI 부설연구소)
Dong-Chan Kim*, The Attached Institute of ETRI, Deukjo Hong, The Attached Institute of ETRI, Jung-Keun Lee, The Attached Institute of ETRI, Daesung Kwon, The Attached Institute of ETRI
With the necessity of cryptographic services in resource-constrained devices such as RFID tag, many hardware-oriented lightweight block ciphers have been proposed for recent years. Similarly, software implementations of cryptographic primitives are widely used in low-resource platforms of smart devices or embedded systems, but there are few block ciphers dedicated to such environments. If an algorithm can be implemented efficiently on resource-constrained platforms, then we call it a software-oriented lightweight algorithm. In this paper, we propose a new block cipher LEA satisfying software-oriented lightweightness in the aspects of low-power consumption and small code size. It has 128-bit block size and 128/192/256-bit key size, and consists of 32-bit ARX operations which are available or easily constructible on most platforms. The SIMD implementation of LEA-128 ECB mode has the performance of 4.38 cycles/byte on Intel Core 2 Quad Q6600 2.4GHz, and the code size of the implementation of LEA-128 ECB mode is optimized to 310 bytes on AT91SAM9G20 400MHz which is an ARM processor. We checked that LEA is secure against existing block cipher attacks.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: block cipher, software-oriented lightweight, low-power encryption, small code size
- Algebras and Representations
- ⋅ 10:30 − 12:10 Chair: Dong-il Lee(Seoul Women's University)
- ⋅ 5th 10:30 − 10:50 Supercategorification of quantum Kac-Moody algebra and its integrable modules (Se-jin Oh, Seok-Jin Kang and Masaki Kashiwara)
- 오세진*(서울대), 강석진(서울대), Masaki Kashiwara(서울대)
Se-jin Oh*, Seoul National University, Seok-Jin Kang, Seoul National University, Masaki Kashiwara, Seoul National University
The quiver Hecke superalgebras can be understood as the ``super"-extension of the Khovanov-Lauda-Rouquier algebras. We study the superalgebra's $\mathbb{Z}$-graded module categories. The categories have a natural supercategory structure induced by the parity involution. We show that the Grothendieck groups of the supercategories are isomorphic to the $I_{{\rm odd}}$-colored symmetrizable quantum Kac-Moody algebras and their integrable modules. We say this a supercategorification. We use the perfect basis theory very crucially when we prove the isomorphisms. This is a joint work with Prof. Seok-Jin Kang and Prof. Masaki Kashiwara.
2010 Mathematics Subject Classification: 05E10, 16G99, 81R10, 17C70
Key Words and Phrases: categorification, quiver Hecke superalgebras, cyclotomic quotients, quantum Kac-Moody algebras
- ⋅ 5th 10:50 − 11:10 Representation type of finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$ (Euiyong Park, Susumu Ariki)
- 박의용*(서울대), Susumu Ariki(Osaka Univ.)
Euiyong Park*, Seoul National University, Susumu Ariki, Osaka University
We study cyclotomic quiver Hecke algebras $R^{\Lambda_0}(\beta)$ in type $A^{(2)}_{2\ell}$, where $\Lambda_0$ is the fundamental weight. The algebras are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras associated with the symmetric group. We give a formula for the dimension of the algebra, and a simple criterion to tell the representation type. The criterion is a natural generalization of Erdmann and Nakano's for the Iwahori-Hecke algebras. This is a jointwork with Susumu Ariki at Osaka university and posted on arXiv:1208.0889.
2010 Mathematics Subject Classification: 16G60, 05E10
Key Words and Phrases: representation type, quiver Hecke algebras
- ⋅ 5th 11:10 − 11:30 Schur expansion of the integral form Macdonald polynomials (Meesue Yoo)
- 유미수(고등과학원)
Meesue Yoo, KIAS
In this talk, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\left<\frac{J_{\lambda}[X;q,q^k]}{(1-q)^n},s_{\mu}(X)\right> \in \mathbb{N}[q]$, we have found explicit combinatorial formula for the Schur coefficients in one row case, two column case and certain hook shape cases. Recent result of Egge-Loehr-Warrington gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonlad polynomials of Haglund-Haiman-Loehr in quasi symmetric functions to prove the Haglund's conjecture in general cases.
2010 Mathematics Subject Classification: 05E06
Key Words and Phrases: Schur coefficients, integral form Macdonald polynomials
- ⋅ 5th 11:30 − 11:50 Mixed Schur-Weyl-Sergeev duality for the queer Lie superalgebra $\mathfrak{q}(n)$ (Ji Hye Jung, Seok-Jin Kang)
- 정지혜*(서울대), 강석진(서울대)
Ji Hye Jung*, Seoul National University, Seok-Jin Kang, Seoul National University
In this talk, we introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$, which we call walled Brauer superalgebras. Let $\mathfrak{q}(n)$ be the queer Lie superalgebra, $V =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and $W$ the dual representation of $V$. We prove that the superalgebra $\overrightarrow{B}_{r,s}$ is anti-isomorphic to the super-centralizer algebra ${\rm End}_{\mathfrak{q}(n)}(V^{\otimes r} \otimes W^{\otimes s})$ of the $\mathfrak{q}(n)$-action on the mixed tensor space $V^{\otimes r} \otimes W^{\otimes s}$ whenever $n \geq r+s$. We also give a presentation of $\overrightarrow{B}_{r,s}$ in terms of generators and relations. This is a joint work with Prof. Seok-Jin Kang at Seoul National University.
2010 Mathematics Subject Classification: 17B10, 05E10
Key Words and Phrases: mixed Schur-Weyl-Sergeev duality, queer Lie superalgebras, walled Brauer superalgebras
- ⋅ 5th 11:50 − 12:10 Crystal base of $q$-deformed Kac-modules over the general linear Lie superalgebra (Jae-Hoon Kwon)
- 권재훈(서울시립대)
Jae-Hoon Kwon, University of Seoul
We introduce the notion of a crystal base of a finite dimensional $q$-deformed Kac module over the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$, and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite dimensional irreducible $U_q(\mathfrak{gl}(m|n))$-module with typical highest weight. We also show that the crystal base of a $q$-deformed Kac module is compatible with that of its irreducible quotient $V(\lambda)$ given by Benkart, Kang and Kashiwara when $V(\lambda)$ is an irreducible polynomial representation.
2010 Mathematics Subject Classification: 17B37
Key Words and Phrases: quantum group, Lie superalgebra
- Number Theory
- ⋅ 09:30 − 10:50 Chair: Byeong-Kweon Oh(Seoul National University)
- ⋅ 6th 09:30 − 09:50 Infinite series relations from generalized non-holomorphic Eisenstein series (Sung-Geun Lim)
- 임성근(목원대)
Sung-Geun Lim, Mokwon University
In this talk, we consider a transformation formula for a certain series with confluent hypergeometric functions of the second kind, which come from a transformation formula for generalized non-holomorphic Eisenstein series, and find some infinite series relations.
2010 Mathematics Subject Classification: 34A25, 11M36
Key Words and Phrases: infinite series, modular transformation
- ⋅ 6th 10:00 − 10:20 $p$-adic periodic zeta functions by a continuously parametrized distribution. (Hae-Sang Sun)
- 선해상(충북대)
Hae-Sang Sun, Chungbuk National University
After introducing a continuously parametrized Bernoulli distribution on $\mathbb{Z}_p$ and discussing its properties, we define a $p$-adic periodic zeta function and discuss the prosperities.
2010 Mathematics Subject Classification: 11Rxx
Key Words and Phrases: Bernoulli distribution, periodic zeta function
- ⋅ 6th 10:30 − 10:50 Normal regular Hermitian lattices over imaginary quadratic fields (Poo-Sung Park)
- 박부성(경남대)
Poo-Sung Park, Kyungnam University
In general, it is not easy to find all regular Hermitian lattices because the local behaviors of Hermitian lattices depends on ground fields. We show that if a normal regular lattice $L$ over $\mathbb{Q}(\sqrt{-m})$ is not universal, $L_p$ is not $\mathbb{Z}_p$-universal for $p \in \{3,5,7\}$ dividing $m$. Besides, we can find normal regular lattices for sufficiently large $m$.
2010 Mathematics Subject Classification: 11E39
Key Words and Phrases: regular Hermitian lattice
- ⋅ 11:10 − 12:30 Chair: Poo-Sung Park(Kyungnam University)
- ⋅ 6th 11:10 − 11:30 Strictly regular quaternary quadratic forms (Ji Young Kim, A. G. Earnest)
- 김지영*(서울대), A. G. Earnest(Southern Illinois Univ.)
Ji Young Kim*, Seoul National University, A. G. Earnest, Southern Illinois University
It will be shown that there exist only finitely many isometry classes of primitive integral positive definite quaternary quadratic $\mathbb{Z}$-lattices that are strictly regular, in the sense that they primitively represent all integers primitively represented by their genus.
2010 Mathematics Subject Classification: 11E12
Key Words and Phrases: regular Hermitian lattice
- ⋅ 6th 11:40 − 12:00 On the abundance of real quadratic fields with large fundamental units (Jeongho Park)
- 박정호(포항공대)
Jeongho Park, POSTECH
On an analytic point of view, the key problem about ideal class groups of number fields is to show that there are lots of number fields with small class group, or equivalently, number fields with large regulator. The difficulty mainly stems from the lack of explicit connection between the field discriminant and the fundamental system of units. We will go round this difficulty in case of real quadratic fields by forgetting about the discriminant and considering the algebraic integers of fixed norms instead. By doing so we can show that there are sufficiently many fields with fundamental unit $\varepsilon_d \gg (\log d)^2$.
2010 Mathematics Subject Classification: 11R29, 11Y40, 11J68
Key Words and Phrases: real quadratic number field, class number, fundamental unit, regulator
- ⋅ 6th 12:10 − 12:30 Congruence of mock theta function (Byungchan Kim, Scott Ahlgren)
- 김병찬*(서울과학기술대), Scott Ahlgren(Univ. of Illinois at Urbana-Champaign)
Byungchan Kim*, Seoul National University of Science and Technology , Scott Ahlgren, University of Illinois at Urbana-Champaign
Ramanujan's striking congruence properties of the partition function $p(n)$ have motivated much research on finding congruences for the partition function or similar arithmetic functions. In this direction, K. Ono proves that there are infinitely many congruences for the partition function $p(n)$ modulo every primes larger than $3$. On the other hand, the behavior of partition function modulo $2$ or $3$ is much more mysterious. In this talk, we examine congruence property of Ramanujan's mock theta function \[ f(q) = 1 + \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \] modulo $2$ and $3$. This talk is based on the ongoing project with Scott Ahlgren (Univ. of Illinois).
2010 Mathematics Subject Classification: 11F33
Key Words and Phrases: partition congruence, mock theta function
- p-Adic Analytic Number Theory and Their Applications
- ⋅ 09:00 − 10:30 Chair: Joohee Jeong(Kyungpook National University)
- ⋅ 5th 09:00 − 09:20 Identities for the $q$-Euler polynomials arising from non-linear differential equations (Hyun-Mee Kim)
- 김현미(국민대)
Hyun-Mee Kim, Kookmin University
Recently, T. Kim has introduced the Frobenius non-linear differential equations (see [T. Kim, {\it Identities involving Frobenius-Euler polynomials arising from non-linear differential equations}, Journal of Number Theory 132 (2012), 2854--2865]). By the same motivation, we give some identities related to $q$-Euler polynomials.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: $q$-Euler polynomials
- ⋅ 5th 09:20 − 09:40 Integral formulae of Bernoulli and Genocchi polynomials (Joung-Hee Jin, Seog-Hoon Rim)
- 진정희*(경북대), 임석훈(경북대)
Joung-Hee Jin*, Kyungpook National University, Seog-Hoon Rim, Kyungpook National University
Recently, some interesting and new identities are introduced in [4]. From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.
2010 Mathematics Subject Classification: HB68
Key Words and Phrases: Bernoulli polynomials, Genocchi polynomials
- ⋅ 5th 09:40 − 10:00 Some relation between $q$-Bernoulli numbers with weight $(\alpha,\beta)$ and $q$-Bernstein polynomials with weight $\alpha$ (Hui Young Lee, Cheon Seoung Ryoo)
- 이희영*(한남대), 유천성(한남대)
Hui Young Lee*, Hannam University, Cheon Seoung Ryoo, Hannam University
In this paper, using bosonic $p$-adic $q$-integral on $\mathbb Z_p$ we give some interesting relationship between $q$-Bernoulli numbers with weight $(\alpha, \beta)$ and $q$-Bernstein polynomials with weight $\alpha$. And we shall extend the $q$-Bernstein polynomials with weight $\alpha$ with two variables.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bernoulli numbers and polynomials, Bernstein polynomials, the $p$-adic $q$-integral on $\mathbb Z_p$
- ⋅ 5th 10:00 − 10:30 On convergence properties of the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication (Lee Chae Jang)
- 장이채(건국대)
Lee Chae Jang, Konkuk University
In this paper, we consider the generalized fuzzy integral which is a fuzzy integral based on the pseudo-addition $S=\vee$(supremum) and a pseudo-multiplication $T$ given in Xie-Fang (2006). The purpose of this study is to define the interval-valued generalized fuzzy integral with respect to a fuzzy measure by means of an interval-representable pseudo-multiplication of measurable interval-valued functions and to investigate some characterizations and convergence properties of them.
2010 Mathematics Subject Classification: 28E10, 28E20, 03E72, 26E50, 11B68
Key Words and Phrases: fuzzy measure, generalized fuzzy integral, interval-representable pseudo-multiplication, interval-valued function
- ⋅ 10:40 − 12:10 Chair: Seog Hoon Rim(Kyungpook National University)
- ⋅ 5th 10:40 − 11:20 Some identities involving Gegenbauer polynomials (Dae San Kim, Taekyun Kim and Seog-Hoon Rim)
- 김대산*(서강대), 김태균(광운대), 임석훈(경북대)
Dae San Kim*, Sogang University, Taekyun Kim, Kwangwoon University, Seog-Hoon Rim, Kyungpook National University
In this paper we derive some interesting identities involving Gegenbauer polynomials arising from the orthogonality of Gegenbauer polynomials for the inner product space ${\mathbb{P}}_n$ with respect to the weighted inner product $=\int_{-1} ^1 p_{1}(x)p_2(x)(1-x^2)^{\lambda-\frac{1}{2}}dx$.
2010 Mathematics Subject Classification: 11S80, 11B68
Key Words and Phrases: Gegenbauer polynomial, Bernoulli polynomial, Euler polynomial
- ⋅ 5th 11:20 − 11:50 Characterization of ergodicity of $T$-adic maps on $F_2[[T]]$ using digit derivatives basis (Jeong Sangtae)
- 정상태(인하대)
Sangtae Jeong, Inha University
Motivated by the work of Anashin et al. on ergodic theory for $p$-adic dynamical systems, Lin et al. recently developed a corresponding theory for function fields. In this paper, we use digit derivatives to present another characterization of the ergodicity of $T$-adic maps over $F_2[[T]]$ that is parallel to Lin et al.
2010 Mathematics Subject Classification: 11S80
Key Words and Phrases: Ergodic functions, 1-Lipschitz, function fields, Carlitz polynomials, digit derivatives
- ⋅ 5th 11:50 − 12:10 Some identities on Bernoulli and Genocchi numbers and polynomials (Sun Jung Lee)
- 이선정(경북대)
Sun Jung Lee, Kyungpook National University
In this paper, we give some interesting identities on the Bernoulli and Genocchi numbers by using the equations of the fermionic and bosonic $p$-adic integral on $\mathbb Z_p$.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: Bernoulli and Genicchi polynomials
- ⋅ 09:30 − 11:00 Chair: Dae San Kim(Sogang University)
- ⋅ 6th 09:30 − 09:50 Convolution sums and Bernoulli numbers (Daeyeoul Kim)
- 김대열(국가수리과학연구소)
Daeyeoul Kim, National Institute for Mathematical Sciences
In this talk, we will present convolution sums of divisor functions. Next, we state the relations of Bernoulli numbers and convolution sums.
2010 Mathematics Subject Classification: 11A67
Key Words and Phrases: convolution sum, Bernoulli number
- ⋅ 6th 09:50 − 10:10 On the Barnes' type related to multiple Genocchi polynomials on $\mathbb Z_p$ (Jung Yoog Kang, Hui young Lee and Nam Soon Jung)
- 강정욱*(한남대), 이희영(한남대), 정남순(한남대)
Jung Yoog Kang*, Hannam University, Hui young Lee, Hannam University, Nam Soon Jung, Hannam University
Using fermionic $p$-adic invariant integral on $\mathbb Z_p$ , we construct the Barnes' type multiple Genocchi numbers and polynomials. From those numbers and polynomials, we derive the twisted Barnes' type multiple Genocchi numbers and polynomials. Moreover, we will find the Barnes' type multiple Genocchi zeta function.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Barnes' type multiple Genocchi polynomials, twisted Barnes' type multiple Genocchi polynomials
- ⋅ 6th 10:10 − 10:40 Some identities involving Euler polynomials and Genocchi polynomials arising from a non-linear differential equation (Joohee Jeong, Seok-Hoon Rim)
- 정주희*(경북대), 임석훈(경북대)
Joohee Jeong*, Kyungpook National University, Seok-Hoon Rim, Kyungpook National University
In this talk, we introduce the method of professor T. Kim which gives identities involving Frobenius-Euler polynomials arising from a non-linear differential equation. Using Kim's idea, we get some interesting identities involving Euler polynomials and then identities involving Genocchi polynomials.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: Euler polynomial, Genocchi polynomial
- ⋅ 6th 10:40 − 11:00 A study on the weighted $q$-Bernoulli numbers and polynomials (Wonjoo Kim, Taekyun Kim)
- 김원주*(광운대), 김태균(광운대)
Wonjoo Kim*, Kwangwoon University, Taekyun Kim, Kwangwoon University
In this talk, we discuss new concept of the $q$-Bernoulli numbers and polynomials with weight $\alpha$. From these $q$-Bernoulli numbers with weight $\alpha$, we establish some interesting identities and relations.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bernoulli numbers and polynomials, $q$-Bernoulli numbers and polynomials, $q$-Bernoulli numbers and polynomials with weight $\alpha$
- ⋅ 11:10 − 12:40 Chair: Lee-Chae Jang(Konkuk University)
- ⋅ 6th 11:10 − 11:40 Generalized $(q, w)$-Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb Z_p$ (Cheon Seoung Ryoo)
- 유천성(한남대)
Cheon Seoung Ryoo, Hannam University
In this talk we generalize the Euler numbers and polynomials by the generalized $(q, w)$-Euler numbers $E_{n, q, w}(a)$ and polynomials $E_{n, q, w}(x:a)$. We observe an interesting phenomenon of 'scattering' of the zeros of the generalized $(q, w)$-Euler polynomials $ E_{n, q, w}(x:a)$ in complex plane.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Euler numbers and polynomials, generalized $(q,w)$-Euler numbers and polynomials
- ⋅ 6th 11:40 − 12:10 A note on Lebegues-Radon-Nikodym theorem with respect to fermionic $p$-adic integrals on $p$-adic integer rings (Seog-Hoon Rim)
- 임석훈(경북대)
Seog-Hoon Rim, Kyungpook National University
In this lecture, we introduce $p$-adic Lebesgues-Radon-Nikodym theorem related to fermionic $p$-adic integrals.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: Lebesgue-Radon-Nikodym theorem, fermionic $p$-adic integrals, $p$-adic integer rings
- ⋅ 6th 12:10 − 12:30 Construction of the generating function of $q$-Bernoulli polynomials, generalized $q$-Bernoulli polynomials and its applications (Min-Soo Kim)
- 김민수(경남대)
Min-Soo Kim, Kyungnam University
In this talk, we introduce the generating function of $q$-Bernoulli numbers, polynomials and generalized $q$-Bernoulli polynomials. The q-analogues of many well known formulas are derived by using several results of $q$-Bernoulli numbers and polynomials. The $q$-analogues of zeta-type functions are given by using generating functions of $q$-Bernoulli, $q$-Euler numbers and polynomials. Finally, their values at non-positive integers are also been computed.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bosonic $p$-adic integrals, generating functions, $q$-analogues of the Dirichlet's $L$-functions
- ⋅ 6th 12:30 − 12:40 A study on $q$-Euler polynomials arising from $p$-adic $q$-integrals on $\mathbb Z_p$ (Taekyun Kim)
- 김태균(광운대)
Taekyun Kim, Kwangwoon University
In this lecture, we introduce some interesting results which is derived from $p$-adic $q$-integrals on $\mathbb Z_p$. Finally, we discuss some applications in the areas of $p$-adic mathematical physics.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: Frobenius, Euler, non-linear differential equation
- Financial Mathematics
- ⋅ 09:00 − 10:20 Chair: Wanmo Kang(KAIST)
- ⋅ 5th 09:00 − 09:20 Stress scenario selection by empirical likelihood (Wanmo Kang, Chulmin Kang and Paul Glasserman)
- 강완모*(카이스트), 강철민(카이스트), Paul Glasserman(Columbia Univ.)
Wanmo Kang*, KAIST, Chulmin Kang, KAIST, Paul Glasserman, Columbia University
We present the selection and analysis of stress scenarios for financial risk assessment, with particular emphasis on identifying sensible combinations of stresses to multiple variables. We consider reverse stress testing - finding the most likely scenarios - using a nonparametric empirical likelihood estimator. We apply the results to marginal expected shortfall, macro stress scenarios, and scenario sampling method.
2010 Mathematics Subject Classification: 60F05
Key Words and Phrases: risk management, stress testing, senario selection, empirical likelihood
- ⋅ 5th 09:20 − 09:40 Market data and application of mathematical model to finance (Seok Hyon Yu)
- 유석현(삼성증권)
Seok Hyon Yu, Samsung Securities Co., Ltd.
시장데이터의 수학적인 구조와 특징을 알아봄. 이러한 특징이 수리적인 모델을 수립하는데 어떠한 영향을 주는지 알아보고 수리 모델의 발전 경향을 살펴봄. 또한 금융회사가 파생상품을 운용하는 과정에서 시장데이터와 모델을 어떻게 활용하고 있는 지, 이 과정에서 발생하는 과제들로는 어떠한 것들이 있는 지 알아봄.
2010 Mathematics Subject Classification: 91G80
Key Words and Phrases: 시장데이터, 수리적인 모델
- ⋅ 5th 09:40 − 10:00 A comparison study of ADI and OSM on option pricing models (Darae Jeong, Junseok Kim)
- 정다래*(고려대), 김준석(고려대)
Darae Jeong*, Korea University, Junseok Kim, Korea Univerisity
In this paper we perform a comparison study of alternating direction implicit (ADI) and operator splitting (OS) methods on multi-dimensional Black-Scholes option pricing models. The ADI method is used extensively in mathematical finance for numerically solving multi-asset option pricing problems. And as can be analyzed by a Fourier method, the standard ADI scheme is stable for L2 initial data such as piecewise linear continuous payoffs and piecewise constant payoffs. However, numerical results from this scheme show oscillatory solution behaviors with nonsmooth pyaoffs or discontinuous derivatives at the exercise price with large time steps. Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price. In the ADI scheme, there are source terms which include y derivatives when we solve x derivative involving equations. Then, due to the nonsmooth payoffs, source term contains abrupt changes which are not in the range of implicit discrete operator and this leads to difficulty in solving the problem. On the other hand, the OS method does not contain the other variable's derivatives in the source term. We provide computational results showing the performance of the methods for two underlying asset option pricing problems. The results show that the OS method is very efficient and gives better accuracy and robustness than the ADI method with large time steps.
2010 Mathematics Subject Classification: 91Gxx
Key Words and Phrases: finite difference method, Black-Scholes equation, ADI, OSM, option pricing
- ⋅ 5th 10:00 − 10:20 Valuing option under the Markov regime-switching beta model (Geon Woo Kim, Sungchul Lee)
- 김건우*(연세대), 이승철(연세대)
Geon Woo Kim*, Yonsei University , Sungchul Lee, Yonsei University
In this paper, we propose a formula for valuing an option where its underlying asset is affected by changing market environments of various factors and business cycle. Specifically, we apply a Markov regime-switching method to a continuous-time Capital Asset Pricing Model. In other words, the beta in continuous-time CAPM model switches between two values,which represent stable and unstable market condition. By measuring the occupation time of the underlying asset process for a given state, we obtain the distribution of occupation time. And using this density, we find analytic pricing formula of European vanilla options and exchange options under this Markov regime-switching Beta model.
2010 Mathematics Subject Classification: 60GXX
Key Words and Phrases: continuous-time CAPM model, Markov regime-switching, beta model
- ⋅ 10:40 − 12:00 Chair: Doobae Jun(Seoul National University)
- ⋅ 5th 10:40 − 11:00 Exact simulation of jump diffusion processes (Jerim Kim, Bara Kim and Hwa-Sung Kim)
- 김제림*(고려대), 김바라(고려대), 김화성(경희대)
Jerim Kim*, Korea University, Bara Kim, Korea University, Hwa-Sung Kim, Kyung Hee University
We present an exact simulation of jump diffusion processes. The algorithm is applied to the analysis of the credit risk models.
2010 Mathematics Subject Classification: 65C05
Key Words and Phrases: jump diffusion, exact algorithm, simulation, credit risk model
- ⋅ 5th 11:00 − 11:20 Martingale techniques under matrix affine models (Chulmin Kang, Wanmo Kang)
- 강철민*(카이스트), 강완모(카이스트)
Chulmin Kang*, KAIST, Wanmo Kang, KAIST
The aim of this talk is to introduce techniques of computing transforms using exponential martingales under matrix affine models. Particularly, we provide some transform formulas for linear functionals of affine processes and their bridges whose state space is a set of positive semidefinite matrices, as well as the conditional Laplace transform formula for a multifactor stochastic volatility model. As an application of the conditional Laplace transform formula, we suggest an exact simulation method of the model.
2010 Mathematics Subject Classification: 60G44
Key Words and Phrases: martingales, affine processes, transform analysis
- ⋅ 5th 11:20 − 11:40 Commodity futures term structure model (Jun Yeol Kim, Hyeong In Choi)
- 김준열*(교보증권), 최형인(서울대)
Jun Yeol Kim*, Kyobo Securities Co., Ltd., Hyeong In Choi, Seoul National University
A new approach to the commodity futures term structure model is introduced. The most salient feature of this model is that, once the interest rate model is given, the commodity futures price volatility is the only quantity that completely determines the model. As a consequence this model enables one to do away with the drudgeries of having to deal with the convenience yield altogether, which has been the most thorny point so far.
2010 Mathematics Subject Classification: 91Gxx
Key Words and Phrases: commodity futures, term structure, convenience yield, volatility
- ⋅ 5th 11:40 − 12:00 Continuity correction for discrete barrier options with two barriers (Doobae Jun)
- 전두배(서울대)
Doobae Jun, Seoul National University
Discrete barrier options are the options whose payoffs are determined by underlying prices at a finite set of times. We consider the discrete barrier option with two barriers. Broadie et al.~(1997) proposed a continuity correction for the discretely monitored barrier option. We extend this idea to barrier option with two barriers. The proof for discrete chained barrier option is provided and numerical results show the continuity correction approximation is remarkably accurate.
2010 Mathematics Subject Classification: 60Hxx
Key Words and Phrases: discrete barrier option, chained option
- Inverse Problems
- ⋅ 10:40 − 12:10 Chair: Mikyoung Lim(KAIST)
- ⋅ 5th 10:40 − 11:10 Non-iterative harmonic $B_z$ algorithm in MREIT with correction (Kiwan Jeon, Chang-Ock Lee)
- 전기완*(국가수리과학연구소), 이창옥(카이스트)
Kiwan Jeon*, National Institute for Mathematical Sciences, Chang-Ock Lee, KAIST
Conductivity imaging became a new medical imaging modality in various ways, for example, EIT, MREPT, and MREIT. MREIT, especially, provides a high resolution conductivity imaging, that utilizes the magnetic flux density generated by injected current through several pairs of electrodes. Conventional conductivity reconstruction methods, for example harmonic $B_z$ algorithm, is designed as an iterative scheme due to the non-linear relation between conductivity and current density. However, iterative methods may not guarantee a reliable conductivity reconstruction in the real experiment, since the measured magnetic flux density may be seriously affected the noise and MR-signal weakness. Local harmonic $B_z$ algorithm is a possible way to obtain a scaled conductivity image for the diagnosis and anatomy studies on the selected region of interest~(ROI), but it still does not give an absolute conductivity. In order to handle such a problem, a direct recovery method, namely {\em non-iterative harmonic $B_z$ algorithm}, are suggested, when the true current density is transversally dominant on $xy$-plane. In this talk, we provide a new non-iterative harmonic $B_z$ algorithm which contains a correction procedure dealing with non-transversal current density inside. In addition, we improve a practical computational issue of the previous non-iterative algorithm. The validation of the proposed algorithm is possible in numerical experiments.
2010 Mathematics Subject Classification: 15A29, 62P10
Key Words and Phrases: MREIT, conductivity imaging, con-iterative algorithm
- ⋅ 5th 11:10 − 11:40 Fast imaging of thin, curve-like electromagnetic inclusions via topological derivative concept (Won-Kwang Park)
- 박원광(국민대)
Won-Kwang Park, Kookmin University
A fast imaging of thin, curve-like electromagnetic inclusions completely hidden in the homogeneous domain with smooth boundaries is considered. By creating an electromagnetic inclusion of a small diameter and applying the asymptotic expansion formula in the existence of such an inclusion, the topological derivative is successfully derived. Based on this derivative, a one-step iteration imaging algorithm is designed by solving an adjoint problem. By exploring a relationship with Bessel function, we can explain why the concept of topological derivative works. Various numerical experiments of single and multiple inclusions demonstrate the viability and limitation of the designed algorithm.
2010 Mathematics Subject Classification: 78A46, 65M32
Key Words and Phrases: topological derivative, thin electromagnetic inclusions, asymptotic expansion formula
- ⋅ 5th 11:40 − 12:10 A gradient estimate for solutions to parabolic equations with discontinuous coefficients (Kyoungsun Kim)
- 김경선(인하대)
Kyoungsun Kim, Inha University
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them {\it manifolds of discontinuities}. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this talk, we gave a parabolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coefficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an applications of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a parabolic operator with piecewise smooth coefficients. The both gradient estimates are independent of the distances between manifolds of discontinuities.
2010 Mathematics Subject Classification: 35K10
Key Words and Phrases: gradient estimates, parabolic equation
- ⋅ 09:30 − 10:30 Chair: Won-Kwang Park(Kookmin University)
- ⋅ 6th 09:30 − 10:00 Generalized polarization tensors for shape description (mikyoung Lim, Habib Ammari, Josselin Garnier, Hyeonbae Kang and Sanghyeon Yu)
- 임미경*(카이스트), Habib Ammari(Ecole Normal Superieure), Josselin Garnier(Uni\-versit´e Paris 7), 강현배(인하대), 유상현(카이스트)
Mikyoung Lim*, KAIST, Habib Ammari, Ecole normal superieure, Josselin Garnier, Universit´e Paris 7, Hyeonbae Kang, Inha University, Sanghyeon Yu, KAIST
With each domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. Recently, a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that high-frequency oscillations of the boundary of a domain are only contained in its high-order GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We provide numerical evidence that GPTs can capture topology and high-frequency shape oscillations.
2010 Mathematics Subject Classification: 35R30, 35B30
Key Words and Phrases: generalized polarization tensors, shape description
- ⋅ 6th 10:00 − 10:30 Numerical methods in diffuse optical tomography (Kiwoon Kwon)
- 권기운(동국대)
Kiwoon Kwon, Dongguk University
가시광선 및 근적외선은 생체에서 흡수, 산란의 과정을 거친다. 이 과정은 생체의 성분에 따라 달라지며 이를 생체 광 흡수/산란 계수로 모델링 하며, 이 광 계수가 주어질 때 각 생체 내의 광밀도 함수는 확산근사방정식이라는 탄성 방정식의 해로 주어진다. 확산광영상법(Diffuse Optical Tomography)은 이 확산방정식에 대한 역문제, 즉 광계수를 구하는 문제로 나타낼 수 있다. 확산광영상법에 대한 수치해법에는 여러가지 역문제해법이 있으며, 이 강연에서는 광밀도함수를 기준 광밀도함수에 대한 Born 전개식에 대한 1차 및 2차 근사를 이용한 선형 및 2차 선형해법을 중심으로 알아본다.
2010 Mathematics Subject Classification: 65Z05
Key Words and Phrases: diffuse optical tomography, Born approximation
- Recent Research of Mathematical Problems Related with Mechanics and Engineering
- ⋅ 08:50 − 10:30 Chair: Gyungsoo Woo(Changwon National University)
- ⋅ 5th 08:50 − 09:10 Mathematical approaches to analysis of thermoelastic characteristics in a rotating composite material circular disk with a graded layer (Soo Ryoun Kong)
- 공수련(창원대)
Soo Ryoun Kong, Changwon National University
Typical composite circular disk is considered for the first type circular disk: an Al2O3 top coat, an Al substrate, a mixture bond coat of Al2O3 and Al. The second type is replaced the top coat with a functionally graded layer (FGL), and third type is replaced the bond coat with a FGL, in the first CM circular disk. A second-order differential equation was derived based on the two-dimensional thermal elastic theory for the thermoelastic characteristics of circular disks. Due to the complexity a finite element method is developed to search numerical approximations.
2010 Mathematics Subject Classification: 65K05
Key Words and Phrases: circular disk, composite material, functionally graded layer, finite element method, thermoelastic c
- ⋅ 5th 09:10 − 09:30 Efficient analytic algorithm for solving nonlinear fractional differential equations (Bongsoo Jang)
- 장봉수(울산과학기술대)
Bongsoo Jang, UNIST
In this work, we propose a new analytic approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of the generalized differential transform associated with the solution in the given fractional problem. However, the recurrent relations of complex nonlinear functions such as exponential, logarithmic and trigonometric functions have not been known. Here we suggest new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve several nonlinear fractional differential equations. Several numerical examples are demonstrated to show the effectiveness of the proposed method. It has been shown that the proposed method is robust and accurate to solve fractional problems.
2010 Mathematics Subject Classification: 65Dxx
Key Words and Phrases: fractional differential equation, Caputo fractional derivative, generalized differential transform method
- ⋅ 5th 09:30 − 09:50 Mathematical modeling of the elasticity analysis on a laminated composite wind turbine blade (Jaegwi Go, Seokchan Kim and Young-Do Choi)
- 고재귀*(창원대), 김석찬(창원대), 최영도(목포대)
Jaegwi Go*, Changwon National University, Seokchan Kim, Changwon National University, Young-Do Choi, Mokpo National University
A 1kW-class horizontal axis wind turbine (HAWT) rotor blade is taken into account to investigate elastic characteristics in 2-D. The elastic blade field is composed of symmetric cross-ply laminated composite material. Blade element momentum theory is applied to obtain the boundary conditions pressuring the blade, and the plane stress elasticity problem is formulated in terms of two displacement parameters with mixed boundary conditions. For the elastic characteristics a fair of differential equations are derived based on the elastic theory. The domain is divided by triangular and rectangular elements due to the complexity of the blade configuration, and a finite element method is developed for the governing equations to search approximate solutions. The results describe that the elastic behavior is deeply influenced by the layered angle of the middle laminate and the stability of the blade can be improved by controlling the layered angle of laminates, which can be evaluated by the mathematical approach.
2010 Mathematics Subject Classification: 74F10
Key Words and Phrases: elastic characteristics, finite element method, horizontal axis wind turbine, laminated composite materials
- ⋅ 5th 09:50 − 10:10 Overlapping Schwarz preconditioners for isogeometric analysis (Durkbin Cho, L. Beirao da Veiga, L. F. Pavarino and S. Scacchi)
- 조덕빈*(동국대), L. Beirao da Veiga(Univ. of Milan), L. F. Pavarino(Univ. of Milan), S. Scacchi(Univ. of Milan)
Durkbin Cho*, Dongguk University, L. Beirao da Veiga, University of Milan, L. F. Pavarino, University of Milan, S. Scacchi, University of Milan
Isogeometric Analysis (IGA) is a non-standard numerical method for partial differential equations (PDEs), which was introduced by T. J. R. Hughes. In the isogeometric framework, the ultimate goal is to adopt the geometry description from a Computer Aided Design (CAD) parametrization, and use it for the analysis, that is, within the PDE solver. Non-uniform rational B-splines (NURBS) are a standard in CAD community mainly because they are extremely convenient of the representation of free-form surfaces and there are very efficient algorithms to evaluate them, to refine and derefine them. In IGA, those same basis functions (that represent the CAD geometry) are also used as the basis for the discrete solution space of PDEs, thus following an isoparametric paradigm. IGA methodologies have been studied and applied in fields as diverse as fluid dynamics, structural mechanics and electromagnetics. Domain decomposition methods are a major area of recent research in numerical analysis for PDEs. They provide robust, parallel and scalable preconditioned iterative methods for the large linear systems arising in discretizaton of the continuous problems. In this talk, we propose overlapping additive Schwarz (OAS) methods for elliptic problems in Isogeometric Analysis. We construct OAS preconditioners both in the parametric space and in the physical space and also prove that our proposed methods in multi-dimensions are scalable. Moreover, we present a set of numerical experiments, including the case with discontinuous coefficients, which is in complete accordance with the theoretical developments.
2010 Mathematics Subject Classification: 65N55, 65N30, 65F10
Key Words and Phrases: domain decomposition methods, overlapping additive Schwarz, scalable preconditioners, isogeometric analysis, finite elements, NURBS
- ⋅ 5th 10:10 − 10:30 Smectic liquid crystals with singularities (Jinhae Park)
- 박진해(충남대)
Jinhae Park, Chungnam National University
Liquid crystal have been attracted many scientists for its fascinating properties and applications. In smectic liquid crystals, centers of mass are arranged locally in one-dimensional layers described by the Chen-Lubensky energy. In this talk, we study a simplified Chen-Lubensky energy for Smectic Liquid Crystals in a physically relevant two-dimensional domain. We discuss the limiting energy and locations of singularities.
2010 Mathematics Subject Classification: 49NXX
Key Words and Phrases: liquid crystal, Chen-Lubensky
- ⋅ 10:40 − 12:20 Chair: Seokchan Kim(Changwon National University)
- ⋅ 5th 10:40 − 11:00 Lagrange multiplier method for solving the quasi-variational inequality in mechanics (Robert V. Namm, Gyungsoo Woo)
- Robert V. Namm*(Pacific National Univ. Russia), 우경수(창원대)
Robert V. Namm*, Pacific National University Russia, Gyungsoo Woo, Changwon National University Korea
For semicoercive elastic contact problem with friction the smooth duality scheme is investigated, which allows on each step of successive approximation method to define simultaneously the displacement vector of elastic body points and normal contact stress defining the friction force on next step of successive approximation method. Therefore on each step of successive approximation method we must solve the auxiliary problem. But we can not use the duality scheme based on classical Lagrangian functional for solving a semicoercive problem. For overcoming these difficulties we investigate the smooth duality scheme based on modified Lagrangian functional. It is shown that the modified smooth duality method converges to saddle point under suitable regularity of auxiliary problem solution.
2010 Mathematics Subject Classification: 65F10, 65K10
Key Words and Phrases: modified Lagrangian functional, quasi-variational inequality
- ⋅ 5th 11:00 − 11:20 Long time existence of smooth solution for the porous medium equation in a bounded domain (Sunghoon Kim)
- 김성훈(포항공대)
Sunghoon Kim, Pohong University of Science and Technology
In this presentation, we will give a talk about the short time and long time existence of smooth solution to the porous medium equation in a bounded domain. The proof is based on the Schauder estimate and H\"older estimate.
2010 Mathematics Subject Classification: 35K20
Key Words and Phrases: porous medium equation, Schauder estimate, H\"older estimate, short time existence, long time existence
- ⋅ 5th 11:20 − 11:40 On the spatially homogeneous Boltzmann equation for relativistic particles (Seok-Bae Yun)
- 윤석배(Academia Sinica)
Seok-Bae Yun, Academia Sinica
The relativistic Boltzmann equation describes the evolution of the velocity distribution function of non-ionized particles in Milkowsky space-time. In this talk, we will consider the Cauchy problem and the propagation of polynomial and exponential moments of the relativistic Boltzmann equation in the case when the dynamics does not depend on the spatial variable. This is a joint work with Robert Strain.
2010 Mathematics Subject Classification: 35Q20
Key Words and Phrases: Boltzmann equations, special relativity
- ⋅ 5th 11:40 − 12:00 Multistability and complex dynamics in a driven nanowire system (Younghae Do)
- 도영해(경북대)
Younghae Do, Kyungpook National University
Nanoscale systems are the subject of forefront interdisciplinary research. The underlying physics renders dynamics of nanosystems nonlinear. Despite extensive research on chaotic dynamics in low-dimensional dynamical systems, little has been done for nanoscale systems. Due to the potentially widespread use of nanoscale systems in science, engineering, and technological applications, it is imperative to investigate and exploit nonlinear dynamics in such systems. This paper addresses this fundamental topic by investigating a prototypical class of nanoelectromechanical systems (NEMS): electrostatically driven Si-nanowires. We find that multistability and complicated structures of basin of attraction are common, and the latter can be attributed to high-dimensional transient chaos. Implications of these phenomena to device operations are explored.
2010 Mathematics Subject Classification: 37D45
Key Words and Phrases: nanowire, multistability, chaos, dynamics
- ⋅ 5th 12:00 − 12:20 On the global existence of strong solutions to the Cucker-Smale-Navier-Stokes system in a periodic domain (Young-Pil Choi)
- 최영필(서울대)
Young-Pil Choi, Seoul National University
In this talk, we provide the global existence of strong solutions to the Cucker- Smale-Navier-Stokes system on a periodic domain. This system has been introduced for the modeling of interacting between Cucker-Smale flocking particles and incompressible viscous fluids. We also show the emergence of timeasymptotic flocking. When viscosity of incompressible fluid is sufficiently large, we show that the velocities of particles and fluid are aligned to the common velocity at least exponentially fast.
2010 Mathematics Subject Classification:
Key Words and Phrases:
- Graduate Student Session I
- ⋅ 09:30 − 10:30 Chair: Junmi Park(Chungnam National University)
- ⋅ 5th 09:30 − 10:00 The asymptotic property of Mittag-Leffler functions (Jaehyun Jeong, Sung Kyu Choi and Namjip Koo)
- 정재현*(충남대), 최성규(충남대), 구남집(충남대)
Jaehyun Jeong*, Chungnam National University, Sung Kyu Choi, Chungnam National University, Namjip Koo, Chungnam National University
In this talk we investigate the monotonic properties of Mittag-Leffler functions and apply these results to study the asymptotic property of solutions of linear fractional systems with coefficient constants.
2010 Mathematics Subject Classification: 24A33, 34D05
Key Words and Phrases: fractional differential equations, Mittag-Leffler Functions, asymptotic property
- ⋅ 5th 10:00 − 10:30 A Gronwall inequality and its application to impulsive fractional differential equations (Chun Mi Ryu, Sung Kyu Choi and Namjip Koo)
- 유춘미*(충남대), 최성규(충남대), 구남집(충남대)
Chun Mi Ryu*, Chungnam National University, Sung Kyu Choi, Chungnam National University, Namjip Koo, Chungnam National University
In this talk we introduce a generalized Gronwall inequality of fractional type. Then we discuss stability of solutions of impulsive fractional differential equations by using this inequality.
2010 Mathematics Subject Classification: 26A33, 34A37
Key Words and Phrases: impulsive fractional differential equations, impulsive fractional integral inequality
- ⋅ 13:40 − 14:40 Chair: Younjin Kim(KAIST)
- ⋅ 5th 13:40 − 14:10 Edge-disjoint decompositions of complete multipartite graphs into gregarious cycles of long length (Jeongmi Park, Jung Rae Cho and Yoshio Sano)
- 박정미*(부산대), 조정래(부산대), Yoshio Sano(National Institute of Informatics)
Jeongmi Park*, Pusan National University, Jung Rae Cho, Pusan National University, Yoshio Sano, National Institute of Informatics
The notion of gregarious cycles in complete multipartite graphs was introduced by Billington and Hoffman in 2003 and was modified by Billington, Smith, and Hoffman in 2007. In this talk, we propose more general definition of "gregarious" cycles which is a common generalization of both the definitions. With our definition, we can consider the problem to characterize complete multipartite graphs which are decomposable into edge-disjoint gregarious long cycles, and we give some results on this problem.
2010 Mathematics Subject Classification: 05C51
Key Words and Phrases: graph decomposition, complete multipartite graph, gregarious cycle
- ⋅ 5th 14:10 − 14:40 Some properties of the permutation tableaux of type B (Kyoung Suk Park)
- 박경숙(아주대)
Kyoung Suk Park, Ajou University
Corteel, Josuat-Verges and Kim introduced some $q$-Eulerian polynomials for symmetry of signed permutation. They proved the symmetry using the pignose diagram. We give an algorithm to prove the polynomial using the permutation tableaux of type B. Moreover, we introduce some interesting results for the permutation tableaux of type B and the Alternating tableaux.
2010 Mathematics Subject Classification: 68R05
Key Words and Phrases: permutation tableaux, alternating tableaux, signed permutation
- Graduate Student Session II
- ⋅ 13:40 − 14:40 Chair: Ji Hun Yoon(Yonsei University)
- ⋅ 5th 13:40 − 14:10 Comparison of numerical schemes on multi-dimensional Black-Scholes equations (Joong Lee Jo, Yongsik Kim and Hyeng Keun Koo)
- 조중리*(아주대), 김용식(아주대), 구형건(아주대)
Joong Lee Jo*, Ajou University, Yongsik Kim, Ajou University, Hyeng Keun Koo, Ajou University
In this paper, we investigate the performance of numerical schemes for solving multi-dimensional option pricing problem. We compare a direct solving method (DSM) and an Operator Splitting Method (OSM) by finite difference approximations. By varying parameters in the Black-Scholes equations for the maximum on the call problem, we observed that there are no significant differences on convergence criterion except huge difference of computational cost. Accordingly, we give numerical examples on OSM including ELS pricing based on either two assets or three assets, with the Monte-Carlo simulation as the benchmark. Finally, we show a huge improvement in speed by Message Passing Interface (MPI) implementation for OSM.
2010 Mathematics Subject Classification: 91G80
Key Words and Phrases: Black-Scholes equations, operator splitting method
- ⋅ 5th 14:10 − 14:40 Spatial effect for interaction between tumor and immune system (Seunghyeon Baek, Inkyung Ahn)
- 백승현*(고려대), 안인경(고려대)
Seunghyeon Baek*, Korea University, Inkyung Ahn, Korea University
In this talk, a mathematical model is proposed that describes interactions among immune effectors and tumor cells. The tumor is assumed to disperse by diffusion with directed movement towards more sufficient nutrition. We study the spatial effect for interaction among activated immune cells, tumor cells and the concentration of IL-2 under various scenario. In addition, we examine the change of the density of tumor cells under the adoptive cellular immunotherapy(ACI) and proposed the an effective treatment for tumor using ACI method. Various simulations are given for the results.
2010 Mathematics Subject Classification: 92C17
Key Words and Phrases: tumor-immune system, directed movement
- Graduate Student Session III
- ⋅ 09:30 − 11:00 Chair: Seung Hee Lee(Chungnam National University)
- ⋅ 6th 09:30 − 10:00 Approximate Euler-Lagrange quadratic mapping in fuzzy Banach spaces (Juri Lee, Hark-Mahn Kim)
- 이주리*(충남대), 김학만(충남대)
Juri Lee*, Chungnam University, Hark-Mahn Kim, Chungnam University
In this paper, we study the Hyers--Ulam stability of an approximate Euler--Lagrange quadratic functional equation: $$f(kx+ly)+f(kx-ly) =kl[f(x+y)+f(x-y)]+2(k-l)[kf(x)-lf(y)]$$ in fuzzy Banach spaces, where $k,l$ are nonzero rational numbers with $k \neq l$.
2010 Mathematics Subject Classification: 39B82, 39B52, 46H99
Key Words and Phrases: stability of Euler-Lagrange type, quadratic mapping, fuzzy Banach space, fixed point
- ⋅ 6th 10:00 − 10:30 Approximate Jensen functions in quasi-Banach spaces (Eunyoung Son, Hark-Mahn Kim)
- 손은영*(충남대), 김학만(충남대)
Eunyoung Son*, Chungnam National University, Hark-Mahn Kim, Chungnam National University
In this article, we investigate the Hyers--Ulam stability of the following Jensen functional equation: $$f(\frac{x-y}{n}+z)+f(\frac{y-z}{n}+x)+f(\frac{z-x}{n}+y)=f(x)+f(y)+f(z)$$ in quasi-Banach {spaces} for any fixed nonzero integer $n$. Let $X$ is a quasi-normed space with quasi-norm $\|\cdot\|_{X}$ and that $Y$ is a $p$-Banach space with $p$-norm $\|\cdot\|$. We define the difference operator $Df: X^3 \rightarrow Y $ by \begin{eqnarray*} Df(x,y,z):=f(\frac{x-y}{n}+z)+f(\frac{y-z}{n}+x)+f(\frac{z-x}{n}+y)-f(x)-f(y)-f(z), \end{eqnarray*} for all $x,y,z \in X$ and for any fixed nonzero integer $n$. Suppose that a mapping $f : X \rightarrow Y $ with $f(0)=0$ satisfies the functional inequality \begin{eqnarray}\label{3.1} \| Df(x,y,z)\|_{Y} \leq \varphi(x,y,z) \end{eqnarray} for all $x,y,z \in X$, and that the perturbing function $\varphi :X^3\rightarrow \mathbb{R}^{+}$ satisfies \begin{eqnarray}\label{3.2} \sum_{i=0}^{\infty}\frac{1}{2^{(i+1)p}}\varphi(2^{i} x,2^{i} y,2^{i} z)^p <\infty \end{eqnarray}for all $x,y,z \in X$. Then, there exists a unique additive mapping $h : X \rightarrow Y$ defined by $h(x)= \lim_{ k \rightarrow \infty } \frac{1}{2^k}f(2^kx) $ such that \begin{eqnarray}\label{3.3} \|f(x)-h(x)\|_{Y} \leq \Big[\sum_{i=0}^{\infty}\frac{1}{2^{(i+1)p}}\varphi(2^{i} x,2^{i} x,(1-n)2^{i} x)^p \Big]^{\frac{1}{p}} \end{eqnarray} for all $x \in X$.
2010 Mathematics Subject Classification: 45J05, 46B03
Key Words and Phrases: stability, Jensen functions, quasi-normed spaces, $p$-Banach spaces
- ⋅ 6th 10:30 − 11:00 Property $(aw)$ and Weyl's theorem (Il Ju An)
- 안일주(경희대)
Il Ju An, Kyung Hee University
In this note we study the property $(aw)$, a variant of Weyl's theorem introduced by Berkani and Zariouh, by means of the localized single valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property $(aw)$ holds. We also relate this property with Weyl's theorem, $a$-Weyl's theorem and property $(w)$. Finally, we show that if $T$ is $a$-polaroid and either $T$ or $T^*$ has SVEP then $f(T)$ satisfies property $(aw)$ for each $f\in H_1(\sigma(T))$.
2010 Mathematics Subject Classification: Primary 47A10, 47A53; Secondary 47B20
Key Words and Phrases: property ($aw$), Weyl's theorem, single valued extension property, $a$-polaroid
- ⋅ 11:10 − 12:40 Chair: Juri Lee(Chungnam National University)
- ⋅ 6th 11:10 − 11:40 Examples of strictly ergodic systems of zero entropy with complexity (Bo Bae Kim)
- 김보배(아주대)
Bo Bae Kim, Ajou University
There are different levels of complexity in entropy zero systems. C.~Grillenberger introduces a construction for strictly ergodic systems of positive entropies using shift spaces and permutations of blocks. We modify the method of Grillenberger, for constructing strictly ergodic systems of zero entropy with complexity. We try to compute the upper and lower entropy dimensions of the examples.
2010 Mathematics Subject Classification: 37A35
Key Words and Phrases: ergodic, entropy
- ⋅ 6th 11:40 − 12:10 $C^1$ persistently expansive homoclinic classes of vector fields with shadowing property (Seunghee Lee, Keonhee Lee)
- 이승희*(충남대), 이건희(충남대)
Seunghee Lee*, Chungnam National University, Keonhee Lee, Chungnam National University
Let $\gamma$ be a hyperbolic closed orbit of a $C^1$ vector field on a compact boundaryless Riemannian manifold $M$, and let $H_X(\gamma)$ be the homoclinic class of $X$ containing $\gamma$. In this talk, we prove that if a $C^1$-persistently expansive homoclinic class $H_X (\gamma)$ has the shadowing property, then $H_X(\gamma)$ is hyperbolic.
2010 Mathematics Subject Classification: 37C10, 37C29, 37C50
Key Words and Phrases: persistently expansive, homoclinic class, vector fields, shadowing property
- ⋅ 6th 12:10 − 12:40 Limit shadowing and hyperbolicity (Junmi Park, Keonhee Lee)
- 박준미*(충남대), 이건희(충남대)
Junmi Park*, Chungnam National University, Keonhee Lee, Chungnam National University
Let $f$ be a diffeomorphism of a closed smooth manifold. We show that (i) the chain recurrent set $\mathcal R (f)$ of $f$ has the $C^1$-stably limit shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and (ii) for the chain component $C_f (p)$ of $f$ containing a hyperbolic periodic point $p$, $f$ has the $C^1$-stably limit shadowing property on $C_f (p)$ if and only if $C_f (p)$ is the hyperbolic homoclinic class of $f$ containing $p$.
2010 Mathematics Subject Classification: 37D05, 37C50
Key Words and Phrases: limit shadowing, hyperbolic
- Graduate Student Session IV
- ⋅ 11:10 − 12:10 Chair: Seong-Deog Yang(Korea University)
- ⋅ 6th 11:10 − 11:40 Curves on the unit 3-sphere $S^3(1)$ in Euclidean 4-space $\mathbb{R}^4$ (Chan Yong Kim, Jeong Hyeong Park and Shinsuke Yorozu)
- 김찬용*(성균관대), 박정형(성균관대), Shinsuke Yorozu(Miyagi Univ. of Education)
Chan Yong Kim*, Sungkyunkwan University, Jeong Hyeong Park, Sungkyunkwan University, Shinsuke Yorozu, Miyagi University of Education
We show many examples of curves on the unit 2-sphere $S^2 (1)$ in $\mathbb{R}^3$ and the unit 3-sphere $S^3 (1)$ in $\mathbb{R}^4$. We study whether its curves are Bertrand curves or spherical Bertrand curves and provide some examples illustrating the resultant curves.
2010 Mathematics Subject Classification: 53A04
Key Words and Phrases: spherical curve, Bertrand curve, spherical Bertrand curve
- ⋅ 6th 11:40 − 12:10 A space classification of a Hopf hypersurface which has recurrent structure Jacobi shape operator in $G_2({\mathbb C}^{m+2})$ (Chang Hwa Woo, Young Jin Suh and Im soon Jung)
- 우창화*(경북대), 서영진(경북대), 정임순(경북대)
Chang Hwa Woo*, Kyungpook National University, Young Jin Suh, Kyungpook National University, Im soon Jung, Kyungpook National University
In this paper, we give a non-existence theorem for Hopf hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$ with recurrent structure Jacobi operator.
2010 Mathematics Subject Classification: 53C15
Key Words and Phrases: complex two-plane Grassmannians, Hopf hypersurface, recurrent structure Jacobi operator
- Poster Session
- ⋅ 10:00 − 15:00 Display Time(게시)\ \ /14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ 5th 14:30 − 15:00 Iterative algorithm for a new system of variational inclusions (Soo Hwan Kim)
- 김수환 (동의대)
Soo Hwan Kim, Dong-Eui University
In this paper, a new iterative scheme based on the extragradient-like method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a strongly positive linear bounded operator and the set of solutions of a mixed equilibrium problem is prosed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Our results extended recent results announced by many others.
2010 Mathematics Subject Classification: 49J40, 90C30
Key Words and Phrases: B-monotone operator, proximal mapping, iterative algorithm, variational inclusion, convergence
- ⋅ 5th 14:30 − 15:00 $L^2$ error analysis for an expanded mixed finite element methods for nonlinear Sobolev equations (Hyun Young Lee, Mi Ray Ohm and Jun Yong Shin)
- 이현영*(경성대), 엄미례(동서대), 신준용(부경대)
Hyun Young Lee*, Kyungsung University, Mi Ray Ohm, Dongseo University, Jun Yong Shin, Pukyong University
Based on an expanded mixed finite element method, we construct semidiscrete solutions of the nonlinear Sobolev equations. We construct the semidiscrete approximations of $u$, grad $u$ as well as $a(u)$ gard$(u) + b(u)$ grad$(u_t)$ and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of $u$, grad $u$ as well as $a(u)$ gard$(u) + b(u)$ grad$(u_t)$ in $L^2$ normed space.
2010 Mathematics Subject Classification: 65M15
Key Words and Phrases: expanded mixed finite element method, nonlinear Sobolev equations
- ⋅ 5th 14:30 − 15:00 An extensive spherical harmony (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
We design an upgrade version of the spherical harmony of horizontality and verticality which was exhibited in the Bridges Conference 2012.
2010 Mathematics Subject Classification: 96U60
Key Words and Phrases: spherical harmony
- ⋅ 09:30 − 12:40 Display Time(게시)\ \ /12:10 − 12:40 Q $\&$ A(질의응답) Chair:
- ⋅ 6th 12:10 − 12:40 The learning effect of learners upon learning for development of multi-metacognitive question (Hyeong Do Heo)
- 허형도(창원대)
Hyeong Do Heo, Changwon National University
In recent years the meta-cognitive instruction has been extensively studied which is considerably effective on enhancement of problem solving skills. This article classified multi-metacognitive questions by their purpose as well as method and states the way to increase effectiveness of learning depending on each cases. In this paper, researchers defined multi-metacognitive question development to reveal differences in terms of students' performance between the fresh approach and traditional forms of teaching.
2010 Mathematics Subject Classification: 97A10
Key Words and Phrases: cognitive, metacognitive