Program and Abstracts

Algebra

⋅ 20th-C-10:40 − 11:20   Chair: Yoonbok Lee (Incheon National University)

⋅ 20th-C-10:40 − 11:00 Rank 3 quadratic generation of projective varieties (Kang Jin Han, Wanseok Lee, Hyunsuk Moon, Euisung Park)

한강진*(대구경북과학기술원), 이완석(부경대), 문현석(국가수리과학연구소), 박의성(고려대)
Kang Jin Han*, DGIST, Wanseok Lee, Pukyung National University, Hyunsuk Moon, NIMS, Euisung Park, Korea University

Let $X$ be any nondegenerate irreducible projective variety over a closed field $\Bbbk$. We say that $X\subset\mathbb{P}^r$ is a quadratic variety of rank k if its homogeneous ideal can be generated by quadrics of rank at most $k$. We also say that for any very ample line bundle $L$ on $X$ the pair $(X,L)$ is a quadratic embedding of rank $k$ if $X\subset\mathbb{P}H^0(X,L)$ is a quadratic variety of rank k. It is known that many classical varieties such as any Segre-Veronese embeddings, rational normal scrolls and curves of high degree are quadratic varieties of rank 4. In this talk, we consider rank 3 quadratic generation among aforementioned varieties. We introduce some ways to generate rank 3 quadrics, apply this to check whether a given variety is quadratic of rank 3 or not, and provide some examples and consequences.

2010 Mathematics Subject Classification: 13D02, 14A25, 14N05, 14M12, 14M17
Key Words and Phrases: low rank quadrics, Veronese variety, Segre variety, Veronese re-embedding, determinantally presented, property $QR(k)$

⋅ 20th-C-11:00 − 11:20 Hypersurface arrangement of aCM type (Edoardo Ballico, Sukmoon Huh)

Edoardo Ballico(Universita di Trento), 허석문*(성균관대)
Edoardo Ballico, Universita di Trento, Sukmoon Huh*, Sungkyunkwan University

To a reduced effective divisor with simple normal crossings, one may associate the logarithmic sheaf of differential 1-forms with logarithmic poles along the divisor. In this talk we report our recent result on when one can expect that the logarithmic sheaf is arithmetically Cohen-Macaulay, specially when the underlying variety is a smooth complete intersection. This is a joint work with E. Ballico.

2010 Mathematics Subject Classification: 14J60
Key Words and Phrases: hypersurface arrangement, logarithmic sheaf, arithmetically Cohen-Macaulay bundle

⋅ 20th-C-11:30 − 12:10   Chair: Sihun Jo (Woosuk University)

⋅ 20th-C-11:30 − 11:50 $3$-folds of general type and canonically of fiber type with high genus (YongJoo Shin)

신용주(고등과학원)
YongJoo Shin, KIAS

Let $X$ be a Gorenstein minimal projective $3$-fold of general type. Chen and Cui gave that a smooth birational model $F$ of the generic irreducible component in the generic fiber of the canonical map has $p_g(F)\le 37$ when $F$ is a surface and $p_g(X)\ge 3890$, and $g(F)\le 91$ when $F$ is a curve and $p_g(X)\ge 183$. Moreover they provided examples of $(p_g(X),p_g(F)=(2,19)$ for a surface $F$, and $(p_g(X),g(F)=(2,13)$ for a curve $F$. And they suggested an open problem to find examples of $p_g(F)$ or $g(F)$ bigger than $19$ or $13$ for each case. In this talk we construct smooth $3$-folds $X$ of general type whose a smooth model $F$ of the generic irreducible component in the generic fiber of the canonical map has $p_g(F)=37$ when $F$ is a surface, and $g(F)=25$ when $F$ is a curve.

2010 Mathematics Subject Classification: 14J29, 14J30
Key Words and Phrases: surface of general type, canonically fibred 3-fold

⋅ 20th-C-11:50 − 12:10 Tropical super Abelian varieties (Hoil Kim)

김호일(경북대)
Hoil Kim, Kyungpook National University

Tropical geometry is a combinatorial approach to understand geometry. It shares much with toric geometry, but is applicable to more subjects. It uses both archimedian and non-archimedian fields, so that it is related to arithmetic geometry, Hodge theory, mirror symmetry, log geometry,...,etc. Tropical geometry has been studied for many varieties including curves, surfaces, and Abelian varieties. We extend the results on tropical Abelian varieties to super Abelian varieties, in particular, constructing super tropical theta functions.

2010 Mathematics Subject Classification: 14K10, 14K25, 14T05
Key Words and Phrases: theta functions, complex torus, super variety, tropical variety

⋅ 20th-D-14:30 − 16:00   Chair: Jaebum Sohn (Yonsei University)

⋅ 20th-D-14:30 − 14:50 The inverses of tails of the Riemann zeta function and related topics (WonTae Hwang, Donggyun Kim, Kyunghwan Song)

황원태(고등과학원), 김동균(고려대), 송경환*(이화여대 수리과학연구소)
WonTae Hwang, KIAS, Donggyun Kim, Korea University, Kyunghwan Song*, Ewha Womans University, Institute of Mathematical Sciences

In this talk, we give some results regarding the Riemann zeta function and its variations. We begin the talk by introducing (1) The Riemann zeta function and its generalized functions. (2) Some properties of the functions related to the Riemann zeta function. (3) A reciprocal sum related to the Riemann zeta function at $s = 2,3,4$ and $5$, as preliminaries. Afterwards, we present a somewhat new result on the reciprocal sum related to the Riemann zeta function at $s=6$, which is a joint work with Dr. WonTae Hwang. Also, we give some bounds of the inverses of tails of the Riemann zeta function on $0 < s < 1$ and compute the integer parts of the inverses of tails of the Riemann zeta function for $s = \frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$, which is a joint work with Prof. Donggyun Kim.

2010 Mathematics Subject Classification: 11M06, 11B83
Key Words and Phrases: Riemann zeta function, Riemann zeta function tail, bounds of Riemann zeta function tails, integer parts of Riemann zeta function tails

⋅ 20th-D-14:50 − 15:10 Minimality of 5-adic polynomial dynamics (Youngwoo Kwon, Donggyun Kim, Kyunghwan Song)

권영우*(고려대), 김동균(고려대), 송경환(이화여대)
Youngwoo Kwon*, Korea University, Donggyun Kim, Korea University, Kyunghwan Song, Ewha Womans University

There are complete characterizations of minimal polynomial maps for p=2 and p=3. In this talk, we characterize the dynamical systems consisting of the set of 5-adic integers and polynomial maps which consist of only one minimal component.

2010 Mathematics Subject Classification: 37E99, 11S85, 65P99
Key Words and Phrases: p-adic polynomial maps, full-cycle, minimality condition, p-adic integers

⋅ 20th-D-15:20 − 15:40 Some properties of $(p, q)$-Euler polynomials of the second kind (Kang Jung Yoog)

강정욱(신라대)
Kang Jung Yoog, Silla University

We use the definition of Euler polynomials of the second kind with $(p, q)$-numbers to identify some identities and properties of these polynomials. We also investigate some relationships between $(p, q)$-Euler polynomials of the second kind, $(p, q)$-Bernoulli polynomials, and $(p, q)$-tangent polynomials by using the properties of $(p, q)$-exponential function.

2010 Mathematics Subject Classification: 11B68, 11B75, 12D10
Key Words and Phrases: $(p, q)$-numbers, $(p, q)$-Euler polynomials of the second kind

⋅ 20th-D-15:40 − 16:00 On a quadratic Waring's problem with congruence conditions (Daejun Kim)

김대준(서울대)
Daejun Kim, Seoul National University

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this talk, we analyze $g_\Delta(n)$ by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We will see that the growth of $g_\Delta(n)$ is at most an exponential of $\sqrt{n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring's problem. We also determine the exact value of $g_\Delta(n)$ for each positive integer less than or equal to $4$.

2010 Mathematics Subject Classification: 11E12, 11E25
Key Words and Phrases: Waring's problem, sums of squares, representation of cosets
Analysis I

⋅ 20th-B-09:00 − 10:35   Chair: Hun Hee Lee (Seoul National University)

⋅ 20th-B-09:00 − 09:20 On the time dependence of the rate of convergence towards Hartree dynamics for interacting Bosons (Jinyeop Lee)

이진엽(한국과학기술원)
Jinyeop Lee, KAIST

We consider interacting $N$-Bosons in three dimensions. It is known that the difference between the many-body Schr$\ddot{o}$dinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order $1/N$. We investigate the time dependence of the difference. To have sub-exponential bound, we use the results of time decay estimate for small initial data. We also refine time dependent bound for singular potential using Strichartz estimate. We consider the interaction potential $V(x)$ of type $\lambda\exp(-\mu|x|)|x|^{-\gamma}$ for $\lambda\in\mathbb{R}$, $\mu\geq0$, and $0<\gamma<3/2$, which covers the Coulomb and Yukawa interaction.

2010 Mathematics Subject Classification: 81V70, 82C10, 81U30
Key Words and Phrases: many body quantum dynamics, Hartree equation, rate of convergence, mean field limit

⋅ 20th-B-09:20 − 09:40 Bishop-Phelps-Bollob\'as property and absolute sums (Mingu Jung, Yun Sung Choi, Sheldon Dantas, Miguel Mart\'in)

정민구*(포항공대), 최윤성(포항공대), Sheldon Dantas(포항공대), Miguel Mart\'in (Univ. of Granadav)
Mingu Jung*, POSTECH, Yun Sung Choi, POSTECH, Sheldon Dantas, POSTECH, Miguel Mart\'in, University of Granada

In this paper we study conditions assuring that the Bishop-Phelps-Bollob\'as property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair $(X,Y)$ of Banach spaces having the BPBp, if $Y_1$ is an absolute summand of $Y$, then $(X,Y_1)$ has the BPBp; if $X_1$ is an absolute summand of $X$ of type $1$ or $\infty$, then $(X_1,Y)$ has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm attaining operators are also given. We also show that the Bishop-Phelps-Bollob\'as property for numerical radius is inherited by absolute summands of type $1$ or $\infty$. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.

2010 Mathematics Subject Classification: 46B04, 46B20, 46E40, 47A12
Key Words and Phrases: Bishop-Phelps theorem, Bishop-Phelps-Bollob\'as property, norm attaining operators, absolute sums

⋅ 20th-B-09:50 − 10:10 Some versions of norm attainment for Lipschitz maps (Geunsu Choi, Yun Sung Choi, Miguel Mart\'in)

최근수*(포항공대), 최윤성(포항공대), Miguel Mart\'in(Univ. of Granada)
Geunsu Choi*, POSTECH, Yun Sung Choi, POSTECH, Miguel Mart\'in, University of Granada

We introduce several versions of the set of norm attaining Lipschitz maps, and present positive results for each set to be dense in ${\mathrm{Lip}}_0(X,Y)$. We mainly deal with compact Lipschitz maps in order to derive more general results such as the pair of Banach spaces $(X,Y)$ having the strong local directional Bishop-Phelps-Bollob\'as property or the local directional Bishop-Phelps-Bollob\'as point property for compact Lipschitz maps, which are stronger results than being dense in ${\mathrm{Lip}}_0(X,Y)$.

2010 Mathematics Subject Classification: 46B04, 26A16, 46B20, 46B25
Key Words and Phrases: Banach space, norm attainment, Lipschitz map, Lipschitz functional, uniformly convex Banach space

⋅ 20th-B-10:10 − 10:30 An extension theorem of holomorphic functions on hyperconvex domains (Seungjae Lee, Yoshikazu Nagata)

이승재*(포항공대), Yoshikazu Nagata(Nagoya Univ.)
Seungjae Lee*, POSTECH, Yoshikazu Nagata, Nagoya University

In this talk, we consider a variation of the Hartogs extension theorem. We obtain the following theorem: Let $\Omega$ be a bounded smooth domain in $\mathbb{C}^n, ~n \geq 3$ which has a smooth plurisubharmonic defining function on $\overline{\Omega}$. Then any holomorphic function on a connected open set in $\overline{\Omega}$ of the closure of $\{ z \in \partial \Omega : \text{the Levi form of} ~\varphi ~\text{at} ~z ~\text{is of rank at least} ~ n-2 \}$ in $ \partial \Omega$ can be extended to the whole domain $\Omega$. To obtain the above theorem, we combine a Donnelly-Fefferman type estimate for $(n,n-1)$ and $(n,n)$ forms and dualities between $L^2-$ Dolbeault cohomologies. Also, we obtain the improvement of Y. Tiba's theorems in arXiv:1706.01441v2 by applying the above theorem.

2010 Mathematics Subject Classification: 32A10, 32D15, 32U10
Key Words and Phrases: Hartogs extension theorem, Plurisubharmonic functions, Donnelly-Fefferman type estimate, Serre duality

⋅ 20th-B-10:30 − 10:35 [Contributed Talk(5min)+Poster Session] Weighted Fock spaces and their induced metric (Hyunil Choi)

최현일(부산대)
Hyunil Choi, Pusan National University

In this presentation, we consider weighted Fock spaces $F_s^2$ with any real number $s$ and sign of holomorphic sectional curvature of a induced metric by a integral kernel function of $F_s^2$. Contrast to positive order cases, we meet some trouble when we deal with negative order cases. So we suggest another weighted Fock spaces which is equivalent to weighted Fock spaces of negative order and convenient to calculate holomorphic sectional curvature of their induced metric.

2010 Mathematics Subject Classification: 32A36, 53C55
Key Words and Phrases: weighted Fock spaces, Bergman metric, K$\ddot{\rm{a}}$hler metric, holomorphic sectional curvature

⋅ 20th-C-10:40 − 11:50   Chair: Dong Hyun Cho (Kyonggi University)

⋅ 20th-C-10:40 − 11:00 The square root problem and the Aluthge transform of unilateral weighted shifts (Jaewoong Kim)

김재웅(육군사관학교)
Jaewoong Kim, Korea Military Academy

In this talk we consider the Square Root Problem for measures: Given a positive probability Borel measure $\mu$ (supported on an interval $[a,b] \subseteq \mathbb{R}_{+}$), does there exist a positive Borel measure $\nu$ such that $\mu=\nu\ast \nu$ holds? (Here $\ast$ denotes the multiplicative convolution, properly defined on $\mathbb{R}_{+}$.) This problem is intimately connected to the subnormality of the Aluthge transform of a unilateral weighted shift. We provide a concrete solution for the case of a finitely atomic measure having at most five atoms. In addition, we sharpen the statement of a previous result on this topic and extend its applicability.

2010 Mathematics Subject Classification: 15B48, 47B20, 47B37
Key Words and Phrases: Aluthge transform, the Square Root Problem for measures, subnormal weighted shifts, finitely atomic measures, multiplicative convolution

⋅ 20th-C-11:00 − 11:20 Sufficient conditions for Carath\'eodory functions and applications (Young Jae Sim, Oh Sang Kwon, Nak Eun Cho)

심영재*(경성대), 권오상(경성대), 조낙은(부경대)
Young Jae Sim*, Kyungsung University, Oh Sang Kwon, Kyungsung University, Nak Eun Cho, Pukyong National University

In this talk we derive several sufficient conditions for a function to be the Carath\'eodory function in the unit disk $\mathbb{D}:=\{ z\in\mathbb{C}: |z|<1 \}$. More precisely, for given $\beta \in (-\pi/2,\pi/2)$, $\gamma \in [0,\cos\beta)$ and $\delta\in(0,\pi/2]$, we find some sufficient conditions for an analytic function $p$ such that $p(0)=1$ to satisfy ${\rm{Re}}\{ {\rm e}^{-{\rm i}\beta} p(z) \} > \gamma$ or $| \arg \{p(z)-\gamma\} |<\delta$ for all $z\in\mathbb{D}$ by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness.

2010 Mathematics Subject Classification: 30C45
Key Words and Phrases: Carath\'eodory functions, differential subordination, starlike functions, spirallike functions, strongly starlike functions

⋅ 20th-C-11:30 − 11:50 Some types of almost periodic functions and measure theory (Hyun Mork Lee, Chan Mi Yun)

이현목*(한밭대), 윤찬미(한밭대)
Hyun Mork Lee*, Hanbat National University, Chan Mi Yun, Hanbat National University

In this talk, we investigate a new concept of Stepanov weighted pseudo almost periodic functions using the method of measure theory established by Ezzinbi et al. recently. This method represent the concept of weighted ergodic functions which is more general than the classical results. Furthermore, we study the uniqueness and existence of mild solutions of some evolution equations with nondense domain on Banach space.

2010 Mathematics Subject Classification: 34A12, 34K06
Key Words and Phrases: Stepanov weighted pseudo almost periodic, measure theory, Ergodicity, $\mu$-pseudo almost periodic, contraction mapping principle

⋅ 20th-D-14:30 − 16:00   Chair: Inbo Sim (University of Ulsan)

⋅ 20th-D-14:30 − 14:50 Four solutions for 2m-Laplacian jumping problem crossing two eigenvalues (Q-Heung Choi, Tacksun Jung)

최규흥*(인하대), 정택선(군산대)
Q-Heung Choi*, Inha University, Tacksun Jung, Kunsan National University

This paper is dealt with 2m-Laplacian jumping problem with nonlinearities crossing eigenvalues by using geometric mapping on the finite dimensional reduced subspace. We get one theorem which shows at least four solutions for 2m-Laplacian jumping problem with nonlinearities crossing two eigenvalues. We obtain this result by finite dimensional reduction method and geometric mapping on the finite reduced subspace.

2010 Mathematics Subject Classification: 35A01, 35A16, 35J30, 35J40, 35J60
Key Words and Phrases: 2m-Laplacian boundary value problem, 2m-Laplacian eigenvalue problem, jumping nonlinearity, finite dimensional reduction method, geometric mapping on the finite reduced subspace

⋅ 20th-D-14:50 − 15:10 Existence of mild solutions in the alpha-norm for some functional integrodifferential equations with nonlocal conditions (Yoon Hoe Goo, Dong Man Im, Chun Mi Ryu)

구윤회*(충남대), 임동만(청주대), 유춘미(충남대)
Yoon Hoe Goo*, Chungnam National University, Dong Man Im, Cheongju University, Chun Mi Ryu, Chungnam National University

In this talk, we investigate the existence of mild solutions in the alpha-norn for some functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space X and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

2010 Mathematics Subject Classification: 34K06, 34K20, 34K30, 47D06
Key Words and Phrases: an analytic semigroup, alpha-norm, nonlocal conditions, partial functional integrodifferential equations

⋅ 20th-D-15:20 − 15:40 A Banach algebra determined by series of measures over paths (Dong Hyun Cho)

조동현(경기대)
Dong Hyun Cho, Kyonggi University

Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$. In this talk, introduce a Banach algebra $\bar{\mathcal S}_{\alpha,\beta;\varphi}^{\prime\prime}$ which is defined over paths in $C[0,T]$ and consists of series of generalized Fourier-Stieltjes transforms of measures on $\Delta_n\times \mathbb R^n$, where $\Delta_n$ is a simplex. Then, we will prove that $\bar{\mathcal S}_{\alpha,\beta;\varphi}^{\prime\prime}$ is continuously embedded in $\bar{\mathcal S}_{\alpha,\beta;\varphi}^{\prime}$ which is the Banach algebra of Fourier-Stieltjes transforms of measures over the functions of bounded variations on $[0,T]$. As an application, we derive the analytic Feynman integrals of functions in $\bar{\mathcal S}_{\alpha,\beta;\varphi}^{\prime\prime}$ which play significant roles in Feynman integration theory and quantum mechanics.

2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: analytic Wiener integral, analytic Feynman integral, Banach algebra, It$\hat{\rm{o}}$ integral, Paley-Wiener-Zygmund integral, Wiener space

⋅ 20th-D-15:40 − 16:00 Fixed point theorem and Ulam type stability in $n$-Banach spaces (Hark-Mahn Kim, Hwan Yong Shin)

김학만(충남대), 신환용*(충남대)
Hark-Mahn Kim, Chungnam National University, Hwan Yong Shin*, Chungnam National University

In the last few decades, many stability results of various functional and differential, difference, integral equations have been investigated by many mathematicians but mainly in classical spaces. However, the notion of an approximate solution of two functions can be understood in a particular situation. One of such non-classical measures of a distance can be introduced by the notion of the n-norm. In this talk, we investigate some known fixed point theorems and stability results of various equations in $n$-Banach spaces.

2010 Mathematics Subject Classification: 39B52, 39B72, 39B82
Key Words and Phrases: Ulam stability, fixed point theorem, $n$-Banach space, difference equation, functional equation
Analysis II

⋅ 20th-C-10:40 − 12:10 Chair: Jinhae Park (Chungnam National University)

⋅ 20th-C-10:40 − 11:00 Weighted norm estimates of the dyadic operators with VMO functions (Daewon Chung)

정대원(계명대)
Daewon Chung, Keimyung University

In this talk, we primary interested in reducing the dependence of the $A_p$-chraracteristic $[w ]_{A_p}$ in the weighted norm estimates for the commutators of the Hilbert transform as well as the dyadic paraproduct. It is now well-known fact that the commutators $[b, H ]$ where $H$ is the Hilbert transform and $b$ is a function in BMO obey the quadratic dependence on the $A_2$ characteristic of the weight and also the dependence is optimal. However we can reduce the dependence of the weight characteristic by choosing a function in VMO which was introduced by D. Sararon. The functions in BMO are characterized by the boundedness of their mean oscillation over interval. The function in VMO on the circle $\mathbb{T}$ are those with the additional property that their mean oscillations over small intervals are small. The space VMO is a closed subspace of BMO and contains all uniformly continuous functions in BMO. The analogue of VMO on the real line was defined by Coifman and Weiss, where they proved that it is the predual of the Hardy space $H^1$. The precise definition of VMO and several alternative characterizations of it will be given in the presentation. During the talk, the new results regarding the weighted norm estimate of the dyadic operators with VMO functions and their brief proof will be delivered.

2010 Mathematics Subject Classification: 42B20, 42B25
Key Words and Phrases: weighted norm estimate, commutator, dyadic paraproduct, vanishing mean oscillation

⋅ 20th-C-11:00 − 11:20 Hyponormal Toeplitz operators with non-harmonic symbols (jongrak Lee, Eungil Ko)

이종락*(이화여대), 고응일(이화여대)
jongrak Lee*, Ewha Womans University, Eungil Ko, Ewha Womans University

In this talk, on the weighted Bergman space, we study the hyponormal Toeplitz operators $T_{\varphi}$ with symbol $\varphi=a_{m} z^{m} +a_N z^N+\overline{{a_{-m}} z}^{m} +\overline{a_{-N} z}^N$. We present necessary and sufficient conditions for the hyponormality of $T_{\varphi}$ under some assumptions about the coefficients of $\varphi$. Next, we consider hyponormality of $T_{\varphi}$ with non-harmonic symbol $\varphi$.

2010 Mathematics Subject Classification: 47B20, 47B35
Key Words and Phrases: Toeplitz operators, hyponormal, weighted Bergman space

⋅ 20th-C-11:30 − 11:50 On complex symmetric Toeplitz operators on the weighted Bergman space (Ji Eun Lee, Eungil Ko, Jongrak Lee)

이지은*(세종대), 고응일(이화여대), 이종락(이화여대)
Ji Eun Lee*, Sejong University, Eungil Ko, Ewha Womans University, Jongrak Lee, Ewha Womans University

In this paper, we give a characterization of a complex symmetric Toeplitz operator $T_{\varphi}$ on the weighted Bergman space $A^2_{\alpha}(\Bbb D)$. First, we state properties of complex symmetric Toeplitz operators $T_{\varphi}$ on $A^2_{\alpha}(\Bbb D)$. Next, we characterize the symbol functions $\varphi$ for which Toeplitz operators $T_{\varphi}$ on $A^2_{\alpha}(\Bbb D)$ are complex symmetric. Finally, we consider differences between complex symmetric Toeplitz operators on $A^2_{\alpha}(\mathbb D)$ and the Hardy space $H^2$.

2010 Mathematics Subject Classification: 47B35, 47B15, 47A05
Key Words and Phrases: complex symmetric operator, Toeplitz operator, normal operator, weighted Bergman space

⋅ 20th-C-11:50 − 12:10 Multiplicity results for $p$-Laplacian boundary value problem with jumping nonlinearities (Tacksun Jung, Q-Heung Choi)

정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University

We investigate multiplicity of solutions for one dimensional $p$-Laplacian Dirichlet boundary value problem with jumping nonlinearites. We obtain three theorems: The first one is that there exists exactly one solution when nonlinearities cross no eigenvalue. The second one is that there exist exactly two solutions, exactly one solutions and no solution depending on the source term when nonlinearities cross one first eigenvalue. The third one is that there exist at least three solutions, exactly one solutions and no solution depending on the source term when nonlinearities cross the first and second eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the $p$-Laplacian eigenvalue problem, and the contraction mapping principle on $p$-Lebesgue space. We obtain the third result by Leray-Schauder degree theory.

2010 Mathematics Subject Classification: 35A01, 35A16, 35J30, 35J40, 35J60
Key Words and Phrases: p-Laplacian problem, p-Laplacian eigenvalue problem, jumping nonlinearity, contraction mapping principle, Leray-Schauder degree theory

⋅ 20th-D-14:30 − 16:00 Chair: Youngae Lee (NIMS)

⋅ 20th-D-14:30 − 14:50 Contact discontinuities for 2-D inviscid compressible flows in infinitely long nozzles (Myoungjean Bae, Hyangdong Park)

배명진(포항공대), 박향동*(포항공대)
Myoungjean Bae, POSTECH, Hyangdong Park*, POSTECH

In this talk, recent results on subsonic weak solutions to steady Euler system with contact discontinuities and nonzero vorticity will be presented.

2010 Mathematics Subject Classification: 35J47, 35J57, 35J66, 35Q31, 35R35, 74J40, 76N10
Key Words and Phrases: asymptotic state, compressible, contact discontinuity, free boundary problem, Helmholtz decomposition, infinite nozzle, inviscid, steady Euler system, subsonic, vorticity

⋅ 20th-D-14:50 − 15:10 Stability of 3-D axisymmetric transonic shock solutions of the full Euler system in divergent nozzles (Yong Park)

박용(포항공대)
Yong Park, POSTECH

R. Courant and K. Friedrich in Supersonic flow and shock waves (1948) describe transonic shock phenomena for an invicid compressible flow in convergent-divergent type nozzle so called de Laval nozzle: when the flow after the throat of the nozzle is supersonic, if an appropriate large pressure $p_c$ is given at the exit, then the flow is compressed and slow down to the subsonic speed. The position and the strength of the shock front are automatically adjusted so that the end pressure at the exit becomes $p_c$. Related with this phenomenon, there was a question that transonic shock flows in divergent nozzles are stable and still it does not have a full answer. In this talk, we present the recent result about the stability of steady 3-D axisymmetric transonic shock flows for the full Euler system in divergent nozzles. The main progress is the development of determining a shock location process for a steady flow for the full Euler system and resolving a singularity issue appearing in our approach to dealing with 3-D axisymmetric transonic shock flow. We will discuss these in this talk.

2010 Mathematics Subject Classification: 35J57, 35M10, 35Q31, 35R35, 76H05, 76N10
Key Words and Phrases: transonic shock, full Euler system, axisymmetric flow, free boundary problem, elliptic system

⋅ 20th-D-15:20 − 15:40 Blow-up solutions to nonlinear parabolic equations with non-autonomous reactions under the mixed boundary conditions (Soon-Yeong Chung, Jaeho Hwang)

정순영(서강대), 황재호*(서강대)
Soon-Yeong Chung, Sogang University, Jaeho Hwang*, Sogang University

In this paper, we study blow-up solutions to nonlinear parabolic equations \begin{equation*} \begin{cases} (b(u(x,t))_{t}=\nabla\cdot(\rho(u(x,t)\nabla u(x,t)+f(u(x,t),t), & (x,t)\in\Omega\times (0,t^{*}),\\ \mu(z)\frac{\partial u}{\partial n}(z,t)+\sigma(z)u(z,t)=0, & (z,t)\in\partial\Omega\times[0,t^{*}),\\ u\left(x,0\right)=u_{0}(x)\geq 0, & x\in \Omega, \end{cases} \end{equation*} In order to obtain blow-up solutions, we introduce a new condition $$\begin{aligned} &(C_{\rho}\,1)\hspace{5mm} \alpha(t) \int_{0}^{u}f(s,t)\rho(s)ds \leq uf(u,t)\rho(u)+\beta(t) u^{2}+\gamma(t),\,\,u>0,\,\,t>0,\\ &(C_{\rho}\,2)\hspace{5mm}\int_{0}^{u}f_{t}(s,t)ds\geq \frac{\alpha(t)\gamma'(t)-\alpha'(t)\gamma(t)}{\alpha^{2}(t)},\,\,\,u>0,\,\,t>0, \end{aligned}$$ for some nonnegative functions $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ with $$\inf_{s> 0}\alpha(s)>2\,\, \text{and}\,\,0\leq\beta(t)\leq\left[\frac{\alpha(t)}{2}-1\right] \lambda_{0}\rho_{m}^{2}, \,\,t>0, $$ where $\lambda_{0}$ is the first eigenvalue for the Laplace operator $\Delta$ and $\rho_{m}:=\inf_{s> 0} \rho(s)$.

2010 Mathematics Subject Classification: 35K55, 35K57, 35B44
Key Words and Phrases: nonlinear parabolic equation, blow-up conditions, mixed boundary

⋅ 20th-D-15:40 − 16:00 On the critical set for Fujita type blow-up of solutions to the discrete Laplacian parabolic equations with nonlinear source on networks (Soon-Yeong Chung, Min-Jun Choi, Jea-Hyun Park)

정순영(서강대), 최민준*(서강대), 박재현(군산대)
Soon-Yeong Chung, Sogang University, Min-Jun Choi*, Sogang University, Jea-Hyun Park, Kunsan National University

In this paper, we are interested in long time behaviors of solutions to the discrete Laplacian parabolic equations $u_t = \Delta_\omega u + \psi f(u)$ with nonnegative and non-trivial initial data. In particular, we assume that the function $f$ is convex only on a short interval and $f(\alpha s) \approx f(\alpha)f(s)$ for $0<\alpha<1$, $s>0$, and we present a critical set depending on the function $\psi$ in the sense that if $f$ is in the critical set, then solutions blow up in finite time for any initial data, and if not, then solutions are global or blow up according to the size of initial data.

2010 Mathematics Subject Classification: 39A12, 39A13, 39A70
Key Words and Phrases: discrete Laplacian, Fujita blow-up, critical set, critical exponent
Geometry

⋅ 20th-D-14:30 − 15:10 Chair: Jinsung Park (KIAS)

⋅ 20th-D-14:30 − 14:50 Equidistant hypersurfaces in the biDisk (Youngju Kim)

김영주(건국대)
Youngju Kim, Konkuk University

Equidistant hypersurfaces between two points are basic objects in a metric space. In many rank $1$ symmetric spaces, completely different sets of points share a common equidistant hypersurface. However, generically, an equidistant hypersurface is determined by a unique pair of points in the biDisk $\mathbf{H}^2 \times \mathbf{H}^2$ which is a rank $2$ geometry.

2010 Mathematics Subject Classification: 51B10, 57M50, 57M60, 53A35
Key Words and Phrases: biDisk, rank 2 geometry, equidistant hypersurface

⋅ 20th-D-14:50 − 15:10 Properties of solitons for inverse mean curvature flow (Daehwan Kim)

김대환(고등과학원)
Daehwan Kim, KIAS

The inverse mean curvature flow has been studied not only the flow itself as a geometric flow, but also for its applications to prove various inequalities. Inverse mean curvature flow is the deformation of a submanifold in the normal direction according to the inverse value of the mean curvature with opposite sign. One of the ways to understand the flow is to analyze its solitons as special solutions, which are the homothetic and translating solitons in this talk. The homothetic soliton and translating soliton for the flow are self-similar solutions deformed by only homothetic and translation under the flow, respectively. To be specific, several examples of such solitons are provided, their incompleteness in particular cases are proved and their area growths are obtained.

2010 Mathematics Subject Classification: 53A10, 53C44
Key Words and Phrases: inverse mean curvature flow, soliton, incompleteness
Topology

⋅ 20th-B-09:00 − 10:30   Chair: Dae-Woong Lee (Chonbuk National University)

⋅ 20th-B-09:00 − 09:20 Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $a,b,c \to \infty$ (Weonmo Lee, Yong-Geun Oh)

이원모*(포항공대 / 기초과학연구원 기하학 수리물리 연구단), 오용근(기초과학연구원 기하학 수리물리 연구단 / 포항공대)
Weonmo Lee*, POSTECH / IBS-Center for Geometry and Physics, Yong-Geun Oh, IBS-Center for Geometry and Physics / POSTECH

Vianna constructed a family of infinitely many non-Hamiltonian isotopic monotone Lagrangian tori in $\mathbb{CP}^2$. Generalizing Mandini and Pabiniak's symplectic embedding result to almost toric manifolds and analyzing base diagrams(in our case a triangle), we prove that some 4-ball of positive radius, independent of triple $(a,b,c)$, symplectically embeddes into $\mathbb{CP}^2$ without touching Vianna's infinitely many tori.

2010 Mathematics Subject Classification: 53D20, 53D12
Key Words and Phrases: almost toric fibration, symplectic rational blow-down, relative Gromov capacity, base diagram

⋅ 20th-B-09:20 − 09:40 Periodic shadowing property for induced maps on hyperspaces (Namjip Koo, Nyamdavaa Tsegmid)

구남집(충남대), Nyamdavaa Tsegmid*(충남대 / Mongolian National Univ. of Education)
Namjip Koo, Chungnam National University, Nyamdavaa Tsegmid*, Chungnam National University / Mongolian National University of Education

Let $f:X\to X$ be a continuous map on compact metric spaces $X$ with metric $d$. We introduce the periodic shadowing property for the induced maps on hyperspaces, in particularly $F(X)$ and the hyperspace $2^X$ of compact subsets of $X$. We prove the following: (i) $f$ has the periodic shadowing property if and only if $f^{<\omega}:F(X)\to F(X)$ has the periodic shadowing property; (ii) if $f$ has the periodic shadowing property, then the induced map $2^f:2^X\to 2^X$ has the periodic shadowing property; (iii) $f$ has the FinASP if and only if $2^f$ has the FinASP. Also, we give some examples to illustrate our results.

2010 Mathematics Subject Classification: 37C50, 54H20, 37B99, 37B50
Key Words and Phrases: pseudo orbit, periodic pseudo orbit, finite average shadowing, average pseudo orbit, hyperspace map

⋅ 20th-B-09:50 − 10:10 On relation between 2nd cohomology groups of a quandle and its inner automorphism group (Yongju Bae, J. Scott Carter, Byeorhi Kim)

배용주(경북대), J. Scott Carter(Osaka City Univ., Advanced Mathematical Institute), 김벼리*(경북대)
Yongju Bae, Kyungpook National University, J. Scott Carter, Osaka City University, Advanced Mathematical Institute, Byeorhi Kim*, Kyungpook National University

In 2003, J. S. Carter, M. Elhamdadi, M. A. Nikiforou and M. Saito introduced the theory of quandle extension by a quandle 2-cocycle, and many researchers have been studying properties of quandle extensions. According to D. Joyce and S. Matveev, every quandle can be represented by using its automorphism group, so, we have a sequence of groups from a quandle extension. In this talk, we begin studying the mod-2 quandle extension of the 4-elements tetrahedral quandle that is defined by a quandle cocycle in terms of the inner automorphism groups of each. We also observe relationship between quandle 2-cocycle and group 2-cocycle in the example. This is a joint work with Y. Bae and J. S. Carter.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: quandle, quandle extension, quandle 2-cocycle, quandle 2nd cohomology group

⋅ 20th-B-10:10 − 10:30 Theta curves and handcuff graphs with small lattice stick numbers (Sungjong No, Seungsang Oh, Hyungkee Yoo)

노성종(고려대), 오승상(고려대), 유형기*(고려대)
Sungjong No, Korea University, Seungsang Oh, Korea University, Hyungkee Yoo*, Korea University

The lattice graph is an embedding of graph into cubic lattice $\mathbb{Z}^3$ of $\mathbb{R}^3$. The lattice stick number $s_L(G)$ of a spatial graph $G$ is defined to be the minimal number of straight line segments required to construct a lattice graph in the cubic lattice. In this paper, we focus on the specific graphs which are theta-curve and handcuff graph. We mathematically prove that there are six types of spatial graphs with less than $14$ lattice sticks corresponding to theta-curve and handcuff graph. We also get the exact lattice stick number for the previous six types of spatial graphs.

2010 Mathematics Subject Classification: 57M15, 57M25, 57M27
Key Words and Phrases: lattice stick number, theta curve, handcuff graph

⋅ 20th-C-10:40 − 12:10   Chair: Sang Youl Lee (Pusan National University)

⋅ 20th-C-10:40 − 11:00 Pairings and mirror symmetry (Cheol-Hyun Cho, Sangwook Lee, Hyungseok Shin)

조철현*(서울대), 이상욱(고등과학원), 신형석(고등과학원)
Cheol-Hyun Cho*, Seoul National University, Sangwook Lee, KIAS, Hyungseok Shin, KIAS

Given a (homological) mirror symmetry between a symplectic manifold and its Landau-Ginzburg mirrors, we may ask whether mirror symmetry preserves pairing structures. For a symplectic manifold, Poincare duality provides pairings for both open and closed theories. A matrix factorization category has the Kapustin-Li pairing and Jacobian ring has a residue pairing. We use a localized mirror formalism (developed in joint works with Hong and Lau) to find an interesting conformal factor arises between these pairings under (homological) mirror symmetry.

2010 Mathematics Subject Classification: 53D37, 14J33
Key Words and Phrases: homological mirror symmetry, Jacobian ring, Kapustin-Li pairing

⋅ 20th-C-11:00 − 11:20 Twisted Fukaya algebras and mirror symmetry of Riemann surfaces (Cheol-Hyun Cho, Sangwook Lee)

조철현(서울대), 이상욱*(고등과학원)
Cheol-Hyun Cho, Seoul National University, Sangwook Lee*, KIAS

Given a symplectic manifold with a weakly unobstructed Lagrangian submanifold $\mathbb{L}$ and a finite abelian group action, we define the twisted Floer homology of $\mathbb{L}$. Then we show that it is an $A_\infty$-algebra and propose a conjecture that its cohomology algebra satisfies the axiom of orbifold Jacobian algebras due to Kaufmann, Basalaev-Takahashi-Werner et al. Finally we show that how it can be employed to see the mirror symmetry of Riemann surfaces.

2010 Mathematics Subject Classification: 53D37
Key Words and Phrases: orbifold Jacobian algebra, twisted Fukaya algebra

⋅ 20th-C-11:30 − 11:50 On symplectic fillings of small Seifert $3$-manifolds (Hakho Choi, Jongil Park)

최학호*(고등과학원), 박종일(서울대)
Hakho Choi*, KIAS, Jongil Park, Seoul National University

In this talk we study a topological surgery description for symplectic fillings of small Seifert $3$-manifolds with a canonical contact structure. As a result, we show that every minimal symplectic filling of small Seifert $3$-manifolds satisfying certain conditions can be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity. This is joint work with Prof. Jongil Park.

2010 Mathematics Subject Classification: 53D05, 57R17, 32S25
Key Words and Phrases: rational blowdown, small Seifert $3$-manifold, symplectic filling

⋅ 20th-C-11:50 − 12:10 Uniqueness of real Lagrangians up to cobordism (Joontae Kim)

김준태(고등과학원)
Joontae Kim, KIAS

We explore the topology of real Lagrangian submanifolds in symplectic manifolds towards their uniqueness and classification. We prove that a real Lagrangian in a closed symplectic manifold is unique up to smooth cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C}P^2$ and $S^2\times S^2$. Finally, we explain why it is tempting to conjecture that a real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus.

2010 Mathematics Subject Classification: 53D12, 57N70, 55M35
Key Words and Phrases: real Lagrangian submanifold, antisymplectic involution, cobordism, Smith theory

⋅ 20th-D-14:30 − 16:00   Chair: Namjip Koo (Chungnam National University)

⋅ 20th-D-14:30 − 14:50 The configuration space of equidistant points in the Heisenberg group (Joonhyung Kim, Ioannis Platis)

김준형*(전주대), Ioannis Platis(Univ. of Crete)
Joonhyung Kim*, Jeonju University, Ioannis Platis, University of Crete

We discuss the configuration space of equidistant points in the Heisenberg group.

2010 Mathematics Subject Classification: 22E40, 32M15, 57M50, 57S30
Key Words and Phrases: Heisenberg group, Koranyi metric, equidistant triples

⋅ 20th-D-14:50 − 15:10 Minor minimal bipartite intrinsically knotted graphs with 23 edges (Hyoungjun Kim, Seungsang Oh, Thomas Mattman)

김형준*(이화여대), 오승상(고려대), Thomas Mattman(California State Univ.-Chico)
Hyoungjun Kim*, Ewha Womans University, Seungsang Oh, Korea University, Thomas Mattman, California State University-Chico

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. It is already known that intrinsically knotted graphs have at least 21 edges, and K7 and the 13 graphs obtained from K7 by ∇-Y moves are the only intrinsically knotted graphs with 21 edges. There are exactly two minor minimal bipartite intrinsically knotted graphs with at most 22 edges. Moreover, there are no minimal intrinsically knotted graphs with 22 edges that are bipartite. The goal of this talk is to show that there are no minor minimal intrinsically knotted graphs with 23 edges that are bipartite.

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: spatial graph, intrinsic knotting, bipartite intrinsically knotted

⋅ 20th-D-15:20 − 15:40 An enumeration of immersed surface-links (Jieon Kim)

김지언(부산대)
Jieon Kim, Pusan National University

A surface-link is a closed surface embedded in $\Bbb R^4$. An immersed surface-link is a closed surface immersed in $\Bbb R^4$ such that the multiple points are transverse double points. Surface-links and immersed surface-links can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams (cf. [1, 2, 4]). K. Yoshikawa enumerated surface-links in $\Bbb R^4$. In this paper [3], we consider the enumeration problem of immersed surface-links in $\Bbb R^4$.
[1] S. Kamada, A. Kawauchi, J. Kim, and S. Y. Lee, Presentation of immersed surface-links by marked graph diagrams, ArXiv e-prints, July 2017.
[2] A. Kawauchi, T. Shibuya, and S. Suzuki, Descriptions on surfaces in four-space, I; Normal forms, Math. Sem. Notes Kobe Univ. $\bf 10$ (1982), 75--125.
[3] J. Kim, An enumeration of immersed surface-links in $\Bbb R^4$, in preperation.
[4] K. Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. $\bf 31$ (1994), 497--522.

2010 Mathematics Subject Classification: 57M20
Key Words and Phrases: immersed surface-links, marked graph diagram, ch-graph

⋅ 20th-D-15:40 − 16:00 Petal presentations and torus knots (Sung Jong No, Hyoung Jun Kim)

노성종*(고려대), 김형준(이화여대)
Sung Jong No*, Korea University, Hyoung Jun Kim, Ewha Womans University

An $n$-crossing projection of knot is a projection that every crossings of the projection are $n$-tuple points. A petal projection is an $n$-crossing projection with single crossing for some $n$ that has no nesting loops. The minimum possible of $n$ is called a petal number, denoted $p(K)$. Adams introduced this concept and proved that every knot has a petal projection. In this talk, we prove that $p(K) \leq 2c(K)$ and find a bounds for some torus knots.

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: knot, link, petal presentation, torus knot
Probability and Statistics

⋅ 20th-C-11:00 − 12:10 Chair: Seo Insuk (Seoul National University)

⋅ 20th-C-11:00 − 11:20 Scaling limits of random normal matrix ensembles at the soft edge (Yacin Ameur, Nam-Gyu Kang, Seong-Mi Seo)

Yacin Ameur(Lund Univ.), 강남규(고등과학원), 서성미*(고등과학원)
Yacin Ameur, Lund University, Nam-Gyu Kang, KIAS, Seong-Mi Seo*, KIAS

In this talk, I will present recent work on the random normal matrix ensembles with a soft edge of the spectrum. This model interpolates between the free boundary case and the weakly confining case. Using the asymptotic expansions of orthonormal polynomials with respect to the exponentially varying weights, we obtain the scaling limit of local eigenvalues near a point at the soft edge and prove edge universality when the underlying potential is radially symmetric.

2010 Mathematics Subject Classification: 60B20, 60G55, 81T40, 30C40
Key Words and Phrases: random normal matrix, orthogonal polynomial, scaling limit

⋅ 20th-C-11:30 − 11:50 Local law and Tracy-Widom limit for sparse stochastic block models (Wooseok Yang, Ji Oon Lee, Jong Yun Hwang)

양우석*(한국과학기술원), 이지운(한국과학기술원), 황종연(한국과학기술원)
Wooseok Yang*, KAIST, Ji Oon Lee, KAIST, Jong Yun Hwang, KAIST

We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy--Widom law after rescaling and shifting, with an explicit formula on the deterministic shift of the spectral edge.

2010 Mathematics Subject Classification: 15B52, 60B20
Key Words and Phrases: local law, Tracy–Widom distribution, sparse random matrices, stochastic block models

⋅ 20th-C-11:50 − 12:10 Estimates and stability of heat kernels for symmetric jump processes with general mixed polynomial growths on metric measure space (Juhak Bae, Jaehoon Kang, Panki Kim, Jaehun Lee)

배주학(서울대), 강재훈(Bielefeld Univ.), 김판기(서울대), 이재훈*(서울대)
Juhak Bae, Seoul National University, Jaehoon Kang, Bielefeld University, Panki Kim, Seoul National University, Jaehun Lee*, Seoul National University

In this talk, we establish the stability of two-sided heat kernel estimates for symmetric jump Markov processes on metric measure spaces that satisfies general volume doubling condition. Our results cover Markov processes whose jumping density has mixed polynomial growths. In particular, our scaling function may not be comparable to the function which gives the growth of jumps. To obtain sharp two-sided heat kernel estimates, we need additional condition on the metric measure space, which is called the chain condition. If underlying metric measure space allows a conservative diffusion process which has the transition density with certain type of sub-Gaussian estimates, our scaling function depends on not only jump density but also walking dimension of metric measure space.

2010 Mathematics Subject Classification: 60J35, 60J75
Key Words and Phrases: Dirichlet form, symmetric Markov process, transition density, heat kernel estimates

⋅ 20th-D-14:30 − 16:00 Chair: Panki Kim (Seoul National University)

⋅ 20th-D-14:30 − 14:50 Cut-off phenomenon for random-cluster models (Seo Insuk, Ganguly Shirshendu)

서인석*(서울대), Ganguly Shirshendu(Univ. of California-Berkeley)
Seo Insuk*, Seoul National University, Ganguly Shirshendu, University of California-Berkeley

In this presentation, we discuss the cut-off phenomenon of the random-cluster model, which is also known as the FK-model. It is known that the mixing time of the Glauber dynamics of random-cluster dynamics on the discrete torus of any dimension is of order $O(N^2)$. Our work refines this result considerably by characterizing exact location of the mixing, and by demonstrating the cutoff-phenomenon around this location.

2010 Mathematics Subject Classification: 60J27, 37A25
Key Words and Phrases: Markov chain, random-cluster model, Mixing time, cutoff

⋅ 20th-D-14:50 − 15:10 Properties of free multiplicative convolution (Hong Chang Ji)

지홍창(한국과학기술원)
Hong Chang Ji, KAIST

For given two Borel probability measures $\mu$ and $\nu$ on $\mathbb R_{+}=[0,\infty)$, we derive properties of the free multiplicative convolution $\mu\boxtimes\nu$ via its Cauchy-Stieltjes transform. In particular we prove that $\mu\boxtimes\nu$ always has no singular continuous part and, under certain conditions, that the density of its absolutely continuous part is bounded by $x^{-1}$. We also consider a special case in which $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty)$ having power law behavior with exponents in $(-1,1)$. In this case, we prove that $\mu\boxtimes\nu$ is another such Jacobi measure whose density has square root decay at the edges of its support.

2010 Mathematics Subject Classification: 46L54
Key Words and Phrases: free multiplicative convolution, analytic functions, Jacobi measures

⋅ 20th-D-15:20 − 15:40 On comparison principles for stochastic heat equations (Le Chen, Kunwoo Kim)

Le Chen(Univ. of Nevada-Las Vegas), 김건우*(포항공대)
Le Chen, University of Nevada-Las Vegas, Kunwoo Kim*, POSTECH

We consider comparison principles for heat equations perturbed by space-time white noise and colored noise. We first show the pathwise comparison principle which compares two solutions for each realization. Then, we show the moment comparison principle which can compare the moments of the solutions. Those principles are very useful in understanding intermittency of the solutions.

2010 Mathematics Subject Classification: 60H15, 35R60, 60G60
Key Words and Phrases: stochastic heat equations, pathwise comparison principle, moment comparison principle

⋅ 20th-D-15:40 − 16:00 Universality for weakly non-Hermitian random matrix ensembles (Gernot Akemann, Yacin Ameur, Sung-Soo Byun)

Gernot Akemann(Bielefeld Univ.), Yacin Ameur(Lund Univ.), 변성수*(서울대)
Gernot Akemann, Bielefeld University, Yacin Ameur, Lund University, Sung-Soo Byun*, Seoul National University

In this talk, I will discuss local bulk statistics of random normal matrix models, in particular in the regime of weak non-Hermiticity. In this regime, the scaling limits for the eigenvalues are described by one parameter family of correlation kernels, which interpolate the celebrated sine and Ginibre kernel. After presenting Ward’s equation satisfied by the Berezin kernel of the weakly non-Hermitian ensemble, I will explain how to characterize the solution of such equation to prove the existence of certain universality class. This is based on joint work with Gernot Akemann and Yacin Ameur.

2010 Mathematics Subject Classification: 82D10, 60G55, 42C05
Key Words and Phrases: random normal matrices, weak non-Hermiticity, microscopic limit, Ward's equation
Applied Mathematics

⋅ 20th-C-11:30 − 12:10 Chair: Younhee Lee (Chungnam National University)

⋅ 20th-C-11:30 − 11:50 Real option pricing on finite time horizon (Sunju Lee, Younhee Lee)

이순주(충남대), 이윤희*(충남대)
Sunju Lee, Chungnam National University, Younhee Lee*, Chungnam National University

We consider a real option under a regime-switching jump-diffusion model. The purpose of the investor in the real option is to decide an optimal investment time to maximize the discounted expectation of a payoff function. In this talk, the project value is evaluated by solving a partial integro-differential equation (PIDE) and it can be expressed as the closed-form solution. Then the objective function and the optimal investment time are computed by solving a linear complementarity problem. Numerical experiments are performed to describe the various phenomena with the regime-switching process.

2010 Mathematics Subject Classification: 91Gxx
Key Words and Phrases: real option, regime-switching jump-diffusion model

⋅ 20th-C-11:50 − 12:10 Analysis on the Korean highway using transportation network and dynamics prediction algorithm of traffic on highway (Kisoeb Park, Gwangyeon Lee)

박기섭*(인천대), 이광연(한서대)
Kisoeb Park*, Incheon National University, Gwangyeon Lee, Hanseo University

This study deals with the accessibility indices of Korean highway network and the dynamic prediction algorithm of highway traffic. For accessibility indices, we find a transportation network that presents Korean highway network in graphs, and then we proposes new transportation network algorithm which can easily get such as associated number, the relative distance, the accessibility, the degree of connectivity, the index of dispersion, the diameter of graph theory. The new transportation network algorithm is easier to obtaining various accessibility indices than the existing methods. We use the cumulative distribution function (CDF) of traffic data as a new method of finding dynamic prediction algorithm for traffic on highway. The random number generation algorithm (RNGA) using CDF is a new prediction algorithm for traffic data of the actual Korean highway. By applying this algorithm, simulation can confirm from the low precision that highway traffic can be predicted very accurately.

2010 Mathematics Subject Classification: 05C85, 60H30
Key Words and Phrases: accessibility indices, transportation network algorithm, cumulative distribution function, random number generation algorithm

⋅ 20th-D-14:30 − 16:00 Chair: Eun-Hee Park (Kangwon National University)

⋅ 20th-D-14:30 − 14:50 A domain decomposition solver for biharmonic problems (Eun-Hee Park, Susanne C. Brenner, Li-yeng Sung, Kening Wang)

박은희*(강원대), Susanne C. Brenner(Louisiana State Univ.), Li-yeng Sung(Louisiana State Univ.), Kening Wang(Univ. of North Florida)
Eun-Hee Park*, Kangwon National University, Susanne C. Brenner, Louisiana State University, Li-yeng Sung, Louisiana State University, Kening Wang, University of North Florida

In this talk we will discuss a non-overlapping domain decomposition (DD) solver for biharmonic problems. There are two key ingredients in the proposed DD solver: one is a subspace decomposition of the finite element space and the other is a procedure based on balancing domain decomposition by constraints. The performance of DD solvers are mainly determined by the condition number of the resulting linear system. Theoretical results on the condition number estimate of the resulting system will be presented along with numerical results.

2010 Mathematics Subject Classification: 65N55, 65N30
Key Words and Phrases: non-overlapping domain decomposition, BDDC preconditioner, biharmonic problems

⋅ 20th-D-14:50 − 15:10 Semi-uniform grid based multigrid method for interface problems (Gwanghyun Jo, Do Y. Kwak)

조광현*(한국과학기술원), 곽도영(한국과학기술원)
Gwanghyun Jo*, KAIST, Do Y. Kwak, KAIST

We develop a new type of multigrid method for interface problems. We use the semi-uniform grid for the discretization which is obtained by refinement of the uniform grid. By using the subspace correction concept, we restrict the residual at the semi-uniform grids to the uniform grid system. Next, we apply V-cycle multigrid on uniform grids. Finally, the correction term are prolongated to the semi-uniform grids. To obtain robust results we design special type of transfer/interpolation operators. We prove the optimal scalability analysis. We show numerical results which supports our algorithms.

2010 Mathematics Subject Classification: 65N55
Key Words and Phrases: finite element method, multigrid algorithms, optimal scalability

⋅ 20th-D-15:20 − 15:40 Dirichlet-to-Neumann boundary conditions for multiple scattering problems in waveguides (Youngho Min, Seungil Kim)

민영호*(경희대), 김승일(경희대)
Youngho Min*, Kyung Hee University, Seungil Kim, Kyung Hee University

In this work, we consider a Helmholtz equation in a domain with multiple obstacles and cavities with straight waveguides placed between them. We derive a multiple DtN condition for solutions in the straight waveguide, which allows us to have a smaller problem by removing the straight waveguide and imposing the MDtN conditions on artificial boundaries instead. In this process, the computational domain is reduced, so a better computational cost is given. We prove well-posedness of this reduced problem. In addition, it is shown that a problem equipped with the truncated MDtN operators is well-posed and approximate solutions converge exponentially in terms of the number of terms of truncated MDtN operators. To demonstrate these analyses, we give several numerical examples.

2010 Mathematics Subject Classification: 65N12
Key Words and Phrases: multiple scattering, Helmholtz equation, Dirichlet-to-Neumann condition, waveguide

⋅ 20th-D-15:40 − 16:00 Graph signal denosing by using data approximation methods (Yeon Ju Lee, Qiyu Sun)

이연주*(고려대), Qiyu Sun(Univ. of Central Florida)
Yeon Ju Lee*, Korea University, Qiyu Sun, University of Central Florida

In this paper we represent a graph signal filtering for graph sinal denoising. Numerical experimental results validate our graph signal processing-based approach for images and graph signals.

2010 Mathematics Subject Classification: 05C62
Key Words and Phrases: graph signal, denoising
Discrete Mathematics

⋅ 20th-C-10:40 − 12:10 Chair: Seog-Jin Kim (Konkuk University)

⋅ 20th-C-10:40 − 11:00 On strong Sidon sets of integers (Yoshiharu Kohayakawa, Sang June Lee, Carlos Gustavo Moreira, Vojt$\check{v}$ech R$\ddot{o}$dl)

Yoshiharu Kohayakawa(Univ. of Sao Paulo), 이상준*(덕성여대), Carlos Gustavo Moreira(Emory Univ.), Vojt$\check{\rm{v}}$ech R$\ddot{\rm{o}}$dl(IMPA)
Yoshiharu Kohayakawa, University of Sao Paulo, Sang June Lee*, Duksung Women's University, Carlos Gustavo Moreira, Emory University, Vojt$\check{\rm{v}}$ech R$\ddot{\rm{o}}$dl, IMPA

Let $\mathbb N$ be the set of natural numbers. A set $A\subset \mathbb N$ is called a \emph{Sidon set} if the sums $a_1+a_2$, with $a_1,a_2\in S$ and $a_1\leq a_2$, are distinct, or equivalently, if \begin{equation*} |(x+w)-(y+z)|\geq 1 \end{equation*} for every $x,y,z,w\in S$ with $x<y\leq z<w$. We define strong Sidon sets as follows: For a constant $\alpha$ with $0\leq \alpha<1$, a set $S\subset \mathbb N$ is called an \emph{$\alpha$-strong Sidon set} if \begin{equation*} |(x+w)-(y+z)|\geq w^\alpha \end{equation*} for every $x,y,z,w\in S$ with $x<y\leq z<w$. The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of $\mathbb N$. In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojt$\check{\rm{v}}$ech R$\ddot{\rm{o}}$dl.

2010 Mathematics Subject Classification: 05A16
Key Words and Phrases: Sidon set

⋅ 20th-C-11:00 − 11:20 Excess considerations on integer partitions (Byungchan Kim, Eunmi Kim, Jeremy Lovejoy)

김병찬*(서울과학기술대), 김은미(고등과학원), Jeremy Lovejoy(CNRS, Paris 7)
Byungchan Kim*, Seoul National University of Science and Technology, Eunmi Kim, KIAS, Jeremy Lovejoy, CNRS, Paris 7

We consider some excess conditions on integer partitions and discuss how $q$-series transformations and combinatorial models work together to study their arithmetic. This talk will be based on an ongoing joint work with Eunmi Kim and Jeremy Lovejoy.

2010 Mathematics Subject Classification: 05A17
Key Words and Phrases: integer partitions, q-series, injection

⋅ 20th-C-11:30 − 11:50 Characterization of weakly bent functions in terms of strongly regular graphs (Jong Yoon Hyun, Yoonjin Lee)

현종윤*(고등과학원), 이윤진(이화여대)
Jong Yoon Hyun*, KIAS, Yoonjin Lee, Ewha Womans University

A p-ary function $f$ in $n$ variables is an $l$-form if $f(tu) = t^l f(u)$ for any nonzero $t$ in $\mathbb{Z}_p$ and $u$ in $\mathbb{Z}^n_p$. Let $n$ be a positive even integer, $p$ an odd prime, and $l$ an element of $\{1,2,\ldots,p−1\}$ provided that $l= p−1$ if $p > 3$. Let $f$ be a p-ary bent function in $n$ variables of $l$-form with $f(0) =0$ and $\gcd(l −1, p−1) = 1$, and let $H_l =\{ t^l : t\in \mathbb{Z}^*_p\}$. We denote by $G_{f,l}$ the Cayley graph Cay$(\mathbb{Z}^n_p,\cup_{s\in H_l} f^{−1}(s)$. Our main results are as follows:
$1)$ if there is weakly regular p-ary bent $f$ which is not regular, then $l$ is $2$;
$2)$ if $l = 2$, then $f$ is weakly regular p-ary bent if and only if the Cayley graph $G_{f,l}$ is strongly regular;
$3)$ if $l= 2$, then $f$ is regular p-ary bent if and only if the Cayley graph $G_{f,l}$ is strongly regular;
$4)$ $G_{f,l}$ can be replaced by Cay$(\mathbb{Z}^n_p, f^{−1}(0)\{0\})$ in $2)$ and $3)$; and
$5)$ amorphic association schemes are derived by using $2)$ and $3)$.
We prove our main results by computing at most four distinct restricted eigenvalues of $G_{f,l}$.

2010 Mathematics Subject Classification: B4605
Key Words and Phrases: p-ary bent functions, strongly regular graph, amorphic association schemes

⋅ 20th-C-11:50 − 12:10 Counting self-conjugate $(s,s+1,s+2)$-core partitions (Hyunsoo Cho, Ji Sun Huh, Jaebum Sohn)

조현수(연세대), 허지선*(성균관대 응용대수 및 최적화 연구센터), 손재범(연세대)
Hyunsoo Cho, Yonsei University, Ji Sun Huh*, Sunkyunkwan University, Applied Algebra and and Optimization Research Center (AORC), Jaebum Sohn, Yonsei University

We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A partition $\lambda$ is called a {\it $t$-core} if none of its hook lengths are multiples of $t$. We use the notation of a $(t_1,...,t_p)$-core if it is simultaneously a $t_1$-core,\dots, and a $t_p$-core.\\ A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.

2010 Mathematics Subject Classification: 05C30, 05A17
Key Words and Phrases: simultaneous core partition, self-conjugate partition, symmetric Motzkin path

⋅ 20th-D-14:30 − 16:00 Chair: Young Soo Kwon (Yeungnam University)

⋅ 20th-D-14:30 − 14:50 Graphs without two disjoint $S$-cycles (Min jeong Kang, O Joung Kwon, Myeonghwan Lee)

강민정(인천대), 권오정*(인천대), 이명환(인천대)
Min jeong Kang, Incheon National University, O Joung Kwon*, Incheon National University, Myeonghwan Lee, Incheon National University

Lov\'asz (1965) characterized graphs with no two cycles, which implies that such graphs have at most 3 vertices hitting all cycles. In this paper, we ask whether such a small bound exists for $S$-cycles, when a graph has no two vertex-disjoint $S$-cycles. For a graph $G$ and a vertex set $S$ of $G$, an $S$-cycle is a cycle containing a vertex of $S$. We provide an example $G$ on $21$ vertices where $G$ has no two vertex-disjoint $S$-cycles, but $3$ vertices are not sufficient to hit all $S$-cycles. On the other hand, we show that $4$ vertices are enough to hit all $S$-cycles whenever a graph has no two vertex-disjoint $S$-cycles.

2010 Mathematics Subject Classification: 05C38
Key Words and Phrases: Erdos-Posa, S-cycles

⋅ 20th-D-14:50 − 15:10 On the claw-bound (Jongyook Park)

박종육(원광대)
Jongyook Park, Wonkwang University

In this talk, we consider a distance-regular graph $\Gamma$ with intersection array $\{80, 54,12; 1, 6, 60\}$. We first show that a local graph $\Delta$ of $\Gamma$ does not contain a coclique with 5 vertices, and then we prove that the graph $\Gamma$ is geometric by showing that $\Delta$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $\Gamma$, and we could obtain that there is no such a distance-regular graph.

2010 Mathematics Subject Classification: 05C50, 05E30
Key Words and Phrases: distance-regular graphs, geometric distance-regular graphs, Delsarte cliques, the claw-bound

⋅ 20th-D-15:20 − 15:40 Kirchhoff index of simplicial networks (Woong Kook, Kang-Ju Lee)

국웅(서울대), 이강주*(서울대)
Woong Kook, Seoul National University, Kang-Ju Lee*, Seoul National University

We introduce a high-dimensional analogue of Kirchhoff index which is also known as total effective resistance. This analogue, which we call simplicial Kirchhoff index $Kf(X)$, is defined to be the sum of the simplicial effective resistances of all $(d+1)$-subsets of the vertex set of a simplicial complex $X$ of dimension $d$. For a $d$-dimensional simplicial complex $X$ with $n$ vertices, we give formulas for the simplicial Kirchhoff index in terms of the pseudoinverse of the Laplacian $L_{X}$ in dimension $d-1$ and its eigenvalues: \[ Kf(X)=n \cdot \operatorname{tr}{L_{X}^+}=n \cdot \sum_{\lambda \in \Lambda_{+}}{\dfrac{1}{\lambda}}, \] where $L_X^+$ is the pseudoinverse of $L_X$, and $\Lambda_{+}$ is the multi-set of non-zero eigenvalues of $L_X$. Using this formula, we obtain an inequality for a high-dimensional analogue of algebraic connectivity and Kirchhoff index, and propose these quantities as measures of \emph{robustness} of simplicial complexes. In addition, we derive its integral formula and relate this index to a simplicial dynamical system.

2010 Mathematics Subject Classification: 94C15, 05C50, 05E45, 35J05
Key Words and Phrases: simplicial Kirchhoff index, effective resistance, robustness, pseudoinverse, Laplacian

⋅ 20th-D-15:40 − 16:00 Coloring squares of graphs with mad constraints (Herve Hocquard, Seog-Jin Kim, Theo Pierron)

Herve Hocquard(Univ. of Bordeaux), 김석진*(건국대), Theo Pierron(Univ. of Bordeaux)
Herve Hocquard, University of Bordeaux, Seog-Jin Kim*, Konkuk University, Theo Pierron, University of Bordeaux

The square $G^2$ of a graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only if the distance between $u$ and $v$ is at most two. We denote by $\chi(G^2)$ the chromatic number of $G^2$, which is the least integer $k$ such that a $k$-coloring of $G^2$ exists. In this paper, we prove that the square of every graph $G$ with $Mad(G)<4$ and $\Delta(G) \geq 8$ is $(3\Delta(G)+1)$-choosable and even correspondence-colorable. Furthermore, we show a family of $2$-degenerate graphs $G$ with $Mad(G)<4$, arbitrarily large maximum degree, and $\chi(G^2)\geq \frac{5\Delta(G)}{2}$, improving a result of Kim and Park. This is joint work with Herv\'{e} Hocquard and Th\'{e}o Pierron (University of Bordeaux, France).

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: graph coloring, square of graph, maximum average degree
Cryptography

⋅ 20th-B-09:00 − 10:30 Chair: Jooyoung Lee (KAIST)

⋅ 20th-B-09:00 − 09:20 A new method to construct threshold schemes for LWE encryption, using ISIS (Jung Hee Cheon, Wonhee Cho, Jinhyuck Jeong, Donggeon Yhee)

천정희(서울대), 조원희(서울대), 정진혁(서울대), 이동건*(서울대)
Jung Hee Cheon, Seoul National University, Wonhee Cho, Seoul National University, Jinhyuck Jeong, Seoul National University, Donggeon Yhee*, Seoul National University

LWE-based encryption draw lots of attention due to its quantum-resistancy and versatile properties resulting in various schemes including homomorphic encryptions and functional encryptions. In many of their applications, threshold decryptions are essential, but the previous works are not sufficient to provide practical threshold schemes. We propose a new method to construct threshold schemes for LWE-based encryptions. Its decryption procedure requires solving Inhomogeous Small Integer Solution (ISIS), which is asymptotically slow but sufficiently efficient for certain parameter ranges.

2010 Mathematics Subject Classification: 94A62
Key Words and Phrases: threshold cryptography, learning with errors, secret sharing schemes

⋅ 20th-B-09:20 − 09:40 Numerical methods for comparison on homomorphically encrypted numbers (Jung Hee Cheon, Dongwoo Kim, Duhyeong Kim, Hun Hee Lee, Keewoo Lee)

천정희(서울대), 김동우(서울대), 김두형(서울대), 이훈희(서울대), 이기우*(서울대)
Jung Hee Cheon, Seoul National University, Dongwoo Kim, Seoul National University, Duhyeong Kim, Seoul National University, Hun Hee Lee, Seoul National University, Keewoo Lee*, Seoul National University

We propose a new method to compare numbers which are encrypted by Homomorphic Encryption (HE). Previously, comparison and min/max functions were evaluated using Boolean functions where input numbers are encrypted bit-wisely. However, the bit-wise encryption methods require relatively expensive computation of basic arithmetic operations such as addition and multiplication. In this paper, we introduce iterative algorithms that approximately compute the min/max and comparison operations of several numbers which are encrypted word-wisely. From the concrete error analyses, we show that our min/max and comparison algorithms have $\Theta(\alpha)$ and $\Theta(\alpha\log\alpha)$ computational complexity to obtain approximate values within an error rate $2^{-\alpha}$, while the previous minimax polynomial approximation method requires the exponential complexity $\Theta(2^{\alpha/2})$ and $\Theta(\sqrt{\alpha}\cdot 2^{\alpha/2})$, respectively. We also show the (sub-)optimality of our min/max and comparison algorithms in terms of asymptotic computational complexity among polynomial evaluations to obtain approximate min/max and comparison results. Our comparison algorithm is extended to several applications such as computing the top-$k$ elements and counting numbers over the threshold in encrypted state. Our new method enables word-wise HEs to enjoy comparable performance in practice with bit-wise HEs for comparison operations while showing much better performance on polynomial operations. Computing an approximate maximum value of any two $\ell$-bit integers encrypted by HEAAN, up to error $2^{\ell-10}$, takes only $1.14$ milliseconds in amortized running time, which is comparable to the result based on bit-wise HEs.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: homomorphic encryption, min/max, comparison, iterative algorithm

⋅ 20th-B-09:50 − 10:10 A survey on almost perfect nonlinear function (Sumin Lee, Jaesung Jung, Namhun Koo, Soonhak Kwon)

이수민*(성균관대), 정재성(성균관대), 구남훈(성균관대), 권순학(성균관대)
Sumin Lee*, Sungkyunkwan University, Jaesung Jung, Sungkyunkwan University, Namhun Koo, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University

APN(Almost Perfect Nonlinear) function has optimal resistance to differential cryptanalysis due to its high differential uniformity, and its properties has been studied intensively. In this paper, we survey some results about APN functions studied on last few years. In particular, we focus on classification and construction of known APN functions.

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: APN function, Sbox

⋅ 20th-B-10:10 − 10:30 On some infinite families of quadratic APN multinomials (Jaeseong Jeong, Sumin Lee, Namhun Koo, Soonhak Kwon)

정재성*(성균관대), 이수민(성균관대), 구남훈(성균관대), 권순학(성균관대)
Jaeseong Jeong*, Sungkyunkwan University, Sumin Lee, Sungkyunkwan University, Namhun Koo, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University

The resistance of differential attacks on block ciphers relies on cryptographic properties of their S-boxes. Particularly, the lower the amount of so-called difference uniformity is, the better the resistance is. So, much research has been done to construct almost perfect nonlinear(APN) function that is of the lowest differential uniformity. Among them, the differential equations of the quadratic function are linear, so it is easier to analyze the properties than the higher order. We propose a method to find the number of roots of the linear equation and find the infinite class of the quadratic APN multinomials.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: S-Box, APN function, differential uniformity, linear polynomial

⋅ 20th-C-10:40 − 12:10 Chair: Jooyoung Lee (KIAS)

⋅ 20th-C-10:40 − 11:00 Generic attacks on Feistel-like ciphers (Yeongmin Lee, Seongkwang Kim, Wonseok Choi, Jooyoung Lee)

이영민*(한국과학기술원), 김성광(한국과학기술원), 최원석(한국과학기술원), 이주영(한국과학기술원)
Yeongmin Lee*, KAIST, Seongkwang Kim, KAIST, Wonseok Choi, KAIST, Jooyoung Lee, KAIST

Feistel-like structures are considered to be suitable for format preserving encryption (FPE) as they can be used as a mode of operation, transforming a block cipher into a permutation on any domain. In this context, the domain size is typically small, while the small domain size permitted various generic attacks. One of the powerful attacks is to recover all the evaluations of the underlying round functions, which is called a round function recovery (RFR) attack. In this paper, we propose RFR attacks on 4-round Feistel-like structures including Feistel, Misty and Lai-Massey structures in the chosen-plaintext attack (CPA) model. Our attacks are optimal as they use $O(N)$ plaintext-ciphertext pairs, running in $O(N)$ time, where $N$ denotes the domain size of each round function.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Feistel-like structures, round function recovery attack, key recovery attack

⋅ 20th-C-11:00 − 11:20 McNie2-Gabidulin: A code-based cryptosystem (Jon-Lark Kim, Young-Sik Kim, Lucky Erap Galvez, Myeong Jae Kim)

김종락(서강대), 김영식(조선대), Lucky Erap Galvez*(서강대), 김명재(서강대)
Jon-Lark Kim, Sogang University, Young-Sik Kim, Chosun University, Lucky Erap Galvez*, Sogang University, Myeong Jae Kim, Sogang University

Code-based cryptosystems are promising quantum resistant candidates because they are based on a well-studied problem of correcting random codes. In this talk, we present McNie2-Gabidulin, an improvement of McNie public key cryptosystem. By using Gabidulin codes, we eliminate the decoding failure. Suggested parameters are also given which provides low key sizes

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: code-based cryptography, Gabidulin code, public key encryption

⋅ 20th-C-11:30 − 11:50 SHECS-PIR: Somewhat homomorphic encryption-based compact and scalable private information retrieval (Jeongeun Park, Mehdi Tibouchi)

박정은*(이화여대), Mehdi Tibouchi(NTT Secure Platform Laboratories)
Jeongeun Park*, Ewha Womans University, Mehdi Tibouchi, NTT Secure Platform Laboratories

A Private Information Retrieval (PIR) protocol allows a client to retrieve arbitrary elements from a database stored on a server without revealing to the server any information about the element she requests. PIR is an important building block of many privacy-preserving protocols, and its efficient implementation is therefore of prime importance. Several concrete, practical PIR protocols have been proposed and implemented so far, particularly based on very low-depth somewhat homomorphic encryption. The main drawback of these protocols, however, is their very large communication size, especially in terms of the server's reply, which grows like $\Omega(\sqrt{n})$ for an $n$-element database.In this work, we describe an efficient PIR protocol called SHECS-PIR,based on deeper circuits and GSW-style homomorphic encryption. SHECS-PIR reduces the communication cost down to $O(\log n)$ while maintaining a high level of efficiency. In fact, for large databases, we achieve faster server processing time in addition to more compact queries.

2010 Mathematics Subject Classification: 94C99
Key Words and Phrases: PIR, CPIR, FHE, FHE based PIR

⋅ 20th-C-11:50 − 12:10 Security and privacy of multi-function verifiable computation against adversaries with verification queries from homomorphic authenticated encryption (Aaram Yun, Kim Jeongsu)

윤아람(이화여대), 김정수*(울산과학기술원)
Aaram Yun, Ewha Womans University, Kim Jeongsu*, UNIST

We study stronger notions of privacy and security of multi-function verifiable computation. We consider adversaries that have access to the verification oracle (the result of the delegated computation). In this stronger model, the client can use the verifiable computation scheme even when some of the client’s information on the past has been leaked to the server. To achieve the stronger security and the stronger privacy, we directly use the IND-CCA secure homomorphic authenticated encryption. Furthermore, we present the first generic construction of IND-CCA secure fully homomorphic authenticated encryption. Our generic construction preserves amortized efficiency, and therefore, resultant verifiable computation becomes efficiently verifiable regardless the complexity of the delegated function.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: verifiable computation, homomorphic authenticated encryption, fully homomorphic authenticated encryption, IND-CCA, UF-sCMA, homomorphic authenticator

Nonlocal Differential Equation: Analysis and Numerics

⋅ 20th-B-09:00 − 09:40   Chair: Young-Pil Choi (Inha University)

⋅ 20th-B-09:00 − 09:20 Synchronous harmony in an ensemble of Hamiltonian mean-field oscillators and inertial Kuramoto oscillators (Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li)

하승열(서울대), 이재승*(서울대), Zhuchun Li(Harbin Institute of Technology)
Seung-Yeal Ha, Seoul National University, Jaeseung Lee*, Seoul National University, Zhuchun Li, Harbin Institute of Technology

We study a dynamic interplay between Hamiltonian mean-field oscillators and inertial Kuramoto oscillators. We present several sufficient frameworks leading to asymptotic complete synchronization for the mixed ensemble. For a two-oscillator system with the same natural frequencies, we prove that the mixed ensemble exhibits asymptotic complete synchronization for any initial data, whereas we also show that the two-oscillator system tends to asymptotic complete synchronization under an a priori assumption on the uniform boundedness on the phase diameter. For the many-body system, we show that asymptotic complete frequency synchronization occurs for Kuramoto oscillators with inertia if the oscillators have the same natural frequencies. Moreover, we show that overall phase concentration can be controlled by increasing the coupling strengths. We also provide several numerical experiments and compare them with analytical results.

2010 Mathematics Subject Classification: 34C15
Key Words and Phrases: Hamiltonian mean-field oscillator, Kuramoto oscillator, Synchronization

⋅ 20th-B-09:20 − 09:40 Emergence of anomalous flocking in the fractional Cucker-Smale model (Seung-Yeal Ha, Jinwook Jung, Peter Kuchling)

하승열(서울대), 정진욱*(서울대), Peter Kuchling(Bielefeld Univ.)
Seung-Yeal Ha, Seoul National University, Jinwook Jung*, Seoul National University, Peter Kuchling, Bielefeld University

In this talk, we study the emergent behaviors of the Cucker-Smale (C-S) ensemble under the interplay of memory effect and flocking dynamics. As a mathematical model incorporating aforementioned interplay, we introduce the fractional C-S model which can be obtained by the Caputo fractional time derivative. For the proposed fractional C-S model, we provide a sufficient framework which admits the emergence of anomalous flocking at the algebraic rate and an $\ell^2$-stability estimate with respect to initial data. We also provide several numerical examples and compare them with our theoretical results.

2010 Mathematics Subject Classification: 70F99, 92D25
Key Words and Phrases: Caputo fractional derivative, collective motion, fractional calculus, fractional Cucker-Smale model, flocking

⋅ 20th-B-09:50 − 10:30   Chair: Jinyeong Park (Hanyang University)

⋅ 20th-B-09:50 − 10:10 A conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation (Sebastiano Boscarino, Seung-Yeon Cho, Giovanni Russo, Seok-Bae Yun)

Sebastiano Boscarino(Univ. of Catania), 조승연*(성균관대), Giovanni Russo(Univ. of Catania), 윤석배(성균관대)
Sebastiano Boscarino, University of Catania, Seung-Yeon Cho*, Sungkyunkwan University, Giovanni Russo, University of Catania, Seok-Bae Yun, Sungkyunkwan University

In this work, we present a high order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation in one and two dimensions. When one implements a semi-Lagrangian scheme, main difficulties arise from the lack of conservation in macroscopic quantities. One main reason for this loss of conservation property is the non-linear weights used in the reconstruction of numerical solutions at characteristic foots. To overcome this, we introduce an interpolation which is fourth order accurate, non-oscillatory and preserves macroscopic moments. Based on this conservative reconstruction, we construct a semi-Lagrangian scheme and check its effectiveness through several numerical tests.

2010 Mathematics Subject Classification: 65M06
Key Words and Phrases: semi-Lagrangian scheme, conservation, BGK model, Boltzmann equation

⋅ 20th-B-10:10 − 10:30 Emergence of bi-polar aggregation of a swarm sphere model with attractive-repulsive couplings (Se Eun Noh)

노세은(명지대)
Se Eun Noh, Myongji University

In this talk, we study a swarm sphere model under attractive-repulsive couplings and present sufficient frameworks leading to complete and practical bi-polar aggregations. In our interaction problem, inter-ensemble and intra-ensemble couplings are assumed to be repulsive and attractive respectively.

2010 Mathematics Subject Classification: 82C22, 35B37, 34C15, 34D06
Key Words and Phrases: aggregation, attractive-repulsive coupling, swarm sphere model, synchronization

⋅ 20th-C-10:40 − 11:20   Chair: Young-Pil Choi (Inha University)

⋅ 20th-C-10:40 − 11:00 Spectral analysis of $\theta$-periodic Schr$\ddot{\rm{o}}$dinger operators and applications to periodic waves (Soyeun Jung, Peter Howard)

정소연*(공주대), Peter Howard(Texas A&M Univ.)
Soyeun Jung*, Kongju National University, Peter Howard, Texas A&M University

In this talk, we consider the associated $\theta$-periodic Schr$\ddot{\rm{o}}$dinger operators $H_\theta$ on intervals $[0, P]$, where in our applications $P$ denotes the period of a stationary periodic solution to a nonlinear evolutionary PDE. We relate the Morse index of $H_\theta$ to certain Maslov indices, and apply our theory to operators obtained when Allen-Cahn equations and systems are linearized about stationary periodic solutions.

2010 Mathematics Subject Classification: 35P05
Key Words and Phrases: spectral analysis, Maslov index

⋅ 20th-C-11:00 − 11:20 Besov and Triebel-Lizorkin space estimates for fractional diffusion (Kozo Yabuta, Minsuk Yang)

Kozo Yabuta(Kwansei Gakuin Univ.), 양민석*(연세대)
Kozo Yabuta, Kwansei Gakuin University, Minsuk Yang*, Yonsei University

We study Besov and Triebel-Lizorkin space estimates for fractional diffusion. We measure the smoothing effect of the fractional heat flow in terms of the Besov and Triebel-Lizorkin scale.

2010 Mathematics Subject Classification: 42B25
Key Words and Phrases: fractional diffusion

⋅ 20th-C-11:30 − 12:10   Chair: Jinyeong Park (Hanyang University)

⋅ 20th-C-11:30 − 11:50 Inexact reweighted algorithms for nonconvex and nonsmooth optimization problems (Myeongmin Kang)

강명민(충남대)
Myeongmin Kang, Chungnam National University

Iterative reweighted algorithms are popular methods for solving nonconvex minimization problems and have been often used in applications such as image processing. They have a convex subproblem which does not have closed form solution in general. In this manuscript, we propose inexact versions of proximal iteratively reweighted algorithms for nonconvex and nonsmooth unconstrained minimization problems. Specifically, we can achieve an approximate solution for the subproblem by applying a computable inexact stopping rule. We prove the convergence of our methods based on inexact unified framework. Numerical applications are also presented to validate the effectiveness of the proposed algorithms.

2010 Mathematics Subject Classification: 65K05, 68U10
Key Words and Phrases: iteratively reweighted algorithm, nonconvex optimization, nonsmooth objective function, Kurdyka-Lojasiewicz property

⋅ 20th-C-11:50 − 12:10 Energy-stability on the finite difference based numerical method for solving the Incompressible Navier-Stokes equations (Byungjoon Lee)

이병준(가톨릭대)
Byungjoon Lee, The Catholic University of Korea

Understanding fluid flow is one of the fundamental tasks in science. For the incompressible fluid flow, the Navier-Stokes equations have been served as the most successful tool for analyzing the behavior of fluids. Despite of its importance in real world, the existence of the solution of Navier-Stokes equation is not known in 3D space. Hence, many researchers have made attempts to analyze the Navier-Stokes equation via its proper numerical solutions. This talk considers on the following incompressible Navier-Stokes(INS) equations: \[ \begin{cases} \rho\left(U_{t}+\left(U\cdot\nabla\right)U\right) & =-p+\mu\nabla\cdot\left(\mu\left(\nabla U+\nabla U^{T}\right)\right)+\rho g\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nabla\cdot U & =0 \end{cases}\,\text{\,in}\,\,\Omega\] In this talk, numerical approaches based on the finite difference method for the above equations will be presented. Above all things, we focus on the construction of the numerical methods for the INS that obey the physical energy stability in PDE sense. Several numerical examples will be given along with the mathematical analysis for verifying that the proposed methodologies are indeed energy-stable method for INS.

2010 Mathematics Subject Classification: 34A45, 76D05
Key Words and Phrases: incompressible Navier-Stokes equations, finite difference method, energy stability
Dynamical Systems and Related Topics

⋅ 19th-A-15:00 − 16:30   Chair: Sangil Kim (Pusan National University)

⋅ 19th-A-15:00 − 15:20 Cauchy problems for arch structures with moving point loads (Jun Hong Ha)

하준홍(한국기술교육대)
Jun Hong Ha, Korea University of Technology and Education

Building long span arch roofs and bridges has been an important practical problem that has occupied structural engineers for many years. The motion of such structures has been studied by engineers and mathematicians since at least 1930s. Our concern is developing a rigorous mathematical framework for short and long time behavior of arch and membrane like structures. In this lecture, we are going to establish the existence, uniqueness and the stability of weak solutions of the following equations under such more general assumptions on the load $f$: \begin{eqnarray*} && y_{tt} +\alpha \Delta^2 y-\left[\beta +\gamma\int_{\Omega}|\nabla y|^2\, dx \right]\Delta y+\mu \Delta^2 y_t +\kappa y_t =f, \ (x,t) \in \Omega \times (0,T), \\ && y=\Delta y =0 \ \mbox{or} \ y=\frac{\partial y}{\partial n} =0,\quad (x,t) \in \partial \Omega\times (0,T), \\ && y(x,0)=u_0(x),\quad y_t(x,0)=v_0(x), \quad x \in \Omega. \end{eqnarray*} Examples simulating a vehicular traffic across a long bridge are considered.

2010 Mathematics Subject Classification: 35Q70
Key Words and Phrases: arch roof, point load, Cauchy problem

⋅ 19th-A-15:20 − 15:40 Optimal harvest strategies of Pacific cod in the South Korea (Giphil Cho, Il Hyo Jung)

조기필*(부산대 산업수학센터), 정일효(부산대)
Giphil Cho*, Pusan National University, Industrial Mathematics Center, Il Hyo Jung, Pusan National University

We propose optimal harvest strategies using a stage-structured fishery model with impulsive system. The economical objective is to maximize the profit of fishing in fisheries management. Fishing effort used to harvest is used as a control to investigate the optimal utilization of the resource in economic sense. We analyze the economical optimal harvest strategy of Pacific cod caused by monthly price change. The optimal control problem is solved numerically using forward backward sweep method. Simulation results show the difference of the harvest strategies of the Pacific cod with and without considering monthly price. We expect that maximum sustainable yield and profit of the Pacific cod can be increased by approximately 20% due to the optimal harvest strategies.

2010 Mathematics Subject Classification: 46N60
Key Words and Phrases: optimal harvest strategy, impulsive model, Pacific cod

⋅ 19th-A-15:50 − 16:10 An numerical approach for interface problems in synaptic cleft (Sat byul Seo)

서샛별(경남대)
Sat byul Seo, Kyungnam University

We provide a new numerical approach, continuous velocity method, to solve the heat diffusion equation with piecewise continuous coefficients. We proposed this method in three dimensional space for a cubic domain, and proved its second order accuracy. We further applied this method to a diffusion process for neurotransmitter release in a synapse, and validated the method for second order accuracy numerically.

2010 Mathematics Subject Classification: 65P99
Key Words and Phrases: finite difference method, discontinuous coefficients, diffuse interface problems. heat diffusion equation

⋅ 19th-A-16:10 − 16:30 Mathematical modeling to capture the bystander killing effect in antibody-drug conjugates (Jong Hyuk Byun)

변종혁(부산대)
Jong Hyuk Byun, Pusan National University

A tumor growth inhibition (TGI) model represents a profile of tumor growth inhibition over time by drug administration. Some payloads have the ability to kill adjacent antigen negative tumor cells. This phenomenon is called the bystander-killing effect in addition to the direct killing of antigen-positive tumor cells. We developed tumor growth inhibition models to account for tumor-drug interaction considering the bystander-killing effect by means of a stochastic process that contains a Markovian and non-Markovian process. We propose that age-structured models are required for capturing the bystander-killing effect reported for some antibody-drug conjugates. Exponential, Erlang, and Mittag-Leffler distributions are discussed for this.

2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: Tumor growth inhibition, Bystander-killing effects, PKPD, Antibody-drug conjugates, Stochastic process

⋅ 20th-B-09:00 − 10:25   Chair: DoYong Kwon (Chonnam National University)

⋅ 20th-B-09:00 − 09:15 Best Diophantine approximations and singularity (Taehyeong Kim)

김태형(서울대)
Taehyeong Kim, Seoul National University

Let $A$ be an $m\times n$ matrix with real entries. For $\epsilon > 0$, let $(y_k)_{k\geq1}$ be a sequence of best Diophantine approximations to $m \times n$ real matrix $A$. Setting $Y_i=\|\mathbf{y}_i\|$ and $M_i=\inf_{{\mathbf{p}\in\mathbb{Z}^m}} \|A\mathbf{y}-\mathbf{p}\|$, it is well known that
(1) A is badly approximable if and only if $\liminf_{k\to\infty}Y_{k}^n M_{k}^m > 0$;
(2) A is singular if and only if $\lim_{k\to\infty}Y_{k+1}^n M_{k}^m=0$.
In this talk, we will give a similar characterization for singular on average property using the notion of the statistical convergence: $A$ is singular on average if and only if \[ \lim_{k\to\infty}Y_{k}^{1/k}=\infty\quad\text{or}\quad \lim_{k\to\infty} \text{stat } Y_{k+1}^n M_{k}^m=0. \]

2010 Mathematics Subject Classification: 11K60, 28A80, 37D20
Key Words and Phrases: inhomogeneous Diophantine approximation, singular on average, fractals

⋅ 20th-B-09:15 − 09:30 Inhomogeneous Diophantine approximation for doubly metric case (Wooyeon Kim, Seonhee Lim)

김우연*(서울대), 임선희(서울대)
Wooyeon Kim*, Seoul National University, Seonhee Lim, Seoul National University

We will present the doubly metric case of inhomogeneous Diophantine approximation for matrices. It amounts to the following theorems: THM1. For any $\epsilon>0$, there exists $\delta>0$ such that for all $b\in R^m$, $\dim_H \mathbb{Bad}^b(\epsilon)<mn-\delta$. THM2. For any $\epsilon>0$, there exists $\delta>0$ such that $\dim_H \mathbb{Bad}(\epsilon)<mn+m-\delta$. THM3. For any $\epsilon>0$, there exists $\delta>0$ such that the Hausdorff dimension of the set of $A$ satisfying $\dim_H \mathbb{Bad}_A(\epsilon)<m-\delta$ is strictly less than $mn$.

2010 Mathematics Subject Classification: 37A15, 11J13
Key Words and Phrases: inhomogeous Diophantine approximation, homogeneous dynamics, badly approximable vectors

⋅ 20th-B-09:35 − 09:50 Martin boundaries of rank one Hadamard surfaces (Jaelin Kim)

김재린(서울대)
Jaelin Kim, Seoul National University

In this talk, we prove that the geometric boundary and $(\Delta+\lambda I)$-Martin boundary on a rank one Hadamard manifold of dimension 2 which admits a compact quotient coincide for every $0\le\lambda\le\lambda_{0}$. We used the uniformization theorem to simplify computations and Gibbs measures of potentials associated to diffusion processes on the hyperbolic plane.

2010 Mathematics Subject Classification: 37A50
Key Words and Phrases: thermodynamic formalism, potential theory, Martin boundary, rank one Hadamard manifolds

⋅ 20th-B-09:50 − 10:05 L$\acute{\rm{e}}$vy constants of Sturmian continued fraction expansions (Yann Bugeaud, Dong Han Kim, Seul Bee Lee)

Yann Bugeaud(Univ. of Strasbourg), 김동한(동국대), 이슬비*(서울대)
Yann Bugeaud, University of Strasbourg, Dong Han Kim, Dongguk university, Seul Bee Lee*, Seoul National University

For the denominator $q_n$ of the convergents of the continued fractions of a real number, we call $\lim_{n\rightarrow \infty}\frac{\log{q_n}}{n}$ L$\acute{\rm{e}}$vy constant. In 1936, Paul L$\acute{\rm{e}}$vy showed that for almost every real number, the L$\acute{\rm{e}}$vy constant is $\frac{\pi^2}{12\log 2}.$ It is investigated by Jager and Liardet in 1988 and Belova and Hazard in 2018 that the L$\acute{\rm{e}}$vy constant of quadratic irrational.\\ On the other hand, a quadratic irrational has a periodic continued fraction expansion and a Sturmian word is a sequence of letters which has the lowest complexity among non-periodic words. We showed that there exists L$\acute{\rm{e}}$vy constant of a real number whose continued fraction expansion is a Sturmian word. We also have some partial results to see the spectrum of such L$\acute{\rm{e}}$vy constants. This is joint work with Yann Bugeaud and Dong Han Kim.

2010 Mathematics Subject Classification: 37B10, 11A55, 11J70
Key Words and Phrases: L$\acute{\rm{e}}$vy constants, Sturmian words, continued fractions

⋅ 20th-B-10:10 − 10:25 On the existence of invariant Gibbs measures for shift spaces and classes of balanced shift spaces (Minkyu Kim, Uijin Jung, Soonjo Hong)

김민규*(아주대), 정의진(아주대), 홍순조(홍익대)
Minkyu Kim*, Ajou University, Uijin Jung, Ajou University, Soonjo Hong, Hongik University

An invariant Gibbs measure is a special equilibrium state and is used in many fields of mathematics including thermodynamic formalism of topological dynamics. \par It is well known that if $X$ is a mixing shift of finite type and $f$ is a H$\ddot{\rm{o}}$lder continuous function from $X$ to $\mathbb{R}$, then there exists a unique invariant Gibbs measure with respect to the potential. Bowen extended to the case where $X$ is a subshift with the specification property and $f$ is a function in the Bowen class, Baker and Ghenciu showed that there exists a Gibbs measure for the potential $0$ if and only if $X$ is (right-)balanced. We study a necessary and sufficient condition for the existence of invariant Gibbs measures for the potential $0$, generalize this result for real-valued continuous functions of $X$ and construct a class of right-balanced shift spaces which is not left-balanced.

2010 Mathematics Subject Classification: 37B10, 37D35
Key Words and Phrases: subshifts, Gibbs measures, equilibrium states, thermodynamic formalism

⋅ 20th-C-10:40 − 12:10   Chair: Seonhee Lim (Seoul National University)

⋅ 20th-C-10:40 − 11:00 Lagrange spectrum of Romik's dynamical system (Byungchul Cha, Dong Han Kim)

차병철(Muhlenberg College), 김동한*(동국대)
Byungchul Cha, Muhlenberg College, Dong Han Kim*, Dongguk University

Let $\mathcal L(S^1)$ be the Lagrange spectrum arising from the intrinsic Diophantine approximation of the unit circle $S^1$ by its rational points. In this talk, we give a complete description of the structure of $\mathcal L(S^1)$ below its smallest accumulation point. First, we prove that the smallest accumulation point of $\mathcal L(S^1)$ is 2. We use a certain digit expansion of points on $S^1$, which was initially introduced by Romik in 2008. This digit expansion is an analogue of simple continued fraction of a real number. We characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of their Romik digit expansions. Our theorem is an analogue of a celebrated theorem of Markoff on badly approximable real numbers. We also adapt our method to the unit sphere $S^2$ and find a sharp Hurwitz bound, that is, the minimum of $\mathcal L(S^2)$.

2010 Mathematics Subject Classification: 11J06, 11J70, 68R15
Key Words and Phrases: Lagrange spectrum, Romik's dynamical system, Diophantine approximation on manifold

⋅ 20th-C-11:00 − 11:20 Quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms of rank 3 and 4 (Jiyoung Han)

한지영(서울대)
Jiyoung Han, Seoul National University

The quantitative Oppenheim conjecture, which was proved by Eskin, Margulis and Mozes in 1998 says that for any irrational isotropic quadratic forms of rank at least 5, the number of integral vectors $v$ such that $q(v)$ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $v$ such that $q(v)$ is contained in the same interval. In dimension $3$ or $4$, there is a measure-zero set of exceptional quadratic forms which fails to satisfy the quantitative Oppenheim conjecture. In this talk, we will introduce the $S$-arithmetic analogy of this result.

2010 Mathematics Subject Classification: 22F30
Key Words and Phrases: Oppenheim conjecture, $S$-arithmetic, homogeneous space, Siegel transform

⋅ 20th-C-11:30 − 11:50 Distribution of extreme values of orbits under geodesic flow on quotients of trees (Sanghoon Kwon, Seonhee Lim)

권상훈*(가톨릭관동대), 임선희(서울대)
Sanghoon Kwon*, Catholic Kwandong University, Seonhee Lim, Seoul National University

We discuss extreme value distributions for geodesic flows on quotient of trees by certain discrete groups, which concerns the limiting behavior of the maximum values over expanding intervals with respect to a one-parameter action. The main ingredient is to find a Markov partition of the dynamical system obtained via taking a quotient by the associated full group of the discrete group. We investigate the equivalent condition for Markov chains with countably many alphabets to be geometrically ergodic and present some examples with calculations.

2010 Mathematics Subject Classification: Primary 37A10, 37D40; Secondary 60G70
Key Words and Phrases: extreme value distribution, Markov chain, geodesic flow, trees

⋅ 20th-C-11:50 − 12:10 Bowen-Margulis measure on hyperbolic graphs (Soon Ki Hong)

홍순기(서울대)
Soon Ki Hong, Seoul National University

Let $\widetilde{X}$ be a locally finite Gromov hyperbolic graph which has the geometric boundary consisting of infinite points. A discrete group $\Gamma$ acts isometrically and geometrically on $\widetilde{X}$.We plan to prove the local limit theorem of Brownian motion on the hyperbolic graphs. The main tool is thermodynamics on the space of geodesic lines on the hyperbolic graphs. Let $\phi$ be a H$\ddot{\rm{o}}$lder continuous function on the space. We construct the Patterson-Sullivan density on Busemann boundary. Using Hopf parametrization on the geodesic,we construct Bowen-Margulis measure that is the equilibrium state of $\phi$. Using the measure, we obtain the measure on the space of geodesic lines. Furthermore. we observe the properties of the measure.

2010 Mathematics Subject Classification: 37D35
Key Words and Phrases: hyperbolic graph, Bowen-Margulis measure, Brownian motion
Trends in Number Theory

⋅ 20th-B-09:20 − 10:30 Chair: Soon-Yi Kang (Kangwon National University)

⋅ 20th-B-09:20 − 09:40 A continued fraction of order six (Yoonjin Lee, Yoon Kyung Park)

이윤진(이화여대), 박윤경*(공주교육대)
Yoonjin Lee, Ewha Womans University, Yoon Kyung Park*, Gongju National University of Education

We study a continued fraction of order six which is an analogue of Rogers-Ramanujan continued fraction. It has studied by Vasuki at el. before. First, we prove its modularity to obtain its modular equations for integer levels. We also present a ray class field modulo 6 over a given imaginary quadratic field by evaluating a continued fraction of order six. This is a joint work with Yoonjin Lee.

2010 Mathematics Subject Classification: 11F03, 11R37, 11R04
Key Words and Phrases: Ramanujan continued fraction, modular function, modular equation, ray class field

⋅ 20th-B-09:50 − 10:10 On moments of Hecke-Maass forms (Sihun Jo)

조시훈(우석대)
Sihun Jo, Woosuk University

In this talk, we investigate the asymptotic behavior of moments of Hecke-Maass forms using probabilistic methods.

2010 Mathematics Subject Classification: 11F12, 11M99
Key Words and Phrases: Maass forms, L-functions

⋅ 20th-B-10:10 − 10:30 1-universal binary and ternary Hermitian lattices over imaginary quadratic fields (Ji Young Kim, Byeong Moon Kim)

김지영*(서울대), 김병문 (강릉원주대)
Ji Young Kim*, Seoul National University, Byeong Moon Kim, Gangneung-Wonju National University

This talk is concerned with the representation of unary non-free Hermitian lattices by another Hermitian lattice over imaginary quadratic fields. A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices which are including both free and non-free Hermitian lattices. We estimate the minimal rank $u_m^1$ of 1-universal Hermitian lattices and we classify all binary and ternary 1-universal Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$ for all positive square-free integers $m$.

2010 Mathematics Subject Classification: 11E39
Key Words and Phrases: 1-universal Hermitian lattices, minimal rank

⋅ 20th-C-10:40 − 11:50 Chair: YoungJu Choie (POSTECH)

⋅ 20th-C-10:40 − 11:00 Elements of finite order in the Brauer group of elliptic curves (Taekyung Kim)

김태경(기초과학연구원 기하학 수리물리 연구단)
Taekyung Kim, IBS-Center of Geometry and Physics

Brauer groups are classical invariants of fields and of algebraic varieties with remarkable importance. Originally coming from the classification problem of division algebras and central simple algebras over a given field, after Azumaya, Grothendieck and Manin et al., Brauer groups also have provided an invaluable tool to investigating Diophantine problems on certain algebraic varieties. Let us consider the case when the given algebraic variety is an elliptic curve. Although Tsen's classical theorem asserts that the elliptic curves have trivial Brauer group, this results hold only when the curve is defined over an algebraically closed base field. When the base field is $\mathbf{Q}$ or more generally a number field, Chernousov--Guletskii determined the elements of order 2 on the Brauer group of certain elliptic curves defined over some number fields. In this talk, I will provide similar results for points of finite order 3, and give some idea to generalize this result.

2010 Mathematics Subject Classification: 11G05
Key Words and Phrases: Brauer group, elliptic curves

⋅ 20th-C-11:00 − 11:20 Distribution of ideals in an arithmetic progression (Lee Jungyun, Jun Byungheup, Sun Haesang)

이정연*(강원대), 전병흡(울산과학기술원), 선해상(울산과학기술원)
Lee Jungyun*, Kangwon National University, Jun Byungheup, UNIST, Sun Haesang, UNIST

In this talk, we consider the distribution of ideals of a number field in an arithmetic progression, equivalently a ray class. We want to see how the distribution is related to nonvanishing properties of L-values. The number of ideals of norm bounded by sufficiently large number in an arithmetic progression is asymptotically computed by Rohrlich. We develop a method to count ideals whose norm is relatively small compared to the conductor. This is a joint work with Haesang Sun and Byungheup Jun.

2010 Mathematics Subject Classification: 11M20
Key Words and Phrases: ideal, number field, arithmetic progression

⋅ 20th-C-11:30 − 11:50 The Lindel$\ddot{\rm{o}}$f hypothesis for primes is equivalent to the Riemann hypothesis (Steve Gonek, Sidney Graham, Yoonbok Lee)

Steve Gonek(Univ. of Rochester), Sidney Graham(Central Michigan Univ.), 이윤복*(인천대)
Steve Gonek, University of Rochester, Sidney Graham, Central Michigan University, Yoonbok Lee*, Incheon National University

We recast the classical Lindel$\ddot{\rm{o}}$f hypothesis as an estimate for the sums $ \sum_{n\leq x}n^{-it}$. This leads us to propose that a more general form of the Lindel$\ddot{\rm{o}}$f hypothesis may be true, one involving estimates for sums of the type $$ \sum_{ \substack{n\leq x \\ n\in \mathscr{N} }}n^{-it},$$ where $\mathscr{N}$ can be a quite general sequence of real numbers. We support this with several examples and show that when $\mathscr{N}=\mathscr{P}$, the sequence of prime numbers, the truth of our conjecture is equivalent to the Riemann hypothesis. These results suggest to us that a general form of the Lindel$\ddot{\rm{o}}$f hypothesis may be both true and more fundamental than the classical Lindel$\ddot{\rm{o}}$f hypothesis and the Riemann hypothesis.

2010 Mathematics Subject Classification: 11M06
Key Words and Phrases: Lindel$\ddot{\rm{o}}$f hypothesis, Riemann hypothesis
Recent Results on Rings and Modules

⋅ 20th-B-09:00 − 09:40   Chair: Gyu Whan Chang (Incheon National University)

⋅ 20th-B-09:00 − 09:20 Endomorphisms of algebras and their semiclassical limits (Sei-Qwon Oh)

오세권(충남대)
Sei-Qwon Oh, Chungnam National University

We construct homomorphisms from endomorphisms of algebras into Poisson endomorphisms of their semiclassical limits.

2010 Mathematics Subject Classification: 17B63, 20K30
Key Words and Phrases: algebras, Poisson algebras, homomorphisms, semiclassical limits

⋅ 20th-B-09:20 − 09:40 Quantum nilpotent subalgebras of classical quantum groups and affine crystals (Il-Seung Jang, Jae-Hoon Kwon)

장일승*(서울대), 권재훈(서울대)
Il-Seung Jang*, Seoul National University, Jae-Hoon Kwon, Seoul National University

We study the crystal of quantum nilpotent subalgebras of quantum group $U_q(\mathfrak{g})$ associated to a maximal Levi subalgebra of type $A_{n-1}$, where $\mathfrak{g}$ is the simple Lie algebra of type $D_n$. We show that the crystal has natural affine crystal structure of type $D_n^{(1)}$, which is isomorphic to a (direct) limit of perfect Kirillov-Reshetikhin crystal $B^{n,s}$ for $s \ge 1$, and give a new polytope realization of $B^{n,s}$. Also, we show that a variation of Robinson–Schensted–Knuth correspondence for type $D$ due to Burge is an isomorphism of affine crystals and give an analogue of Greene's formula for type $D$ in terms of double paths on a lattice.

2010 Mathematics Subject Classification: 17B37, 22E46, 05E10
Key Words and Phrases: quantum groups, quantum nilpotent subalgebra, crystal graphs

⋅ 20th-B-09:50 − 10:30   Chair: Gangyong Lee (Chungnam National University)

⋅ 20th-B-09:50 − 10:10 Factorization in commutative semigroup (Chun Sangmin)

천상민(중앙대)
Chun Sangmin, Chung-Ang University

In this talk, we consider factorization properties in commutative semigroup. In particular, we explain unique factorization, finite factorization, bounded factorization.

2010 Mathematics Subject Classification: 20M12, 20M14
Key Words and Phrases: irreducible, prime element, unique factorization, finite factorization, bounded factorization

⋅ 20th-B-10:10 − 10:30 On the twisted semigroup rings (Dong Yeol Oh, Gyu Whan Chang)

오동렬*(조선대), 장규환(인천대)
Dong Yeol Oh*, Chosun University, Gyu Whan Chang, Incheon National University

Let $R$ be an integral domain, and $\Gamma$ be a torsion-free commutative cancellative monoid. We introduce a twisted function $t$ of $\Gamma$ on $R$ and the twisted semigroup ring $R^{t}[X; \Gamma]$ of $\Gamma$ over $R$ with respect to a twisted function $t$. And then we study the algebraic properties of the twisted semigroup rings.

2010 Mathematics Subject Classification: 13A02, 13A15, 13F15
Key Words and Phrases: twisted function, twisted semigroup ring, GCD-domain

⋅ 20th-C-10:40 − 11:20   Chair: Tai Keun Kwak (Daejin University)

⋅ 20th-C-10:40 − 11:00 Relative Loewy modules and relative Artinian modules (Hwankoo Kim, Jung Wook Lim, Dechuan Zhou)

김환구*(호서대), 임정욱(경북대), Dechuan Zhou(Southwest Univ. of Science and Technology)
Hwankoo Kim*, Hoseo University, Jung Wook Lim, Kyungpook National University, Dechuan Zhou, Southwest University of Science and Technology

In this talk, we present a theory for the structure of $\tau_w$-Loewy series of modules over commutative rings, where $\tau_w$ is the hereditary torsion theory induced by the so-called $w$-operation, and explore the relationship between $\tau_w$-Loewy modules and $w$-Artinian modules. More precisely, among other things, it is shown that for a $w$-module $M$, $M$ is $w$-Artinian if and only if $M$ is a $w$-locally Artinian module and $w$-$Sp(M)$ is finite, if and only if $M$ is a $\tau_w$-Loewy $R$-module and all its $\tau_w$-Loewy invariants are finite.

2010 Mathematics Subject Classification: 13E10, 13D30
Key Words and Phrases: $w$-operation, $w$-Artinian module, $w$-Loewy module, $w$-simple module

⋅ 20th-C-11:00 − 11:20 Dimension problems of power series rings over integral domain related with star operations (Minjae Kwon, Jungwook Lim)

권민재*(경북대), 임정욱(경북대)
Minjae Kwon*, Kyungpook National University, Jungwook Lim, Kyungpook National University

In this talk, $D$ is an integral domain and $D[\![X]\!]$ a power series ring over $D$. Star operations is one of the important tools used to classify the class of domains. Important examples of star operations are $v$-operation, $t$-operation and $w$-operation. I will discuss the dimension problems related with the star operations and power series rings.

2010 Mathematics Subject Classification: 13A15
Key Words and Phrases: power series rings, star operations, dimension theory

⋅ 20th-C-11:30 − 12:10   Chair: Hwankoo Kim (Hoseo University)

⋅ 20th-C-11:30 − 11:50 Module-theoretic characterizations of $t$-almost Dedekind domains (Dechuan Zhou, Hwankoo Kim, Fanggui Wang, Kui Hu)

Dechuan Zhou*(Southwest Univ. of Science and Technology), 김환구(호서대), Fanggui Wang(Sichuan Normal Univ.), Kui Hu(Southwest Univ. of Science and Technology)
Dechuan Zhou*, Southwest University of Science and Technology, Hwankoo Kim, Hoseo University, Fanggui Wang, Sichuan Normal University, Kui Hu, Southwest University of Science and Technology

Let $R$ be a commutative ring. In this paper, the concepts of almost $w$-projective modules and almost projective modules are provided. Let $M$ be an $R$-module. Then $M$ is said to be {\it almost $w$-projective} (resp., {\it almost projective}) if $\textrm{Ext}_R^1(M, N)=0$ for any $R_\frak{m}$-module $N$, where $\frak{m}$ is a maximal $w$-ideal (resp., a maximal ideal) of $R$. It is shown that an $R$-module $M$ satisfying that $M_\frak{m}$ is free over $R_\frak{m}$ for any maximal $w$-ideal (resp., any maximal ideal) $\frak{m}$ of $R$ is exactly almost $w$-projective (resp., almost projective). As applications, it is proved that a domain $R$ is $t$-almost Dedekind (resp., almost Dedekind) if and only if every submodule of projective modules is almost $w$-projective (resp., almost projective).

2010 Mathematics Subject Classification: 13F05
Key Words and Phrases: almost $w$-projective module, almost projective module, $t$-almost Dedekind domain, almost Dedekind domain

⋅ 20th-C-11:50 − 12:10 Strongly Gorenstein hereditary rings and example of domains whose ideals are SG-projective (Kui Hu)

Kui Hu(경북대)
Kui Hu, Kyungpook National University

In our recent two notes, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of SG-projective modules and the class of G-projective modules coincide if and only if the class of SG-projective modules is closed under extension. From this we get that a ring is a SG-hereditary ring if and only if every ideal is G-projective and the class of SG-projective modules is closed under extension. We also give some examples of domains whose ideals are SG-projective. Let $R$ be a 1-dimensional Noetherian domain with quotient field $K$ and $T$ be its integral closure in $K$. We prove that if $T$ is a PID and if the ideal $I = (R :_K T)$ is a 2-generated prime ideal, then every ideal of $R$ is SG-projective and further, every finitely generated torsion-free $R$-module is SG-projective.

2010 Mathematics Subject Classification: 13F05
Key Words and Phrases: strongly Gorenstein projective module, strongly Gorenstein hereditary ring, strongly Gorenstein Dedekind domain

⋅ 20th-D-14:30 − 14:50   Chair: Nam Kyun Kim (Hanbat National University)

⋅ 20th-D-14:30 − 14:50 On rings whose right regular elements are left regular (Tai Keun Kwak)

곽태근(대진대)
Tai Keun Kwak, Daejin University

A ring is said to be right (resp., left) regular-duo if every right (resp.,left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings,and the relations between one-sided regular-duo rings and related ring theoretic properties.

2010 Mathematics Subject Classification: 16U80, 16U20
Key Words and Phrases: right (left) regular element, right (left) regular-duo ring, upper triangular matrix ring, right (left) Ore domain

⋅ 20th-D-14:50 − 15:40   Chair: Dong Yeol Oh (Chosun University)

⋅ 20th-D-14:50 − 15:10 $\mathfrak{L}$-principally quasi-Baer modules (Gangyong Lee)

이강용(충남대)
Gangyong Lee, Chungnam National University

Right (left) principally quasi-Baer rings have been studied as these properties play an important role in the study of a quasi-Baer ring, introduced by Birkenmeier, Kim, and Park [1]. $\mathfrak{L}$-principally quasi-Baer (simply, $\mathfrak{L}$-p.q.-Baer) modules also play an important role in the study of a quasi-Baer module. The notion of quasi-Baer modules as in a general module theoretic setting was introduced by Rizvi and Roman [5], and then Lee and Rizvi studied the direct sum property for quasi-Baer modules, recently [3]. In addition, many other mathematicians investigated an $\mathfrak{L}$-p.q.-Baer module in a general module theoretic setting for a left p.q.-Baer ring (see [2]) as investigating that for quasi-Baer rings. In this talk, we obtain characterizations and properties of $\mathfrak{L}$-p.q.-Baer modules. Examples which show that the notion of an $\mathfrak{L}$-p.q.-Baer module is distinct from that of a p.q.-Baer module are provided. It is shown that every direct summand of an $\mathfrak{L}$-p.q.-Baer module inherits the property. Furthermore, we obtain that every direct sum of copies of an $\mathfrak{L}$-p.q.-Baer module is an $\mathfrak{L}$-p.q.-Baer module. We provide conditions when $\mathfrak{L}$-p.q.-Baer modules become quasi-Baer modules. In particular, if every direct sum of copies of a module $M$ is p.q.-Baer then the module $M$ is a quasi-Baer module.
[1] G.F. Birkenmeier, J.Y. Kim and J.K. Park, Principally quasi-Baer rings, Comm. Algebra $\bf 29$ (2001), no. 2, 638--660.
[2] P. Amirzadeh Dana and A. Moussavi, Endo-principally quasi-Baer modules, J. Algebra Appl. $\bf 15$ (2016), no. 2, 1550132, 19 pp.
[3] G. Lee and S.T. Rizvi, Direct sums of quasi-Baer modules, J. Algebra $\bf 456$ (2016), 76--92.
[4] G. Lee, Principally quasi-Baer modules and their generalizations, Comm. Algebra, Accepted.
[5] S.T. Rizvi and C.S. Roman, Baer and quasi-Baer modules, Comm. Algebra $\bf 32$ (2004) no. 1, 103--123.

2010 Mathematics Subject Classification: 16D40, 16D70, 16S50
Key Words and Phrases: principally quasi-Baer rings and modules, quasi-Baer rings and modules

⋅ 20th-D-15:20 − 15:40 Radicals in skew polynomial and skew Laurent polynomial rings (Nam Kyun Kim)

김남균(한밭대)
Nam Kyun Kim, Hanbat National University

In this paper, we first characterize the Levitzki radical of a skew (Laurent) polynomial ring by the prime ideals and skewed prime ideals in the base ring. We next provide formulas for the strongly prime radical and the uniformly strongly prime radical of skew (Laurent) polynomial rings.

2010 Mathematics Subject Classification: 16N40, 16S36
Key Words and Phrases: Levitzki radical, strongly prime radical, uniformly strongly prime radical
Elliptic and Parabolic Partial Differential Equations with Applications

⋅ 20th-B-09:00 − 09:40   Chair: Sun-Sig Byun (Seoul National University)

⋅ 20th-B-09:00 − 09:20 On eigenvalue problems with singular weight (Inbo Sim)

심인보(울산대)
Inbo Sim, University of Ulsan

Introducing Cuesta's eigenvalue problem for $p$-Laplacian with a singular weight, we study more singular cases on $N=1$ and $N \ge 2.$ We focus on $C^1$-regularity of solutions for the case $N=1$ and for the case $N \ge 2,$ we give a stronger singular weight condition to keep standard properties of the 1st eigenvalue and its corresponding eigenfunctions for (fractional) $p$-Laplacian.

2010 Mathematics Subject Classification: 35B32, 35B45, 35B50, 35J20, 35P15
Key Words and Phrases: singular weight, $C^1$-regularity, $p$-Laplacian, eigenvalue problem

⋅ 20th-B-09:20 − 09:40 Fundamental embedding results on the fractional Sobolev spaces with variable exponents and applications to the fractional $p(\cdot)$-Laplacian problems (Yun-Ho Kim, Ky Ho)

김연호*(상명대), Ky Ho(Duy Tan Univ.)
Yun-Ho Kim*, Sangmyung University, Ky Ho, Duy Tan University

In this talk, we first refine a fractional Sobolev space with variable exponent, as investigated in some recent works, and obtain fundamental imbeddings in our space that is new or is an improvement of known results. With these imbeddings, we then provide a sufficient condition guaranteeing global a-priori bounds for weak solutions of nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian. It is worth mentioning that such sufficient condition is new and our result is the first regularity result of the fractional $p(\cdot)$-Laplace problems to the best of authors' knowledge. The existence of infinitely many solutions of a class of problems involving the fractional $p(\cdot)$-Laplacian is also obtained as an application of our regularity result.

2010 Mathematics Subject Classification: 35B45, 35D30, 35J20, 35J60, 35J92, 46E35
Key Words and Phrases: fractional $p(\cdot)$-Laplacian, fractional Sobolev spaces with variable exponents, a-priori bounds, De Giorgi iteration, variational methods

⋅ 20th-B-09:50 − 10:30   Chair: Inbo Sim (University of Ulsan)

⋅ 20th-B-09:50 − 10:10 H$\ddot{\rm{o}}$lder continuity of weak type minimizers for functionals with generalized Orlicz growth (Mikyoung Lee, Petteri Harjulehto, Peter H$\ddot{\rm{a}}$st$\ddot{\rm{o}}$)

이미경*(부산대), Petteri Harjulehto(Univ. of Turku), Peter H$\ddot{\rm{a}}$st$\ddot{\rm{o}}$(Univ. of Turku)
Mikyoung Lee*, Pusan National University, Petteri Harjulehto, University of Turku, Peter H$\ddot{\rm{a}}$st$\ddot{\rm{o}}$, University of Turku

In this talk, we discuss H$\ddot{\rm{o}}$lder regularity of weak type minimizers of functionals with non-standard growth. Compared with previous results, it covers more general minimizing functionals and need fewer assumptions. This talk is based on the joint work with Petteri Harjulehto and Peter H$\ddot{\rm{a}}$st$\ddot{\rm{o}}$.

2010 Mathematics Subject Classification: 35B65, 35J60, 35A15
Key Words and Phrases: generalized Orlicz space, quasiminimizer, omega-minimizer, Harnack's inequality, H$\ddot{\rm{o}}$lder continuity

⋅ 20th-B-10:10 − 10:30 Higher integrability result for nonlinear elliptic systems with conormal boundary conditions (Youchan Kim, Seungjin Ryu, Pilsoo Shin)

김유찬*(서울시립대), 유승진(서울시립대), 신필수(서울대)
Youchan Kim*, University of Seoul, Seungjin Ryu, University of Seoul, Pilsoo Shin, Seoul National University

We prove a boundary higher integrability result for nonlinear elliptic systems with conormal boundary conditions in locally uniform domains. To do it, we derive a boundary version of Gehring-Giaquinta-Modica Lemma and Sobolev-Poincar$\acute{\rm{e}}$ type inequality in locally uniform domains. Our result plays a key role for handling the coefficients when obtaining Calder$\acute{\rm{o}}$n-Zygmund type estimates with conormal boundary conditions in nonsmooth domains.

2010 Mathematics Subject Classification: 35J47, 35J60
Key Words and Phrases: nonlinear elliptic system, higher integrability

⋅ 20th-C-10:40 − 11:20   Chair: Yun-Ho Kim (Sangmyung University)

⋅ 20th-C-10:40 − 11:00 Global H$\ddot{\rm{o}}$lder continuity for general elliptic equations involving measure (Sun-Sig Byun, Dian K. Palagachev, Pilsoo Shin)

변순식(서울대), Dian K. Palagachev(Politecnico di Bari), 신필수*(서울대)
Sun-Sig Byun, Seoul National University, Dian K. Palagachev, Politecnico di Bari, Pilsoo Shin*, Seoul National University

We deal with general quasilinear divergence-form coercive equations $$ \mathrm{div}\ a(x,u,Du) = b(x,u,Du) + \nu $$ whose prototype is the m-Laplacean equations. The nonlinear terms are given by Carath$\acute{\rm{e}}$odory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. We further assume that the Borel measure $\nu$ also belongs to suitable Morrey spaces. The fairly irregular boundary of the underlying domain is supposed to satisfy a capacity density condition.

2010 Mathematics Subject Classification: 35J60, 35B65
Key Words and Phrases: quasilinear elliptic operator, Morrey space; H$\ddot{\rm{o}}$lder continuity, measure data, non-smooth domain

⋅ 20th-C-11:00 − 11:20 Existence of suitable weak solutions to the incompressible MHD equations in time varying domains (Yunsoo Jang, Dugyu Kim)

장윤수*(연세대 응용해석 및 계산센터), 김두규(연세대 응용해석 및 계산센터)
Yunsoo Jang*, Yonsei University, Center for Mathematical Analysis and Computation (CMAC), Dugyu Kim, Yonsei University, Center for Mathematical Analysis and Computation (CMAC)

In cylindrical domains, suitable weak solutions are important for the regularity theory due to the local energy inequality. Suitable weak solutions to the incompressible MHD are obtained by approximate Stokes equations and heat equations. We apply the Schauder theory to time varying domains for the approximate equations whose solutions satisfy a uniform localized energy estimate including boundary. By the Leray-Schauder fixed point theorem, existence of solutions for the approximate equations follows and from compactness in Lebesgue and Sobolev spaces we prove the existence of suitable weak solutions in time varying domains.

2010 Mathematics Subject Classification: 35K20, 35A01, 35D30
Key Words and Phrases: MHD equation, time varying domain, suitable weak solution, Schauder theory, Leray-Schauder fixed point theorem

⋅ 20th-C-11:30 − 12:10   Chair: Jongmin Han (Kyung Hee University)

⋅ 20th-C-11:30 − 11:50 Formation of radial patterns via mixed attractive and repulsive interactions for Schr$\ddot{\rm{o}}$dinger systems (Youngae Lee, Jaeyoung Byeon, Zhi-Qiang Wang)

이영애*(경북대), 변재형(한국과학기술원), Zhi-Qiang Wang(Utah State Univ.)
Youngae Lee*, Kyungpook National University, Jaeyoung Byeon, KAIST, Zhi-Qiang Wang, Utah State University

This talk is concerned with the asymptotic behavior of least energy vector solutions for nonlinear Schr$\ddot{\rm{o}}$dinger systems with mixed couplings of attractive and repulsive forces. We focus on the radially symmetric case. Though there is still the general phenomenon of component-wise pattern formation with co-existence of partial synchronization and segregation for positive least energy vector solutions, in our case of radially symmetric domains, it turns out that the energy of synchronization part may be concentrated either on the center of the domain or on the boundary of the domain depending on the spatial dimension of the domain. This is a distinct new feature from the previous results due to the radially symmetric property. Our approach develops techniques of multi-scale asymptotic estimates.

2010 Mathematics Subject Classification: 35N05, 58J10, 58J20
Key Words and Phrases: vector solution, least energy solution, component-wise pattern formation

⋅ 20th-C-11:50 − 12:10 Semiclassical limit of solitons between Schr$\ddot{\rm{o}}$dinger-Poisson systems and Vlasov-Poisson systems (Jinmyoung Seok, Woocheol Choi, Younghun Hong)

석진명*(경기대), 최우철(인천대), 홍영훈(중앙대)
Jinmyoung Seok*, Kyonggi University, Woocheol Choi, Incheon National University, Younghun Hong, Chung-Ang University

Statistical dynamics of a system of collisionless particles in the gravitational field they generate is described by a Vlasov-Poisson system. By quantizing it, we get a Hartree system which describes statistical dynamics of a system of bosons. It has been an one of central problems of mathematical physics to rigorously prove the convergence of the semiclassical limit between IVP from Hatree systems to Vlasov-Poisson systems. In this talk, we turn to semiclassical limits of solitons between them. We will see that the semiclassical problems for solitons are independent of problems for IVP so completely different approaches, for example, variational approaches are required to prove it.

2010 Mathematics Subject Classification: 35J50, 35Q83
Key Words and Phrases: classical limit, soliton, variational method

⋅ 20th-D-14:30 − 15:10   Chair: Jinmyoung Seok (Kyonggi University)

⋅ 20th-D-14:30 − 14:50 String solutions of the gravitational Maxwell-Higgs model on compact surfaces (Jongmin Han)

한종민(경희대)
Jongmin Han, Kyung Hee University

In this talk, we consider a gravitational Maxwell-Higgs model on compact surfaces which describes the electromagnetic dynamics under the effect of gravity. Under a suitable restriction on the string number, we present the existence of maximal solutions by super- and sub-solution method and show the multiplicity by the degree argument. We also discuss the possibility of removing the restriction on the string number.

2010 Mathematics Subject Classification: 35J61, 35Q75, 81T13
Key Words and Phrases: gravitational Maxwell-Higgs model equation, super- and sub-solution method, multiplicity of solutions

⋅ 20th-D-14:50 − 15:10 Fine phase mixtures in 1-D hyperbolic-elliptic problem (Hyung Jun Choi, Seonghak Kim)

최형준(한국기술교육대), 김성학*(경북대)
Hyung Jun Choi, Korea University of Technology and Education, Seonghak Kim*, Kyungpook National University

In this talk, we present fine phase mixtures of weak solutions to the initial-boundary value problem for a class of hyperbolic-elliptic equations in one space dimension. Such solutions are constructed through a carefully modified method of convex integration to capture fine scale oscillations of spatial derivatives of solutions. We also include a numerical simulation via FEM to assert that our solutions are indeed reasonable candidates from infinitely many solutions to the problem.

2010 Mathematics Subject Classification: 35M13
Key Words and Phrases: fine phase mixtures, hyperbolic-elliptic problem, convex integration

⋅ 20th-D-15:20 − 16:00   Chair: Jinhae Park (Chungnam National University)

⋅ 20th-D-15:20 − 15:40 Periodic Maxwell-Chern-Simons vortices with concentrating property (Weiwei Ao, Oh Sang Kwon, Youngae Lee)

Weiwei Ao(Wuhan Univ.), 권오상*(충북대), 이영애(경북대)
Weiwei Ao, Wuhan University, Oh Sang Kwon*, Chungbuk National University, Youngae Lee, Kyungpook National University

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell-Chern-Simons (MCS) model was introduced by [Lee, Lee, Min (1990)] as a unified system of the classical Abelian-Higgs model (AH) and the Chern-Simons (CS) model. In this talk, one of our main goals is to improve and complete the (CS) limit result of (MCS) model without the restriction on either a particular class of solutions, the number of vortex points, or the Chern-Simons parameter. This result also provides the critical clue to answer the open problems raised by [Ricciardi, Tarantello (2000)] and [Tarantello (2004)], and we succeed to establish the existence of periodic Maxwell-Chern-Simons vortices satisfying the concentrating property.

2010 Mathematics Subject Classification: 35J47
Key Words and Phrases: bubbling solutions, asymptotic behaviors, Maxwell-Chern-Simons model

⋅ 20th-D-15:40 − 16:00 Singular vortex patches (Injee Jeong, Tarek M. Elgindi)

정인지*(고등과학원), Tarek M. Elgindi(Univ. of California-San Diego)
Injee Jeong*, KIAS, Tarek M. Elgindi, University of California-San Diego

Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for t>0.

2010 Mathematics Subject Classification: 35Q35
Key Words and Phrases: Euler equation, vortex patch, Yudovich theory
Kinetic Equations and Particle Dynamics

⋅ 20th-B-09:00 − 09:40   Chair: Injo Hur (Chonnam National University)

⋅ 20th-B-09:00 − 09:20 Well-posedness and stability result for a dispersion managed nonlinear Schr$\ddot{\rm{o}}$dinger equation (Mi-Ran Choi, Young-Ran Lee)

최미란*(서강대), 이영란(서강대)
Mi-Ran Choi*, Sogang University, Young-Ran Lee, Sogang University

We consider a dispersion managed nonlinear Schr$\ddot{\rm{o}}$dinger (DMNLS) equation with a power-law nonlinearity. This equation describes the propagation of an optical pulse in a dispersion managed fiber. In this talk, we show that the Cauchy problem of the DMNLS equation is well-posed in the energy space. We also present the orbital stability result for positive and zero average dispersion. Our approach is based on a suitable adjustment of NLS techniques to the dispersion management setting.

2010 Mathematics Subject Classification: 35Q55, 35A01
Key Words and Phrases: dispersion management, nonlinear Schr$\ddot{\rm{o}}$dinger equation, well-posedness, stability

⋅ 20th-B-09:20 − 09:40 Probabilistic well-posedness of the mass-critical NLS with radial data below $L^2(\mathbb{R}^d)$ (Gyeong Ha Hwang)

황경하(영남대)
Gyeong Ha Hwang, Yeungnam University

In this talk, we consider the Cauchy problem of the mass-critical nonlinear Schr$\ddot{\rm{o}}$dinger equation (NLS) with radial data below $L^2(\mathbb R^d)$. We prove almost sure local well-posedness along with small data global existence and scattering. Furthermore, we also derive conditional almost sure global well-posedness of the defocusing NLS under the assumption of a probabilistic a priori energy bound. The main ingredient is to establish the probabilistic radial Strichartz estimates.

2010 Mathematics Subject Classification: 35Q55, 42B37
Key Words and Phrases: nonlinear Schr$\ddot{\rm{o}}$dinger equation, almost sure well-posedness, Strichartz estimates, radial data, mass-critical nonlinearity

⋅ 20th-B-09:50 − 10:30   Chair: Hyung Ju Hwang (POSTECH)

⋅ 20th-B-09:50 − 10:10 The Boltzmann equation in bounded domains (Donghyun Lee)

이동현(포항공대)
Donghyun Lee, POSTECH

Mathematical theory for rarefied gas is described by kinetic theory. The Boltzmann equation is one of the fundamental equation for collisional kinetic theory. In particular, the equation explains how the gas particle system converges to an equilibrium. In this talk, we introduce recently results and mathematical tools for the Boltzmann equation in general bounded domains.

2010 Mathematics Subject Classification: 82B40
Key Words and Phrases: Kinetic theory, Boltzmann equation, equilibrium, boundary condition

⋅ 20th-B-10:10 − 10:30 Mean-field limit for collective behavior models with sharp sensitivity regions (Young-Pil Choi)

최영필(인하대)
Young-Pil Choi, Inha University

Emergent aggregation and flocking phenomena appearing in many biological systems are simple instances of collective behavior. Recently, they have been an active research in applied mathematics, biology, engineering, and physics. In this talk, we discuss the mean-field limit for a large class of swarming individual based models with local sharp sensitivity regions. We provide a quantitative error estimate between the solutions to the differential inclusion system corresponding to the particle descriptions and weak solutions to the expected limiting kinetic equation by employing tools from optimal transport theory.

2010 Mathematics Subject Classification: 92D25, 74A25, 76N10
Key Words and Phrases: mean-field limit, weak-strong stability, sharp sensitivity region

⋅ 20th-C-10:40 − 11:20   Chair: Young-Ran Lee (Sogang University)

⋅ 20th-C-10:40 − 11:00 1D Schr$\ddot{\rm{o}}$dinger operators and their generalization in spectral theory (Injo Hur)

허인조(전남대)
Injo Hur, Chonnam National University

In this talk, as a big picture, we explore 1D Schr$\ddot{\rm{o}}$dinger operators and their generalization in the viewpoint of spectral theory. For this we use so-called Weyl-Titchmarsh m-function, which are Herglotz functions and contain all spectral information to given operators. In the reverse way, inverse spectral theory will also be discussed.

2010 Mathematics Subject Classification: 34L55
Key Words and Phrases: canonical system, Herglotz function, Schr$\ddot{\rm{o}}$dinger operator, Weyl m-function

⋅ 20th-C-11:00 − 11:20 Global Kato type smoothing estimates via local ones for dispersive equations (Jungjin Lee)

이정진(울산과학기술원)
Jungjin Lee, UNIST

In this talk I will present that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schr$\ddot{\rm{o}}$dinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Recently, Sun, Trelat, Zhang and Zhong have shown it only in spacetime $\mathbb R \times \mathbb R$. Our result resolves this issue in higher dimensions. The other one is the sharp global-in-time maximal Schr$\ddot{\rm{o}}$dinger estimates. Recently, the pointwise convergence conjecture of the Schr$\ddot{\rm{o}}$dinger equation has been settled by Du–Guth–Li–Zhang and Du–Zhang. For this they proved related sharp local maximal Schr$\ddot{\rm{o}}$dinger estimates. By our result, these lead to the sharp global-in-time maximal Schr$\ddot{\rm{o}}$dinger estimates.

2010 Mathematics Subject Classification: 35B65, 42B10, 42B15, 42B37
Key Words and Phrases: Kato smoothing, maximal Schr$\ddot{\rm{o}}$dinger

⋅ 20th-C-11:30 − 12:10   Chair: Donghyun Lee (POSTECH)

⋅ 20th-C-11:30 − 11:50 The Einstein-Boltzmann equations with an accelerated cosmological expansion (Ho Lee)

이호(경희대)
Ho Lee, Kyung Hee University

We consider the Einstein-Boltzmann system as a cosmological model with an accelerated expansion. The simplest way of obtaining accelerated expansion is to use a positive cosmological constant. In this talk we consider a nonlinear scalar field to describe a more general situation. We make suitable assumptions on the spacetime, the scattering kernel, and the scalar field, and obtain global existence and asymptotic properties of solutions.

2010 Mathematics Subject Classification: 35Q20
Key Words and Phrases: Einstein, Boltzmann, cosmology

⋅ 20th-C-11:50 − 12:10 Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation (Jin Woo Jang, Robert M. Strain, Seok-Bae Yun)

장진우*(기초과학연구원 기하학수리물리연구단), Robert M. Strain(Univ. of Pennsylvania), 윤석배(성균관대)
Jin Woo Jang*, IBS-Center for Geometry and Physics, Robert M. Strain, University of Pennsylvania, Seok-Bae Yun, Sungkyunkwan University

We establish the propagation of the uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. For this, we will establish two types of estimates for the gain part of the collision operator: namely, a potential type estimate and a relativistic hyper-surface integral estimate. We then combine them using the relativistic counter-part of the Carlemann representation to derive a uniform control of the gain part, which gives the desired propagation of the uniform bounds of the solution. Two applications of the propagation of the uniform upper bound will also be presented : the celebrated $H$-theorem and asymptotic convergence of solutions to equilibrium.

2010 Mathematics Subject Classification: 35Q20, 76P05, 82C40, 35B65, 83A05
Key Words and Phrases: special relativity, Boltzmann equation, Carlemann representation, Uniform upper bounds, Boltzmann H-theorem
PDEs Related with Fluid Mechanics

⋅ 19th-A-15:00 − 16:30   Chair: Kyung Keun Kang (Yonsei University)

⋅ 19th-A-15:00 − 15:20 On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary (TongKeun Chang, Kyungkeun Kang)

장통근*(연세대), 강경근(연세대)
TongKeun Chang*, Yonsei University, Kyungkeun Kang, Yonsei University

We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.

2010 Mathematics Subject Classification: 35Q30
Key Words and Phrases: Caccioppoli's inequality, Stoke equations, Navier-Stokes equations, boundary

⋅ 19th-A-15:20 − 15:40 A remark on vortex stretching and anomalous dissipation for 3D Navier-Stokes (Injee Jeong, Tsuyoshi Yoneda)

정인지*(고등과학원), Tsuyoshi Yoneda(Univ. of Tokyo)
Injee Jeong*, KIAS, Tsuyoshi Yoneda, University of Tokyo

By DNS of Navier-Stokes turbulence, Goto-Saito-Kawahara (2017) showed that turbulence consists of a self-similar hierarchy of anti-parallel pairs of vortex tubes, in particular, stretching in larger-scale strain fields creates smaller-scale vortices. Inspired by their numerical result, we examine the Goto-Saito-Kawahara type of vortex-tubes behavior using the 3D incompressible Euler equations, and show that such behavior induces energy cascade in the absence of nonlinear scale-interaction. From this energy cascade, we prove a modified version of the zeroth-law.

2010 Mathematics Subject Classification: 35Q35, 35B30, 76F99
Key Words and Phrases: Euler equations, vortex-stretching, energy cascade, zeroth law

⋅ 19th-A-15:50 − 16:10 $L^{3}$ - solutions for the stationary Navier-Stokes equations in the exterior of a rotating obstacle (Dugyu Kim)

김두규(연세대 응용해석 및 계산센터)
Dugyu Kim, Yonsei University, Center for Mathematical Analysis and Computation (CMAC)

Consider the stationary motion of an incompressible Navier-Stokes fluid around a rotating body $\mathbb{R}^{3} \setminus \Omega$ which is also moving in the direction of the axis of rotation with constant velocity $- k e_{1}$. We assume that the angular velocity $\omega = |\omega| e_{1}$ is also constant and the external force is given by $f = {\rm div} F$. Then the motion is described by a variant of the stationary Navier-Stokes equations with the velocity $k e_{1}$ at infinity. Our main result is the existence of at least one solution $u$ satisfying $u - k e_{1} \in L^{3}(\Omega)$ for arbitrarily large $F \in L^{3/2}(\Omega)$. Moreover, we establish several regularity results to obtain an existence theorem for weak solutions $u$ satisfying $\nabla u \in L^{3/2}(\Omega)$ and $u - k e_{1} \in L^{3}(\Omega)$.

2010 Mathematics Subject Classification: 35D30, 76D05
Key Words and Phrases: stationary Navier-Stokes equations, weak and very weak solutions, exterior domain, rotating obstacle

⋅ 19th-A-16:10 − 16:30 On the growth of the support of positive vorticity for 2D Euler equation in an infinite cylinder (Kyudong Choi, Sergey Denisov)

최규동*(울산과학기술원), Sergey Denisov(Univ. of Wisconsin-Madison)
Kyudong Choi*, UNIST, Sergey Denisov, University of Wisconsin-Madison

We consider the incompressible 2D Euler equation in an infinite cylinder $R\times T$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d\le Ct^{1/3}\log^2 t$ when $t$ is large.

2010 Mathematics Subject Classification: 35Q31
Key Words and Phrases: 2d Euler, infinite cylinder, positive vorticity, support, growth

⋅ 20th-B-09:00 − 10:30   Chair: Jihoon Lee (Chung-Ang University)

⋅ 20th-B-09:00 − 09:20 Fluids with free-boundary and their vanishing viscosity limits (Donghyun Lee)

이동현(포항공대)
Donghyun Lee, POSTECH

The free-boundary problem in fluid studies fluid-vacuum interaction problem. Because of its complex boundary condition structure, their well-posedness is very hard for inviscid Euler, in particular. In this talk, we introduce how to obtain local solution of the free-boundary Euler through vanishing viscosity limit for with/without surface tension. And then we apply these methods to the free-boundary Magnetohydrodynamics.

2010 Mathematics Subject Classification: 74F10
Key Words and Phrases: free-boundary, Euler, Magnetohydrodynamics

⋅ 20th-B-09:20 − 09:40 On the Cauchy problem for the Hall MHD equations without resistivity (In-Jee Jeong, Sung-Jin Oh)

정인지(고등과학원), 오성진*(고등과학원)
In-Jee Jeong, KIAS, Sung-Jin Oh*, KIAS

In this talk, I will describe recent work with I.-J. Jeong on the Cauchy problem for the Hall-MHD equation without resistivity. This PDE, first investigated by Lighthill, is a one-fluid description of magnetized plasma with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. We demonstrate both ill and wellposedness of the Cauchy problem depending on the initial data. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character.

2010 Mathematics Subject Classification: 76W05, 35Q35, 35Q85
Key Words and Phrases: magnetohydrodynamics, Hall magnetohydrodynamics, electron magnetohydrodynamics, quasilinear dispersive equation

⋅ 20th-B-09:50 − 10:10 Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces (Young-Pil Choi)

최영필(인하대)
Young-Pil Choi, Inha University

In this talk, we study the convergence to global equilibrium of damped Euler equations under the influences of confining and nonlocal interaction forces. We construct Lyapunov functionals on the space of probability measures in order to investigate the convergence with respect to the 2-Wasserstein distance.

2010 Mathematics Subject Classification: 49K20, 76N99, 35L40
Key Words and Phrases: convergence to equilibrium, Euler equations, overdamped limit, Wasserstein distance

⋅ 20th-B-10:10 − 10:30 Compressible Navier--stokes system with general inflow-outflow boundary data on piecewise regular domains (Hi Jun Choe, Antonin Novotny, Minsuk Yang)

최희준(연세대), Antonin Novotny(Universite du Sud Toulon-Var), 양민석*(연세대)
Hi Jun Choe, Yonsei University, Antonin Novotny, Universite du Sud Toulon-Var, Minsuk Yang*, Yonsei University

We prove existence of weak solutions to the compressible Navier-Stokes system in barotropic regime with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded Lipschitz piecewise regular domain.

2010 Mathematics Subject Classification: 35Q35
Key Words and Phrases: compressible Navier-Stokes system

⋅ 20th-C-10:40 − 12:10   Chair: Kyung Keun Kang (Yonsei University)

⋅ 20th-C-10:40 − 11:00 2-D detached shocks past a blunt body (Myoung Jean Bae)

배명진(포항공대)
Myoung Jean Bae, POSTECH

The shock polar analysis shows that if a weak solution of steady Euler system for inviscid compressible flow has a shock past a blunt body, then the shock cannot be attached to the blunt body. This observation naturally raises a question on the existence of a detached shock solution past a blunt body. In this talk, I will demonstrate how a shock polar analysis is used to analyze two dimensional shocks past wedges or blunt body, and to review the recent result on the existence of detached shocks past a blunt body with a asymptotic state at far field, And, further open questions on detached shocks are discussed. This talk is based on collaboration with Wei Xiang at CUHK.

2010 Mathematics Subject Classification: 35A01, 35J25, 35J62, 35M10, 35Q31, 35R35, 76H05, 76L05, 76N10
Key Words and Phrases: blunt body, detached shock, Euler system, free boundary problem, inviscid compressible flow, irrotational, shock polar, strong shock, transonic shock

⋅ 20th-C-11:00 − 11:20 Nematic liquid crystal flows in an applied magnetic field (Soojung Kim)

김수정(숭실대)
Soojung Kim, Soongsil University

In this talk, we consider global solutions to the Ericksen-Leslie system for the Oseen-Frank model describing the hydrodynamics of nematic liquid crystals. In the presence of an applied magnetic field, the liquid crystals can be easily aligned along the direction of the external field. We first discuss magnetic field-induced instability of global solutions to the Ericksen-Leslie system for isotropic liquid crystals in dimension two. We also study the anisotropic liquid crystal flows.

2010 Mathematics Subject Classification: 82D30, 35K10, 35K59, 35K61
Key Words and Phrases: liquid crystal, Ericksen-Leslie system, dynamical instability

⋅ 20th-C-11:30 − 11:50 Existence and temporal decay of regular solutions to non-Newtonian fluids combined with the modified Maxwell equations (Kyungkeun Kang, Hwa Kil Kim, Jae Myoung Kim)

강경근(연세대), 김화길(한남대), 김재명*(연세대 응용해석 및 계산센터)
Kyungkeun Kang, Yonsei University, Hwa Kil Kim, Hannam University, Jae Myoung Kim*, Yonsei University, Center for Mathematical Analysis and Computation (CMAC)

We consider the Cauchy problem of a certain type of non-Newtonian fluids combined with the modified Maxwell equations in three dimensions. We establish local existence of unique regular solutions for sufficiently smooth initial data. In addition, the regular solutions are globally extended in time, provided that the $H^3$-norm of the initial data is small enough. Lastly, using the Fourier splitting method, we show that $H^l$-norms of the global regular solution decay with the rate of $(1+t)^{-(\frac{3}{4}+\frac{l}{2})}$ for $l \geq 0$, as time tends to infinity.

2010 Mathematics Subject Classification: 35Q30, 76A05, 76N10, 76W05
Key Words and Phrases: non-Newtonian fluid, Navier-Stokes equations, MHD equations, regular solution

⋅ 20th-C-11:50 − 12:10 Local well-posedness in Wasserstein space for a chemotaxis model coupled to Navier-Stokes equations (Kyungkeun Kang, Hwa Kil Kim)

강경근(연세대), 김화길*(한남대)
Kyungkeun Kang, Yonsei University, Hwa Kil Kim*, Hannam University

We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work, we established the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space using the optimal transportation technique. Exploiting this result, we constructed solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space. In this work, we refine the result on the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations under weaker assumptions on the initial data.

2010 Mathematics Subject Classification: 35K55, 75D05, 35Q84, 92B05
Key Words and Phrases: chemotaxis, Navier-Stokes equations, Fokker-Plank
Functional Analysis Related to Quantum Phenomenon

⋅ 20th-C-10:40 − 12:10 Chair: Hun Hee Lee (Seoul National University)

⋅ 20th-C-10:40 − 11:00 Quantum systems associated to projective representations on abelian groups (Hun Hee Lee)

이훈희(서울대)
Hun Hee Lee, Seoul National University

We will make a survey on quantum systems associated to projective representations on abelian groups, including Bosonic/Fermionic systems.

2010 Mathematics Subject Classification: 20C25
Key Words and Phrases: quantum system, projective representation

⋅ 20th-C-11:00 − 11:20 Spectral gap of the generator of quadratic open quantum harmonic oscillator (Ameur Dhahri, Franco Fagnola, Hyun Jae Yoo)

Ameur Dhahri(Politecnico di Milano), Franco Fagnola(Politecnico di Milano), 유현재*(한경대)
Ameur Dhahri, Politecnico di Milano, Franco Fagnola, Politecnico di Milano, Hyun Jae Yoo*, Hankyong National University

We introduce a quantum dynamical system called quadratic open quantum harmonic oscillator. It is a quantum open system model for an oscillator by means of a quantum Markov semigroup with formal Lindblad generator with operators arising in Fock representations of the $sl_2$ Lie algebra. We describe explicitly the invariant states and investigate the spectral gap of the generator. This is a joint work with Ameur Dhahri and Franco Fagnola.

2010 Mathematics Subject Classification: 46L55, 82C10, 60J27
Key Words and Phrases: quantum harmonic oscillator, quantum Markov semigroup, Fock representation, spectral gap

⋅ 20th-C-11:30 − 11:50 Noncommutative Ising sigma model and gauge actions (Hyun Ho Lee)

이현호(울산대)
Hyun Ho Lee, University of Ulsan

In this presentation, we consider a sigma model from noncommutative time-space to two-points space, having only two states. In particular our noncommutative time-space is a two dimensional space called a noncommutative torus. By taking Lagrangian approach to a field theory, we explain a class of fields corresponds to sections on a vector bundle are gauge invariant.

2010 Mathematics Subject Classification: 58B34
Key Words and Phrases: noncommutative torus, Ising model, noncommutative harmonic maps

⋅ 20th-C-11:50 − 12:10 Hellinger distance between quantum channels (Un Cig Ji)

지운식(충북대)
Un Cig Ji, Chungbuk National University

We start with the well-known Monge-Kantrovich optimal mass transportation problem. Motivated by the explicit form of Wasserstein distance between two Gaussian measures in terms of their covariance operators, we discuss Bures distance and fidelity between quantum states and then we study the Hellinger distance between two certain quantum channels.

2010 Mathematics Subject Classification: 81S25
Key Words and Phrases: Wasserstein distance, Gaussian measure, quantum state, Bures distance, fidelity, quantum channel, Hellinger distance
Submanifolds and Conformal Geometry

⋅ 20th-C-10:40 − 11:20 Chair: Young Jin Suh (Kyungpook National University)

⋅ 20th-C-10:40 − 11:00 On conformal transformation and curvature tensors (Byung Hak Kim, Sang Deok Lee, Jin Hyuk Choi)

김병학*(경희대), 이상덕(단국대), 최진혁(경희대)
Byung Hak Kim*, Kyung Hee University, Sang Deok Lee, Dankook University, Jin Hyuk Choi, Kyung Hee University

In this talk, we will report some recent results for conformal transformations on Riemannian manifolds and curvature related problems. We also talk about classification of Riemannian manifolds admitting conformal transformation or a certain curvature conditions.

2010 Mathematics Subject Classification: 53A30
Key Words and Phrases: conformal transformation, curvature tensor

⋅ 20th-C-11:00 − 11:20 A Kobayashi pseudo-distance for holomorphic bracket generating distributions (Aeryeong Seo)

서애령(경북대)
Aeryeong Seo, Kyungpook National University

In this talk, a generalization of Kobayashi pseudo-distance on complex manifolds with holomorphic bracket generating distributions will be presented. For a semisimple Lie group G, a G-homogeneous complex manifold M with an invariant holomorphic bracket generating distribution is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain with the superhorizontal distribution.

2010 Mathematics Subject Classification: 32Q45, 32M10
Key Words and Phrases: flag domain, Kobayashi hyperbolicity, negative holomorphic curvature

⋅ 20th-C-11:30 − 12:10 Chair: Byung Hak Kim (Kyung Hee University)

⋅ 20th-C-11:30 − 11:50 Hopf real hypersurface with singular normal vector field in the complex quadric (Hyunjin Lee, Young Jin Suh)

이현진*(경북대), 서영진(경북대)
Hyunjin Lee*, Kyungpook National University, Young Jin Suh, Kyungpook National University

The complex quadric $Q^{m}=SO_{m+2} /SO_{m} SO_{2}$ is a kind of Hermitian symmetric space with rank 2 of compact type, which is a complex hypersurface in complex projective space $\mathbb C P^{m}$. Also, it can be regard as a kind of real Grassmann manifold of compact type with rank 2. Accordingly, $Q^{m}$ admits both a complex conjugation structure $A$ and a K$\ddot{\rm{a}}$hler structure $J$, with anti-commutes with each other. By using these geometric structures of $Q^{m}$, which is given by an ambient space, we want to give some characterizations for a Hopf real hypersurface $M$ with the singular normal vector field $N$ in $Q^{m}$. In fact, there are two types of singular vector field $N \in TQ^{m}$, $\mathfrak A$-isotropic and $\mathfrak A$-principal. Motivated this, we give a characterization for Hopf real hypersurface $M$ with $\mathfrak A$-principal normal vector field in $Q^{m}$, $m \geq 3$. Furthermore, by using this result we give another characterization with respect to the generalized parallelism for the shape operator of $M$.

2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: real Grassmannians, complex quadric, real hypersurface, singular normal vector field, $\eta$-parallelism

⋅ 20th-C-11:50 − 12:10 Real hypersurface in the complex hyperbolic quadric with parallel Ricci tensor (Kim Gyu Jong, Imsoon Jeong, Young Jin Suh)

김규종*(우석대), 정임순(배재대), 서영진(경북대)
Kim Gyu Jong*, Woosuk University, Imsoon Jeong, Pai Chai University, Young Jin Suh, Kyungpook National University

We introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex hyperbolic quadric ${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$ . According to the $\frak A$-principal or the $\frak A$-isotropic unit normal vector field $N$, we give a complete classification of real hypersurfaces in ${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$ with Ricci parallelism.

2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: parallel Ricci tensor, K$\ddot{\rm{a}}$hler structure, complex conjugation, complex hyperbolic quadric
Mathematics for Data and Machine Learning

⋅ 20th-B-09:00 − 10:30 Chair: Young Ho Park (Kangwon National University)

⋅ 20th-B-09:00 − 09:20 Using an encoder-decoder model for a signal prediction (Kim Jongeun)

김종은(연세대)
Kim Jongeun, Yonsei University

The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder–Decoder. This is called sequence-to-sequence learning. It’s very powerful technique that be used to solve many kinds problems. We apply this algorithm to predict a time series.

2010 Mathematics Subject Classification: 68T50, 62P30
Key Words and Phrases: recurrent neural network, RNN encoder-decoder, sequence to sequence model, LSTM, GRU, attention

⋅ 20th-B-09:20 − 09:40 Brain network analysis using dynamical approach (Jaemin Park, Seonhee Lim, Jaejin Song)

박재민*(서울대), 임선희(서울대), 송재진(서울대 분당 병원)
Jaemin Park*, Seoul National University, Seonhee Lim, Seoul National University, Jaejin Song, Seoul National University Bundang Hospital

The brain network is represented by a binary graph, a weighted graph or a metric graph. Combinatorial and graph-theoretic approaches have been used widely to analyze brain networks. In this talk, we will concentrate on dynamical analysis for a metric graph. More precisely, we will introduce volume entropy and weights on vertices associated with it. The volume entropy is related to the quantity of information flow through the whole graph and weights on vertices gives local importance to each vertex for information flow. We will discuss implications on the tinnitus patients with/without hearing loss. This talk is based on joint works with Seonhee Lim and Jaejin Song.

2010 Mathematics Subject Classification: 05C82, 28D20
Key Words and Phrases: brain network, volume entropy, metric graph

⋅ 20th-B-09:50 − 10:10 An unpaired deep learning approach to image denoising in X-ray CT (Park Hyoung Suk, Jineon Baek, Sun Kyoung You, Jae Kyu Choi, Jin Keun Seo)

박형석*(국가수리과학연구소), 백진언(국가수리과학연구소), 유선경(충남대병원), 최재규(동제대), 서진근(연세대)
Park Hyoung Suk*, NIMS, Jineon Baek, NIMS, Sun Kyoung You, Chungnam National University Hospital, Jae Kyu Choi, Tongji University, Jin Keun Seo, Yonsei University

We propose a deep learning method for noisy low-dose computerized tomography (CT) images in the absence of paired training data. The proposed method approximately estimates the Maximum a Posteriori, which can be expressed as minimizing the Kullback-Leibler divergence in the generative adversarial network framework. This enables to training with unpaired noisy and noise-free CT images. Training datasets in our deep learning framework reflect prior information of target CT image. We performed numerical simulation and clinical experiments to show the validity of the proposed approach.

2010 Mathematics Subject Classification: 68T05
Key Words and Phrases: computerized tomography, denoising, low-dose, generative adversarial network, unsupervised learning

⋅ 20th-B-10:10 − 10:30 Data analysis based on GARCH model (Yong-Ki Ma)

마용기(공주대)
Yong-Ki Ma, Kongju National University

Starting out by observing realtime information analysis of data, we can understand characteristics of systems using optimal stochastic models. By using these models, we know theoretical basis about forecast, reasoning, response, and application. In this lecture, we introduce the GARCH model in order to analyze the data.

2010 Mathematics Subject Classification: 62M10
Key Words and Phrases: data analysis, GARCH model, dependency structure

⋅ 20th-C-10:40 − 12:10 Chair: Yong-Ki Ma (Kongju National University)

⋅ 20th-C-10:40 − 11:00 Interference fringe pattern denoising based on deep learning (Young Ho Park, Sunggoo Cho)

박영호(강원대), 조성구*(세명대)
Young Ho Park, Kangwon National University, Sunggoo Cho*, Semyung University

Interferometry is a high precision measurement technology using the phenomenon of interference of light to recover the phase distribution from the interference fringe pattern. However, noise in the interference fringe pattern can cause severe errors in estimating the phase distribution, and denoising is a necessary preprocessing step to analyze the interference fringe pattern. In this work, we reduce the noises of interference fringe patterns by applying deep learning. Training data samples are generated automatically by a mathematical algorithm, while most other training data samples for deep learning are usually collected through experimental methods.

2010 Mathematics Subject Classification: 68T99
Key Words and Phrases: interferometry, denoising, deep learning

⋅ 20th-C-11:00 − 11:20 Estimating the proper number of science museum visitors (Jin-Hwan Cho, Minjung Gim)

조진환*(국가수리과학연구소), 김민중(국가수리과학연구소)
Jin-Hwan Cho*, NIMS, Minjung Gim, NIMS

Mathematicians are unfamiliar with the process of problem solving in mathematics for industry, and they face up to unexpected difficulties in this process. At first, the clients do not explain their problem in a concrete way. They often do not know what the problem is exactly. The clients also do not provide data sets that are necessary to solve the problem. They even do not know what kind of data sets they possess. On the contrary, mathematicians do not have domain knowledge and engineering techniques including coding and visualization. Moreover, they often lack communication and presentation skills that are crucial to interpret the result for the clients. In this talk, we present the difficulties in the process of solving the problem that a science museum has commissioned recently.

2010 Mathematics Subject Classification: 62P25, 62H12, 62H30, 62M20
Key Words and Phrases: proper number of visitors, science museum, estimation, classification, clustering, linear regression

⋅ 20th-C-11:30 − 11:50 Text feature extraction from National Competency Standards (Jin-Hwan CHO, Dong Heon Choe, Yunyoung Park)

조진환(국가수리과학연구소), 최동헌(국가수리과학연구소), 박윤영*(국가수리과학연구소)
Jin-Hwan CHO, NIMS, Dong Heon Choe, NIMS, Yunyoung Park*, NIMS

Wikipedia says "Competency standards are a set of benchmarks used to assess the skills and knowledge that a person must demonstrate in the workplace to be seen as competent." The government of Korea has developed NCS (National Competency Standards) since 2013, that contains 948 units of competency up to now. NCS, a huge text based database, can be applied to many real world problems, e.g., job matching and education. In this talk we discuss how to use the text datasets from NCS, automatic dataset collection, data processing, and text feature extraction algorithms including tf-idf, latent Dirichlet allocation, and word2vec.

2010 Mathematics Subject Classification: 62T10, 68T50, 62M45, 62H30, 62M20
Key Words and Phrases: text feature extraction, natural language processing, National Competecy Standards

⋅ 20th-C-11:50 − 12:10 What and how to teach deep learning for math major? (Young Ho Park)

박영호(강원대)
Young Ho Park, Kangwon National University

Recently, mathematics community in Korea is emphasizing the role of mathematicians in the industries and trying to develop several subjects of industrial mathematics. One of the main concern lies on the deep learning. There are quite a lot of books for the deep learning. However, no standard text for the students majoring mathematics has been developed yet. The content of such text must be different from that for the students majoring the computer science or the engineering. We want to share the experience of the actual teaching of deep learning course in math department and discuss a sample syllabus with audiences.

2010 Mathematics Subject Classification: 92B20, 97R40
Key Words and Phrases: machine learning, deep learning, mathematics