Program and Abstracts

Algebra

⋅ 21st-C-14:50 − 16:20 Chair: Jinwon Choi (Sookmyung Women's University)

⋅ 21st-C-14:50 − 15:10 Minimality criterion for rational maps with good reduction on the projective line over $\mathbb{Q}_{p}$ (Sangtae Jeong, Dohyun Ko, Yongjae Kwon, Youngwoo Kwon)

정상태((인하대)), 고도현((인하대)), 권용재((인하대)), 권영우*((인하대))
Sangtae Jeong, Inha University, Dohyun Ko, Inha University, Yongjae Kwon, Inha University, Youngwoo Kwon*, Inha University

In this talk, we provide the minimality criterion for rational maps of degree at least degree 2 with good reduction on the projective line over $\mathbb{Q}_{p}$. This criterion enables us to obtain a complete description of minimal conditions for such a map on $\mathbb{P}^{1}(\mathbb{Q}_{p})$ in terms of its coefficients for $p=2$ or $3$. For an arbitrary prime $p\ge 5$, we present a method of characterizing minimal rational maps of degree$\ge2$ on $\mathbb{P}^{1}(\mathbb{Q}_{p})$, provided that the prescribed conditions for the reduction of the map on $\mathbb{P}^{1}(\mathbb{F}_{p})$ to be transitive are known. This is a joint work with Sangtae Jeong, Dohyun Ko and Yongjae Kwon.

2010 Mathematics Subject Classification: 37P05, 11S82, 37B05
Key Words and Phrases: $p$-adic dynamical systems, projective line, minimal, rational maps, good reduction

⋅ 21st-C-15:10 − 15:30 Recent developments on Higman's PORC conjecture and the number of finite $p$-groups (Seungjai Lee)

이승재((서울대))
Seungjai Lee, Seoul National University

For a positive integer $m$, let $f(m)$ denote the number of (isomorphism classes) of finite groups of order $m$. In 1960, Higman conjectured that for a given prime $p$ and an integer $n$, the number of finite $p$-groups of order $p^n$ will be polynomial on residue classes: there exist $N\in\mathbb{N}$ and finitely many polynomials $W_i(X)\in\mathbb{Q}[X]$ for $0\leq i \leq N-1$ such that for almost all prime $p$, it $p\equiv i\mod N$, then $f(p^n)=W_i(p)$. This is now known as Higman's PORC conjecture, and is still open for $n\geq8$. In this talk, we introduce and discuss some recent developments made on this ongoing old conjecture using certain tools from number theory. Also, we show how this problem is connected to the study of nilpotent Lie algebras.

2010 Mathematics Subject Classification: 20D15, 17B30
Key Words and Phrases: Finite $p$-groups, Higman's PORC conjecture, nilpotent Lie algebras

⋅ 21st-C-15:40 − 16:00 Essential dimension of semisimple groups of type $B$ (Sanghoon Baek, Yeongjong Kim)

백상훈((카이스트)), 김영종*((카이스트))
Sanghoon Baek, KAIST, Yeongjong Kim*, KAIST

We determine the essential dimension of an arbitrary semisimple group of type $B$ of the form \[G=\big({\bf Spin}(2n_{1}+1)\times\cdots \times {\bf Spin}(2n_{m}+1)\big)/{\bf \mu}\] over a field of characteristic $0$, for all $n_{1},\ldots, n_{m}\geq 7$, and a central subgroup $\bf \mu$ of \linebreak ${\bf Spin}(2n_{1}+1)\times\cdots \times {\bf Spin}(2n_{m}+1)$ not containing the center of ${\bf Spin}(2n_i+1)$ as a direct factor. We also find the essential dimension of $G$ for each of the following cases, where either $n_{i}=1$ for all $i$ or $m=2$, $n_{1}=1$, $2\leq n_{2}\leq 3$, $\bf\mu$ is the diagonal central subgroup for both cases.

2010 Mathematics Subject Classification: 14L24, 14L30
Key Words and Phrases: Linear algebraic group, essential dimension, generically free representation

⋅ 21st-C-16:00 − 16:20 Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two (Dae-Won Lee)

이대원((연세대))
Dae-Won Lee, Yonsei University

Due to the existence of phantom category, not every exceptional collection of line bundles is full. Nonetheless, some varieties are known to satisfy Kuznetsov's conjecture, e.g., $\mathbb{P}^n$, (weak) del Pezzo surfaces, Hirzebruch surfaces, smooth projective toric surfaces of Picard rank 3 or 4 and blow-ups of $\mathbb{P}^3$ at a point, line or a cubic curve. In this talk, we show that for any smooth toric Fano threefolds or fourfolds with Picard rank 2, exceptional collections consisting of line bundles of maximal length are full. We will briefly explain the strategy of the proof and examine in detail by an example.

2010 Mathematics Subject Classification: 14J26, 14M05
Key Words and Phrases: Derived category of coherent sheaves, semiorthogonal decomposition, full exceptional collection

⋅ 21st-D-16:40 − 17:50 Chair: Jinwon Choi (Sookmyung Women's University)

⋅ 21st-D-16:40 − 17:00 Values of inhomogeneous forms at S-integral points (Jiyoung Han, Anish Ghosh)

한지영*((Tata Institute of Fundamental Research)), Anish Ghosh((Tata Institute of Fundamental Research))
Jiyoung Han*, Tata Institute of Fundamental Research, Anish Ghosh, Tata Institute of Fundamental Research

The S-arithmetic space is one of generalization of the real vector space with the natural lattice structure with the linear group action. Hence Oppenheim conjecture, which is one of the famous problems about the distribution of the integral lattice with respect to a given quadratic form. We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogs of Rogers' moment formulae for S-arithmetic congruence quotients as well as for the space of affine lattices.

2010 Mathematics Subject Classification: 11P21, 11H55
Key Words and Phrases: Oppenheim conjecture, Siegel transform, variation of Rogers' higher moment formulas

⋅ 21st-D-17:00 − 17:20 Regularity properties of $k$-Brjuno and Wilton functions (Seul Bee Lee, Stefano Marmi, Izabela Petrykiewicz, Tanja I. Schindler)

이슬비*((Scuola Normale Superiore di Pisa)), Stefano Marmi((Scuola Normale Superiore di Pisa)), Izabela Petrykiewicz((Scuola Normale Superiore di Pisa)), Tanja I. Schindler((Scuola Normale Superiore di Pisa))
Seul Bee Lee*, Scuola Normale Superiore di Pisa, Stefano Marmi, Scuola Normale Superiore di Pisa, Izabela Petrykiewicz, Scuola Normale Superiore di Pisa, Tanja I. Schindler, Scuola Normale Superiore di Pisa

An irrational number is called a Brjuno number if $$\sum_{n=0}^\infty\frac{\log(q_{n+1})}{q_n}<\infty,$$ where $q_n$'s are the denominators of the principal convergents of the regular continued fraction. Yoccoz introduced the Brjuno function which characterizes Brjuno numbers to study analytic small divisors problems. In this talk, we introduce $k$-Brjuno functions and the Wilton function which are related to the classical Brjuno function. We will see their BMO properties and their behavior near rational numbers of their finite truncations. We then complexify the $k$-Brjuno and Wilton function and study their boundary behavior using an extension of the continued fraction algorithm to the complex plane. This is a joint work with Stefano Marmi, Izabela Petrykiewicz, and Tanja I. Schindler.

2010 Mathematics Subject Classification: 37F50, 11A55, 32A40
Key Words and Phrases: Small divisors, continued fractions, Bruno functions, complex boundary behaviour

⋅ 21st-D-17:30 − 17:50 PIR property in a subring of the Nagata ring (Hyungtae Baek, Jungwook Lim)

백형태*((경북대)), 임정욱((경북대))
Hyungtae Baek*, Kyungpook National University, Jungwook Lim, Kyungpook National University

Let $R$ be a commutative ring with identity, $R[X]$ the polynomial ring over $R$ and $$B = \{f \in R[X] \,|\, {\rm \ the \ coefficient \ of \ the \ least \ degree \ term \ of \ } f {\rm \ is \ } 1 \}.$$ Obviously, $B$ is a multiplicative subset of $R[X]$ and the quotient ring $R[X]_B$ is a subring of the Nagata ring of $R$ which contains $R[X]$. In this talk, we investigate the PIR property in $R[X]_B$. For this purpose, we examine the ideal theory, dimension theory and factorization theory in $R[X]_B$.

2010 Mathematics Subject Classification: 13A15
Key Words and Phrases: Nagata ring, Serre's conjecture ring, PIR
Analysis

⋅ 21st-B-10:30 − 12:00 Chair: Kyung Keun Kang (Yonsei University)

⋅ 21st-B-10:30 − 10:50 Lyapunov stability of planar waves to the reaction-diffusion equation with a non-Lipschitzian reaction term (Soyeun Jung, Eunkyung Ko)

정소연*((공주대)), 고은경((계명대))
Soyeun Jung*, Kongju National University, Eunkyung Ko, Keimyung University

We investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on $\mathbb R^n$, $n\geq 2$, with a $\alpha$-H\"older continuous ($0<\alpha<1$), but not necessarily smooth reaction term. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.

2010 Mathematics Subject Classification: 35B35
Key Words and Phrases: Stability, reaction-diffusion equation

⋅ 21st-B-10:50 − 11:10 On the Stability of the sine type $p$-radical functional equation (Gwang Hui Kim)

김광휘((강남대))
Gwang Hui Kim, Kangnam University

In this paper, we will find solutions and investigate the stability for the $p$-radical functional equations as follows: \begin{align*} f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2}&=f(x)f(y),\\ f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2}&=g(x)f(y),\\ f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2}&=f(x)g(y),\\ f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2}&=g(x)g(y).\\ \end{align*} with respect to the sine functional equation, where $p$ is an odd positive integer and $f$ is a complex valued function. Furthermore, the results are extended to Banach spaces.

2010 Mathematics Subject Classification: 39B82, 39B62, 39B52
Key Words and Phrases: Stability, superstability, sine functional equation, cosine functional equation

⋅ 21st-B-11:20 − 11:40 Bochner-Riesz means for the Hermite and special Hermite expansions (Jaehyeon Ryu, Sanghyuk Lee)

유재현*((전북대)), 이상혁((서울대))
Jaehyeon Ryu*, Jeonbuk National University, Sanghyuk Lee, Seoul National University

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions. Developing further Thangavelu's approach, we study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions, we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.

2010 Mathematics Subject Classification: 42B99, 42C10
Key Words and Phrases: Bochner-Riesz means, Hermite and special Hermite functions

⋅ 21st-B-11:40 − 12:00 On the first eigenvalue of the discrete $(p,q)$-Schr\"odinger operators (Jaeho Hwang, Soon-Yeong Chung)

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

In this paper, we discuss the existence, multiplicity and lower bounds of the first eigenvalue of the following $(p,q)$-Schr\"odinger systems \[ \begin{cases} -\Delta_{p,\omega}\phi+V_{\phi}(x,\phi,\psi)=\lambda a_{1}(x)|\phi|^{p-2}\phi+\lambda\alpha b(x)|\phi|^{\alpha-2}\phi|\psi|^{\beta},&\mbox{in }S,\\ -\Delta_{q,\omega}\psi+V_{\psi}(x,\phi,\psi)=\lambda a_{2}(x)|\psi|^{q-2}\psi+\lambda\beta b(x)|\phi|^{\alpha}|\psi|^{\beta-2}\psi,&\mbox{in }S, \end{cases} \] under the mixed boundary conditions, where $p>1$, $q>1$, $\alpha>0$, $\beta>0$ with $\frac{\alpha}{p}+\frac{\beta}{q}=1$, $a_{1}>0$, $a_{2}>0$, $b\geq 0$, $S$ is a network with a boundary $\partial S$, and $V$ is a real-valued quasi-homogeneous $C^{1}$-function of weights $(p,q)$. Using the results of the first eigenvalue, we discuss the existence of the following discrete $(p,q)$-Schr\"odinger systems \[ \begin{cases} -\Delta_{p,\omega}\phi+V_{\phi}(x,\phi,\psi)=F_{\phi}(x,\phi,\psi),&\mbox{in }S,\\ -\Delta_{q,\omega}\psi+V_{\psi}(x,\phi,\psi)=F_{\psi}(x,\phi,\psi),&\mbox{in }S. \end{cases} \]

2010 Mathematics Subject Classification: 35P15, 35P30
Key Words and Phrases: $(p,q)$-Laplace operator, eigenvalue

⋅ 21st-C-14:50 − 15:30 Chair: Seick Kim (Yonsei University)

⋅ 21st-C-14:50 − 15:10 Relationships among analogue of Wiener spaces over paths in abstract Wiener space with their applications (Dong Hyun Cho)

조동현((경기대))
Dong Hyun Cho, Kyonggi University

Let $C^{\mathbb B}[a,b]$ denote an analogue of Weiner space over paths in abstract Wiener space $\mathbb B$, that is, the space of continuous $\mathbb B$-valued functions on the interval $[a,b]$. In this talk, we investigate the translation of time interval $[a,b]$ defining the space $C^{\mathbb B}[a,b]$. As applications of the result, we derive various relationships between the analogue of Wiener space and its product spaces. Especially, we establish the relationships among $C^{\mathbb B}[a,b]$, $C^{\mathbb B}[a,s]\times C_0^{\mathbb B}[s,b]$ and $C^{\mathbb B}[a,s]\times C^{\mathbb B}[s,b]$, where $a<s<b$ and $C_0^{\mathbb B}[s,b]$ is the space of paths $x$ in $C^{\mathbb B}[s,b]$ with $x(s)=0$. Finally, we express the analogue of Wiener measures on $C^{\mathbb B}[a,b]$ as the analogue of Wiener measures on $C^{\mathbb B}[a,s]$ and $C^{\mathbb B}[s,b]$.

2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: Abstract Wiener space, analogue of Wiener space, Brownian motion, Gaussian measure, Wiener space, Wiener space over paths in abstract Wiener space

⋅ 21st-C-15:10 − 15:30 Scattering of the Hartree-type nonlinear Dirac system at critical regularity (Yonggeun Cho, Seokchang Hong, Kiyeon Lee)

조용근((전북대)), 홍석창*((서울대)), 이기연((이화여대))
Yonggeun Cho, Jeonbuk National University, Seokchang Hong*, Seoul National University, Kiyeon Lee, Ewha Womans University

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4\pi}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. We prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-data with additional angular regularity. We also show that only small amount of angular regularity is required to obtain global existence of solutions.

2010 Mathematics Subject Classification: 35Q55, 35Q40
Key Words and Phrases: Dirac equation, global well-posedness, scattering, Yukawa potential
Geometry

⋅ 21st-A-09:00 − 10:10   Chair: Jae Won Lee (Gyeongsang National University)

⋅ 21st-A-09:00 − 09:20 Greatest Ricci lower bounds of horospherical manifolds of Picard number one (Dongseon Hwang, Kyeong-Dong Park, Shin-Young Kim)

황동선((아주대)), 박경동((고등과학원)), 김신영*((기초과학연구원))
Dongseon Hwang, Ajou University, Kyeong-Dong Park, KIAS, Shin-Young Kim*, IBS

A horospherical variety is a normal G-variety such that a connected reductive algebraic group G acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The automorphishm groups of nonhomogenous projective horospherical manifolds of Picard number one are non-reductive, which implies that they admit no Kahler–Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kahler–Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian can be arbitrarily close to zero as its dimension grows.

2010 Mathematics Subject Classification: 14M27, 32Q20
Key Words and Phrases: Greatest Ricci lower bounds, horospherical varieties, algebraic moment polytopes

⋅ 21st-A-09:30 − 09:50 A theorem on quasi contact metric manifolds (Jihong Bae, JeongHyeong Park, Kouei Sekigawa)

배지홍*((성균관대)), 박정형((성균관대)), Kouei Sekigawa((Niigata University))
Jihong Bae*, Sungkyunkwan University, JeongHyeong Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University

A quasi contact metric manifold is an almost contact metric manifold if the corresponding almost Hermitian cone is a quasi K\"ahler manifold. We discuss basic properties of a quasi contact metric manifold. Then, we consider a quasi contact metric manifold with Killing characteristic vector field, and we shall prove that the quasi contact metric manifold is a K-contact manifold. This extends the definition of K-contact manifold.

2010 Mathematics Subject Classification: 53B20, 53C25
Key Words and Phrases: Quasi contact metric manifold, Killing vector field

⋅ 21st-A-09:50 − 10:10 Liouville type theorem for the transversally p-harmonic map on foliations (Xueshan Fu, Seoung Dal Jung)

Xueshan Fu*((제주대)), 정승달((제주대))
Xueshan Fu*, Jeju National University, Seoung Dal Jung, Jeju National University

In this paper, we define the concepts of the transversal p-energy and the transversally p-harmonic map. The first and second variational formulas for the transversally p-harmonic map are investigated explicitly according to the transversal p-energy. Simultaneously, the generalized Weitzenbock type formula is given and the Liouville type theorem for the transversally p-harmonic map is illustrated precisely.

2010 Mathematics Subject Classification: 53C
Key Words and Phrases: Transversally p-harmonic map, Riemannian foliations, Liouville type theorem

⋅ 21st-B-10:30 − 12:00   Chair: Sung-Eun Koh (Konkuk University)

⋅ 21st-B-10:30 − 10:50 Contact hypersurfaces in Hermitian symmetric spaces and CR-symmetry (Jong Taek Cho)

조종택((전남대))
Jong Taek Cho, Chonnam National University

In this talk, we give a realization of some class of contact Riemannian manifolds by real hypersurfaces of the complex quadric and its non-compact dual space, which enables us to prove the classification theorem of CR-symmetric pseudo-Hermitian manifolds.

2010 Mathematics Subject Classification: 53C40, 53C25, 53C35
Key Words and Phrases: Contact hypersurface, complex quadric, non-compact dual of complex quadric, CR-symmetry

⋅ 21st-B-10:50 − 11:10 A generalization of eta-Einstein structure on contact manifolds (Sun Hyang Chun, Jong Taek Cho, Yunhee Euh)

전선향*((조선대)), 조종택((전남대)), 어윤희((성균관대))
Sun Hyang Chun*, Chosun University, Jong Taek Cho, Chonnam National University, Yunhee Euh, Sungkyunkwan University

In this talk, we introduce the notion of weakly eta-Einstein structure as the generalization of eta-Einstein. Then we obtain the characteristic equation for a non-Sasakian contact $(k,\mu)$-space to be weakly eta-Einstein, which provides many interesting examples.

2010 Mathematics Subject Classification: 53C20, 53C25, 53D10
Key Words and Phrases: Weakly eta-Einstein, $(k,\mu)$-space, unit tangent sphere bundle

⋅ 21st-B-11:20 − 11:40 Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds (Kwang Soon Park)

박광순((서울시립대))
Kwang Soon Park, University of Seoul

As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic. Finally, we give some examples of such maps.

2010 Mathematics Subject Classification: 53C15, 53C26, 53C43
Key Words and Phrases: Horizontally conformal submersion, quaternionic manifold, totally geodesic

⋅ 21st-B-11:40 − 12:00 The Yamabe soliton with boundary (Jinwoo Shin, Pak Tung Ho)

신진우*((고등과학원)), Pak Tung Ho((서강대))
Jinwoo Shin*, KIAS, Pak Tung Ho, Sogang University

In this talk, we consider the Yamabe soliton with boundary and the conformal mean curvature soliton, which are natural generalizations of the Yamabe soliton. We define them in equation point of view and discuss their geometric properties. This is a joint work with Pak Tung Ho.

2010 Mathematics Subject Classification: 53E10, 53E20
Key Words and Phrases: Conformal flow, Yamabe problem, manifolds with boundary

⋅ 21st-C-14:50 − 16:00   Chair: Gye-Seon Lee (Sungkyunkwan University)

⋅ 21st-C-14:50 − 15:10 Real hypersurfaces in the complex quadric with cyclic Ricci semi-symmetric (Chang Hwa Woo, Young Jin Suh, Gyu Jong Kim)

우창화*((부경대)), 서영진((경북대)), 김규종((우석대))
Chang Hwa Woo*, Pukyong National University, Young Jin Suh, Kyungpook National University, Gyu Jong Kim, Woosuk University

First we introduce the notion of cyclic Rioci semi-symmetric real hypersurfaces in the complex quadric $Q^{m}=S O_{m+2} / S O_{m} S O_{2}$. Next we give a classification of real hypersurfaces in the complex quadric $Q^{\mathrm{m}}=\mathrm{SO}_{\mathrm{m}+2} / \mathrm{SO}_{\mathrm{m}} \mathrm{SO}_{2}$ with cyclic semisymmetric Ricci tensor.

2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C55
Key Words and Phrases: Cyclic Ricci semi-symmetric, A-isotropic, A-principal, Kaehler structure, complex conjugation, complex quadric

⋅ 21st-C-15:10 − 15:30 Ruled real hypersurfaces in real Grassmannians of compact type with rank 2 (Makoto Kimura, Hyunjin Lee, Juan de Dios P\'erez, Young Jin Suh)

Makoto Kimura((Ibaraki University)), 이현진*((경북대)), Juan de Dios P\'erez((University of Granada)), 서영진((경북대))
Makoto Kimura, Ibaraki University, Hyunjin Lee*, Kyungpook National University, Juan de Dios P\'erez, University of Granada, Young Jin Suh, Kyungpook National University

First we introduce the notions of $\eta$-parallel and $\eta$-commuting shape operator for real hypersurfaces in the complex quadric $Q^{m} = SO_{m+2}/SO_{m}SO_{2}$. Next we give a classification of real hypersurfaces in the complex quadric with such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex totally geodesic hyperplanes $Q^{m-1}$ in $Q^{m}$ whose unit normal vector field in $Q^{m}$ is $\mathfrak A$-principal.

2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: $\eta$-parallel shape operator, singular vector field, ruled real hypersurface, complex conjugation, complex quadric

⋅ 21st-C-15:40 − 16:00 Nonexistence of Einstein hypersurfaces in Damek-Ricci spaces (Sinhwi Kim, Yuri Nikolayevsky, JeongHyeong Park)

김신휘*((성균관대)), Yuri Nikolayevsky((La Trobe University)), 박정형((성균관대))
Sinhwi Kim*, Sungkyunkwan University, Yuri Nikolayevsky, La Trobe University, JeongHyeong Park, Sungkyunkwan University

Einstein hypersurfaces are scarce in rank-one symmetric spaces. Damek-Ricci spaces are harmonic Einstein Hadamard manifolds, and the natural generalizations of noncompact rank-one symmetric spaces. We prove that a Damek-Ricci space does not admit Einstein hypersurfaces.

2010 Mathematics Subject Classification: 53C25, 53C30, 53B25
Key Words and Phrases: Damek-Ricci space, Einstein hypersurface, Rank-one symmetric space
Topology

⋅ 21st-A-08:40 − 10:10   Chair: Hwa Jeong Lee (Dongguk University)

⋅ 21st-A-08:40 − 09:00 Mittag-Leffler inverse limit dynamics in generic sense (Namjip Koo, Hyunhee Lee)

구남집((충남대)), 이현희*((충남대))
Namjip Koo, Chungnam National University, Hyunhee Lee*, Chungnam National University

In this talk we study the inverse limit dynamical system satisfying the Mittag-Leffler condition and consisting of shifts of finite types. More precisely, we prove that if $f$ is conjugated to a Mittag-Leffler inverse limit system consisting generically of continuous maps with the shadowing property on compact Hausdorff spaces, then $f$ has shadowing property. Also we discuss preservation of stability in discrete dynamical systems under inverse limits.

2010 Mathematics Subject Classification: 34Cxx
Key Words and Phrases: Inverse limit, shadowing property, stability, generic

⋅ 21st-A-09:00 − 09:20 Gromov-Hausdorff stability of global attractors under Lipschitz perturbations of the domain and equation (Nguyen Thanh Nguyen, Keonhee Lee)

Nguyen Thanh Nguyen*((충남대)), 이건희((충남대))
Nguyen Thanh Nguyen*, Chungnam National University, Keonhee Lee, Chungnam National University

In this talk, we study the Gromov-Hausdorff stability of the global attractors under Lipschitz perturbations of the domain and equation. More precisely, we first give a sufficient condition for Gromov-Hausdorff stability of a semiflow on its global attractor. The result is applied to show that generically the semiflow induced by a reaction diffusion equation is Gromov-Hausdorff stable under Lipschitz perturbations of the domain and equation. This is a joint work with Keonhee Lee.

2010 Mathematics Subject Classification: 37L15
Key Words and Phrases: Stability, semiflow

⋅ 21st-A-09:30 − 09:50 Reducibility of product spaces with self-closeness number (Hyung Seok Oh, Howon Choi)

오형석*((고려대)), 최호원((고려대))
Hyung Seok Oh*, Korea University, Howon Choi, Korea University

For a CW-complex $X$, $\mathcal A_{\sharp}^{n}(X)$ is the set of all homotopy classes, which induce automorphism on the homotopy groups of $X$ up to $n$-dimension. We consider a quotient monoid $\bar{\mathcal A}_{\sharp}^{n}(X) = \mathcal A_{\sharp}^{n}(X)/\simeq_{n}$ where $\simeq_{n}$ is an equivalence relation defined as that $f\simeq_{n} g$ if $\pi_{k}(f)=\pi_{k}(g)$ for all $k\leq n$. Then, each element in $\bar{\mathcal A}_{\sharp}^{n}(X)$ has two types: the first type contains a self-homotopy equivalence class and the second type does not contain any self-homotopy equivalence class. In this paper, we introduce a subset $\mathcal A_{\sharp}^{n}(X ; f)$ of $\mathcal A_{\sharp}^{n}(X)$ and a group $\mathcal AT_{\sharp}^{n}(X)$. We study the properties of $\bar{\mathcal A}_{\sharp}^{n}(X)$ and $\mathcal AT_{\sharp}^{n}(X)$ to classify them in terms of $\mathcal A_{\sharp}^{n}(X ; f)$.

2010 Mathematics Subject Classification: 55P10, 55Q05
Key Words and Phrases: Self-homotopy equivalence, homotopy group, self-closeness number

⋅ 21st-A-09:50 − 10:10 Eilenberg--Moore spectral sequences converging to the homology of the iterated loop spaces of compact simple Lie groups (Younggi Choi)

최영기((서울대))
Younggi Choi, Seoul National University

The Eilenberg--Moore spectral sequences for the path loop fibrations converging to the mod $p$ (co)homology of the double and the triple loop spaces of any simply connected finite $H$-space collapse at the $E^{2}$-term. But the Eilenberg--Moore spectral sequences of the path loop fibrations converging to the mod $p$ (co)homology of the single loop spaces of compact simple Lie groups do not collapse at the $E_{2}$-term in some exceptional Lie group cases. In this talk, we will show that the Eilenberg--Moore spectral sequences of the path loop fibration converging to the mod $p$ (co)homology of the four fold loop spaces of any compact simple Lie group collapse at the $E^{2}$-term. The four fold loop spaces are related to the based gauge group.

2010 Mathematics Subject Classification: 55P35, 55R20, 55T20
Key Words and Phrases: Compact simple Lie group, iterated loop space, Eilenberg--Moore spectral sequence, based gauge group

⋅ 21st-B-10:30 − 11:10   Chair: Sangyop Lee (Chung-Ang University)

⋅ 21st-B-10:30 − 10:50 Anti-symplectic involutions for Lagrangian spheres in a symplectic quadric surface (Joontae Kim, Jiyeon Moon)

김준태*((고등과학원)), 문지연((서울대))
Joontae Kim*, KIAS, Jiyeon Moon, Seoul National University

We explore the topology of the space of anti-symplectic involutions of a monotone symplectic quadric $S^2\times S^2$. In particular, we show that any two anti-symplectic involutions of $S^2\times S^2$ whose fixed point set is a Lagrangian sphere is connected. This is a joint work with Jiyeon Moon.

2010 Mathematics Subject Classification: 53D12, 55M35, 32Q65
Key Words and Phrases: Anti-symplectic involution, symplectic quadric surface

⋅ 21st-B-10:50 − 11:10 Twisted Jacobian algebras as endomorphisms of matrix factorizations (Sangwook Lee)

이상욱((숭실대))
Sangwook Lee, Soongsil University

Given a polynomial $W$ with an isolated singularity, we can consider the Jacobian ring as its algebraic invariant. If in addition we have a group action on the polynomial ring fixing $W$, we are led to consider the twisted Jacobian ring which reflects the equivariant structure as well. Our main result is to show that the twisted Jacobian ring is isomorphic to the endomorphism ring of the ``twisted diagonal" matrix factorization. As an application, we suggest a way to investigate Floer theory of Lagrangian submanifolds which represent homological mirror functors.

2010 Mathematics Subject Classification: 53D37
Key Words and Phrases: Twisted Jacobian algebras, matrix factorizations, Floer theory

⋅ 21st-D-16:40 − 18:10   Chair: Jung Hoon Lee (Chonbuk National University)

⋅ 21st-D-16:40 − 17:00 On the invariants via Gauss diagrams (Sera Kim)

김세라((해군사관학교))
Sera Kim, Republic of Korea Naval Academy

Prof. Y. H. Im and I investigated the applications of the intersection index via the Gauss diagram for virtual knot and link diagrams. I used this method in order to define invariants for knot diagrams on the cylinder. This way could be applied to define the group presentations and the enhanced winding index through its Gauss diagram. This talk will show you how to apply this method to define invariants for other type of knots and links.

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: Virtual knot, Gauss diagram, intersection index, index polynomial, $n$-th polynomial

⋅ 21st-D-17:00 − 17:20 An explicit formula for the $A$-polynomial of the knot with Conway's notation $C(2n, 4)$ (Ji-Young Ham, Joongul Lee)

함지영*((건국대)), 이준걸((홍익대))
Ji-Young Ham*, Konkuk University, Joongul Lee, Hongik University

An explicit formula for the $A$-polynomial of the knot with Conway's notation $C(2n,4)$ is presented.

2010 Mathematics Subject Classification: 57M27, 57M25
Key Words and Phrases: $A$-polynomial, explicit formula, knot with Conway's notation $C(2n,4)$

⋅ 21st-D-17:30 − 17:50 Twisted 1-loop invariant and twisted Alexander polynomial (Seokbeom Yoon, Stavros Garoufalidis)

윤석범*((Universitat Autonoma de Barcelona)), Stavros Garoufalidis((SUSTech))
Seokbeom Yoon*, Universitat Autonoma de Barcelona, Stavros Garoufalidis, Southern University of Science and Technology

For an ideal triangulation of a cusped 3-manifold, the Neumann-Zagier matrices can be used to construct the so-called 1-loop invariant, which conjecturally equals to the adjoint Reidmeister torsion. In this talk, we would like to introduce the notion of twisted NZ matrices and define a twisted version of the 1-loop invariant which determines the 1-loop invariant of the cyclic covers, and conjecturally equals to the adjoint twisted Alexander polynomial.

2010 Mathematics Subject Classification: 57K10, 57K31, 57K32
Key Words and Phrases: Neumann-Zagier matrices, twisted 1-loop invariant, twisted Alexander polynomial

⋅ 21st-D-17:50 − 18:10 Discrete rational length-spectrum on a hyperbolic graph (Hyungryul Baik, Donggyun Seo, Hyunshik Shin)

백형렬((카이스트)), 서동균*((서울대)), 신현식((University of Georgia))
Hyungryul Baik, KAIST, Donggyun Seo*, Seoul National University, Hyunshik Shin, University of Georgia

The stable translation length of an isometry $g$ on a metric space is defined by $$\tau(g) := \lim_{n \to \infty}\frac{d(x, g^n(x))}{n}$$ for some element $x$. This gives an invariant of a conjugacy class of an isometry. Bowditch showed the action of the mapping class group of a surface on the curve complex has a discrete and rational length spectrum. In this talk, I will explain what criterion gives a discrete rational length spectrum. As an application, we found the action of a right-angled Artin group on an extension graph also satisfies this criterion so it also has a discrete rational length spectrum. This is a joint work with Hyungryul Baik and Hyunshik Shin.

2010 Mathematics Subject Classification: 57M60, 20F65, 20F67, 05C25, 20E08
Key Words and Phrases: Gromov-hyperbolic space, mapping class group, curve complex, right-angled Artin group, extension graph
Probability and Statistics

⋅ 21st-B-10:30 − 12:00 Chair: Kyeong-Hun Kim (Korea University)

⋅ 21st-B-10:30 − 10:50 Scaling limits of 2D symplectic ensembles (Seong-Mi Seo)

서성미((카이스트))
Seong-Mi Seo, KAIST

This talk will be about a class of ensembles of complex eigenvalues with symplectic symmetry. A basic model is the system of eigenvalues of symplectic Ginibre matrices, $n \times n$ matrices with i.i.d. quaternion Gaussian entries. The model forms a Pfaffian point process determined by a matrix-valued kernel of skew-orthogonal polynomials. I will discuss bulk and edge scaling limits of the system on the real line as $n \to \infty$ and explain how to obtain the universality of the limits for a class of radially symmetric potentials beyond the gaussian case.

2010 Mathematics Subject Classification: 60B20
Key Words and Phrases: 2D symplectic ensembles, skew-orthogonal polynomials, universality

⋅ 21st-B-10:50 − 11:10 Higher order fluctuations of extremal eigenvalues of sparse random matrices (Jaehun Lee)

이재훈((카이스트))
Jaehun Lee, KAIST

In this talk, we shall consider higher-order fluctuations of extremal eigenvalues of sparse random matrices on the regime $N^{\epsilon}\ll q \ll N^{1/2}$ where $q$ is the sparsity parameter. In the case $N^{1/9}\ll q\ll N^{1/6}$, Huang-Landon-Yau showed that eigenvalue rigidity can be recovered by removing asymptotically Gaussian fluctuations. We consider the regime $N^{\epsilon} \ll q\ll N^{1/6}$ and apply a higher-order random correction to the spectral edge in order to capture sub-leading order fluctuations of extremal eigenvalues. We establish local semicircle law near the edge under corrections and recover the eigenvalue rigidity by removing asymptotically Gaussian fluctuations arising from higher-order random corrections. Our proof relies on the method developed by Lee-Schnelli (Probab. Theory Related Fields, 171(1-2): 543--616, 2018), Huang-Landon-Yau (Ann. Probab., 48(2): 916--962, 2020) and He-Knowles (Probab. Theory Related Fields, 180: 985--1056, 2021).

2010 Mathematics Subject Classification: 60B20
Key Words and Phrases: Sparse random matrices, extremal eigenvalues, higher-order self-consistent equation, recursive moment estimates

⋅ 21st-B-11:20 − 11:40 Real eigenvalues of elliptic random matrices (Sung-Soo Byun, Nam-Gyu Kang, Ji Oon Lee, Jinyeop Lee)

변성수*((고등과학원)), 강남규((고등과학원)), 이지운((카이스트)), 이진엽((Ludwig Maximilian University of Munich))
Sung-Soo Byun*, KIAS, Nam-Gyu Kang, KIAS, Ji Oon Lee, KAIST, Jinyeop Lee, Ludwig Maximilian University of Munich

In this talk, I will discuss the real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the maximally non-Hermitian regime, which corresponds to the matrix model with real i.i.d. Gaussian entries, it was pioneered by Edelman, Kostlan, and Shub that the number of real eigenvalues is of order $\sqrt{N}$, where $N$ is the size of the matrix. Moreover, it can be heuristically conjectured that as a real random matrix becomes more symmetric, it gets more real eigenvalues. I will demonstrate that such a statement can be made rigorous by presenting the large-$N$ expansion of the mean and the variance of the number of real eigenvalues in the almost-Hermitian regime, where one can observe a non-trivial transition between real i.i.d. and real symmetric random matrices. Furthermore, I will explain the limiting empirical distributions of the real eigenvalues which interpolate the Wigner semicircle law and the uniform distribution.

2010 Mathematics Subject Classification: 60B20, 33C45
Key Words and Phrases: Real elliptic Ginibre matrices, real eigenvalues, almost-Hermitian regime, skew-orth\-ogonal polynomials

⋅ 21st-B-11:40 − 12:00 Fractal geometry of the valleys of the parabolic Anderson equation (Jaeyun Yi, Promit Ghosal)

이재윤*((포항공대)), Promit Ghosal((MIT))
Jaeyun Yi*, POSTECH, Promit Ghosal, MIT

In this talk, we study the macroscopic fractal properties of the deep valleys of the solution of the $(1+1)$-dimensional parabolic Anderson equation $$ \frac{\partial}{\partial t}u(t,x) =\frac{1}{2} \frac{\partial^2}{\partial x^2} u(t,x) + u(t,x)\dot{W}(t,x),~ t>0,~ x\in {\bf R},\quad u(0,x) \equiv u_0(x), \quad x\in {\bf R}, $$ where $\dot{W}$ is the time-space white noise and $0<\inf_{x\in {\bf R}} u_0(x)\leq \sup_{x\in {\bf R}} u_0(x)<\infty.$ Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are macroscopically monofractal. In fact, the macroscopic Hausdorff dimension of the valleys undergoes a phase transition at a point which does not depend on the initial data. The key tool of our proof is a lower bound to the lower tail probability of the parabolic Anderson equation. Such lower bound is obtained for the first time in this paper and will be derived by utilizing the connection between the parabolic Anderson equation and the Kardar-Parisi-Zhang equation. Our techniques of proving this lower bound can be extended to other models in the KPZ universality class including the KPZ fixed point.

2010 Mathematics Subject Classification: 60H15, 35R60, 60K37
Key Words and Phrases: Parabolic Anderson models, KPZ equation, macroscopic Hausdorff dimension
Applied Mathematics(including AI, Data Science)

⋅ 21st-A-09:00 − 10:10   Chair: Junseok Kim (Korea University)

⋅ 21st-A-09:00 − 09:20 Invertibility of circulant matrices and wavelets (Youngmi Hur)

허영미((연세대))
Youngmi Hur, Yonsei University

This talk will present sufficient conditions to guarantee the invertibility of circulant matrices of an arbitrary size with rational entries. These conditions are given as linear combinations of the entries in the first row with integer coefficients. Using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in $\mathbb Z_p$. Also discussed is how these matrices are related to a multivariate wavelet construction method. The talk is mostly based on work with Jeong-Ok Choi.

2010 Mathematics Subject Classification: 15B05, 15B36, 11A07, 42C40
Key Words and Phrases: Circulant matrix, cyclotomic polynomial, Ramanujan's sum, wavelets

⋅ 21st-A-09:30 − 09:50 Mathematical modeling batch adsorber containing adsorbents with various morphologies (Young-Sang Cho)

조영상((한국산업기술대))
Young-Sang Cho, Korea Polytechnic University

Adsorption phenomena in batch adsorber were interpreted for adsorbents with various shapes such as spherical, cylindrical, and slab-type particles as well as their core-shell structures. To this end, reaction-diffusion equations were solved by Laplace transform, assuming linear Henry's isotherm. Abundant calculation results were obtained from mathematical solutions for bulk concentration in adsorber and uptake profile inside adsorbents to study the effect of adjustable parameters such as adsorbent loading, Biot number, and inert core thickness. The results were compared for the shape of adsorbents. Rectangular isotherm was also considered for modeling using shrinking core model to predict the change of concentration. Time-dependent diffusion coefficient was also considered during modeling, and resultant reaction-diffusion equations could be solved by eigenfunction expansion method. The modeling results were compared with experimental data obtained from adsorbents like electro-spun fibers, which can be considered as infinitely long cylinders. \noindent Acknowledgement. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1F1A1047451).

2010 Mathematics Subject Classification: 35Q92
Key Words and Phrases: Reaction-diffusion equation, eigenfunction expansion, Laplace transform, adsorption

⋅ 21st-A-09:50 − 10:10 Effect of adaptive mutation on coexistence in the spatial cyclic competition system (Junpyo Park)

박준표((경희대))
Junpyo Park, Kyung Hee University

In the absence of mutation, a classic cyclic competition system on spatially extended systems can have a critical threshold of mobility to hamper coexistence. In this talk, we introduce the spatial cyclic competition system with adaptive mutation. By means of local expected fitness, we define the adaptive mutation that can be regarded as imitation behavior. From Monte-Carlo simulations, we find that species coexistence can be promoted by the intensification of adaptive mutation at certain high mobility regimes which lead the extinction traditionally. Such a phenomenon can be obtained even if extremely high mobility values are assumed. We also find that the critical mobility to break coexistence is shifted according to the frequency of adaptive mutation. To summarize, we may conclude that adaptive mutation can paly a beneficial role in coexistence of cyclically competing species by resisting mobility.

2010 Mathematics Subject Classification: 37N25, 91A22, 91A20
Key Words and Phrases: Rock-paper-scissors game, adaptive mutation, biodiveristy, critical mobility

⋅ 21st-B-10:30 − 11:40   Chair: Kwang-Yeon Kim (Kangwon National University)

⋅ 21st-B-10:30 − 10:50 $\alpha$-Stable convergence of heavy-tailed infinitely-wide neural networks (Paul Jung, Hoil Lee, Jiho Lee, Hongseok Yang)

Paul Jung((카이스트)), 이호일*((카이스트)), 이지호((카이스트)), 양홍석((카이스트))
Paul Jung, KAIST, Hoil Lee*, KAIST, Jiho Lee, KAIST, Hongseok Yang, KAIST

We consider infinitely-wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with i.i.d. samples from either a light-tailed (finite variance) or heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$-stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$-stable distribution having the same $\alpha$ parameter of that layer. We then extend a recent result of Favaro, Fortini, and Peluchetti (2020), to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$-stable distributions.

2010 Mathematics Subject Classification: 60F05, 60F17
Key Words and Phrases: Heavy-tailed distribution, stable process, multi-layer perceptrons, infinitely-wide neural networks, weak convergence

⋅ 21st-B-10:50 − 11:10 Linear convex splitting schemes to the gradient flows with a high-order polynomial potential (Seunggyu Lee, Sungha Yoon, Junseok Kim)

이승규((고려대)), 윤성하*((고려대)), 김준석((고려대))
Seunggyu Lee, Korea University, Sungha Yoon*, Korea University, Junseok Kim, Korea University

In this talk, we present linear convex splitting schemes to the gradient flows of Ginzburg-Landau free energy functional with a high-order polynomial potential. We prove that the proposed schemes are unconditionally energy stable and uniquely solvable. And the properties of equations and error estimates of numerical solutions are discussed. Several numerical simulations are illustrated to verify the theoretical results.

2010 Mathematics Subject Classification: 65M06, 65M22
Key Words and Phrases: Gradient flow, convex splitting, unconditionally energy stable

⋅ 21st-B-11:20 − 11:40 A locally calculable $P^3$-pressure using a $P^4$-velocity for incompressible Stokes equations (Chun Jae Park)

박춘재((건국대))
Chun Jae Park, Konkuk University

We will suggest a new finite element method to find a $P^4$-velocity and a $P^3$-pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a $P^4$-velocity. Then, using the calculated velocity, a locally calculable $P^3$-pressure will be defined component-wisely. The resulting $P^3$-pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the $P^4$-velocity. Besides, the method overcomes the problem of singular vertices or corners.

2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: FEM, Stokes, decoupled

⋅ 21st-C-14:50 − 16:20   Chair: Jin-Hwan Cho (NIMS)

⋅ 21st-C-14:50 − 15:10 Damped oscillation phenomenon in cucker-smale model with the discrete $p$-Laplacian (Jea Hyun Park)

박재현((국립군산대))
Jea Hyun Park, Kunsan National University

In this talk, we first introduce the Cucker-Smale model with the discrete $p$-Laplacian, which represents the nonlinear interaction of agents in the model, and we discuss the effect on the trajectories of agents according to the parameter $p$, which is related to the smoothness of the trajectory. In particular, we see that a new behavior of agents is generated by the parameter $p$. It is not shown in the traditional Cucker-Smale model. Second, we also introduce damped oscillation phenomena in the singular Cucker-Smale model with decentralized formation control, and then it was shown that the damped oscillation phenomenon can be effectively controlled by the parameter $p$ when the discrete $p$-Laplacian is applied to the singular Cucker-Smale model with decentralized formation control.

2010 Mathematics Subject Classification: 70B05, 92C17, 34D05
Key Words and Phrases: Synchronization, Cucker-Smale model, discrete p-Laplacian, damped oscillation

⋅ 21st-C-15:10 − 15:30 Quadrupole last-passage algorithm for charge densityon an L-shaped conducting surface (Chi-Ok Hwang, Hoseung Jang, Jongmin Park, Unjong Yu)

황치옥*((광주과학기술원)), 장호승((광주과학기술원)), 박종민((광주과학기술원)), 유운종((광주과학기술원))
Chi-Ok Hwang*, GIST, Hoseung Jang, GIST, Jongmin Park, GIST, Unjong Yu, GIST

We further develop the last-passage (LP) Monte Carlo algorithms for the charge density on an L-shaped conducting surface by deriving a quadrupole LP Green's function on the L-shaped flat surface. To demonstrate the algorithm, we compute charge densities on an L-shaped conductor in three-dimensional space and find that our results agree very well with the ones from the Given-Hwang's original last-passage algorithm on a flat surface. Compared with the Given-Hwang's LP algorithm, the quadrupole LP one is very suitable for charge density near the L-shaped edge boundary.

2010 Mathematics Subject Classification: 78M31
Key Words and Phrases: Monte Carlo, quadruple, last-passage, charge density

⋅ 21st-C-15:40 − 16:00 A geometric structure of acceleration and its role in making gradients small fast (Jongmin Lee, Chanwoo Park, Ernest K. Ryu)

이종민*((서울대)), 박찬우((서울대)), 류경석((서울대))
Jongmin Lee*, Seoul National University, Chanwoo Park, Seoul National University, Ernest K. Ryu, National University

Since Nesterov's seminal 1983 work, many accelerated first-order optimization methods have been proposed, but their analyses lacks a common unifying structure. In this work, we identify a geometric structure satisfied by a wide range of first-order accelerated methods. Using this geometric insight, we present several novel generalizations of accelerated methods. Most interesting among them is a method that reduces the squared gradient norm with $\mathcal{O}(1/K^4)$ rate in the prox-grad setup, faster than the $\mathcal{O}(1/K^3)$ rates of Nesterov's FGM or Kim and Fessler's FPGM-m.

2010 Mathematics Subject Classification: 90C26
Key Words and Phrases: Acceleration, convex optimization, Euclidean geometry, gradient norm, small gradients, making gradients small, composite optimization, OGM, FISTA, OGM-G, potential function-based, Lyapunov analysis

⋅ 21st-C-16:00 − 16:20 Extension of tumor perturbed model: data analysis with Markovian and non-Mar\-kovian distribution (Jong Hyuk Byun, In-Soo Yoon, Song Yi Lee, Hyun-Jong Cho, Il Hyo Jung)

변종혁*((부산대)), 윤인수((부산대)), 이송이((강원대)), 조현종((강원대)), 정일효((부산대))
Jong Hyuk Byun*, Pusan National University, In-Soo Yoon, Pusan National University, Song Yi Lee, Kangwon National University, Hyun-Jong Cho, Kangwon National University, Il Hyo Jung, Pusan National University

An extended model for a famous perturbed tumor model in pharmacokinetics and pharmacodynamics was developed. The newly established model demonstrated the transition rate of damaged cells expressed by a convolution of drug rate and age distribution. The previous model was derived using a specific phase-type distribution, and a more generalized model was proposed based on the published data. In addition, a fractional derivative model using non-Markovian distribution had the difference of end-behavior of dynamics compared to the models based on phase-type distribution. The result was accomplished by comparing with the existing model using newly studied data.

2010 Mathematics Subject Classification: 92B10,92C45
Key Words and Phrases: Pharamcokinetics and pharmacodynamcis, age-structured model, delay differential equations, phase-type distribution, fractional derivative model
Mathematical Education

⋅ 21st-C-14:50 − 15:10 Chair: Young Rock Kim (Hankuk University of Foreign Studies)

⋅ 21st-C-14:50 − 15:10 A study on the smallest rectangle including the planar figure of a cube (Kyoung Il Park)

박경일((한국과학영재학교))
Kyoung Il Park, KSA of KAIST

There are many planar figures representing a cube cut open and laid flat. In this paper, we study the smallest rectangle, including the planar figure of a cube, to improve the student creativity and space recognizing ability in geometry and mathematics education. We introduced the net as planer figure of cube via surgery and gluing, and studied the rectangles containing the net and its computational methods. And we provided some results and conjecture for the smallest rectangle including the net. We look forward to using this assignment to enhance student creativity in the classroom.

2010 Mathematics Subject Classification: 97G50
Key Words and Phrases: Planar figure, net, net surgery
Discrete Mathematics

⋅ 21st-A-09:30 − 10:10   Chair: Jeong-Ok Choi (GIST)

⋅ 21st-A-09:30 − 09:50 The Alon-Tarsi number of $K_5$-minor-free graphs (Toshiki Abe, Seog-Jin Kim, Kenta Ozeki)

Toshiki Abe((Miyakonojo College of Technology)), 김석진*((건국대)), Kenta Ozeki((Yoko\-hama National University))
Toshiki Abe, Miyakonojo College of Technology, Seog-Jin Kim*, Konkuk University, Kenta Ozeki, Yokohama National University

In this paper, we show the following three theorems. Let $G$ be a $K_5$-minor-free graph. Then Alon-Tarsi number of $G$ is at most $5$, there exists a matching $M$ of $G$ such that the Alon-Tarsi number of $G-M$ is at most $4$, and there exists a forest $F$ such that the Alon-Tarsi number of $G-E(F)$ is at most $3$.

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: Alon-Tarsi number, K5-minor-free

⋅ 21st-A-09:50 − 10:10 A dichotomy of list-switch homomorphism for signed graphs (Hyobin Kim, Mark H. Siggers)

김효빈*((경북대)), Mark H. Siggers((경북대))
Hyobin Kim*, Kyungpook National University, Mark H. Siggers, Kyungpook National University

The switch homomorphism problem Switch$(H)$ for a signed graph $H$ is known to be polynomial time solvable if $H$ has an switch-core with at most two edges and is otherwise $NP$-complete. We present results towards a similar dichotomy classification for the list version of the problem.

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: Signed graph, homomorphism complexity, switching

⋅ 21st-B-10:30 − 12:00   Chair: Young Soo Kwon (Yeungnam University)

⋅ 21st-B-10:30 − 10:50 The homomorphism reconfiguration problem for triangle free graphs (Mark H. Siggers)

21st-B-10:30 − 10:50
Mark H. Siggers, Kyungpook National University

We show that for triangle free reflexive graphs $G$ and $H$, two homomorphisms of the Hom-graph Hom$(G,H)$ are in the same component if and only if for every cycle in $G$ they induce cycles of the same homotopy type in $H$, and they meet another obvious girth condition. This yields a polynomial time solvable algorithm for the homomorphism reconfiguration problem for the graph $H$. The joint work with Jon Noel and Jae-baek Lee strengthens the link between topological obstructions and a dichotomy classification for the homomorphsim reconfiguration.

2010 Mathematics Subject Classification: 05C10
Key Words and Phrases: Graph homomorphism, reconfiguration, recolouring, algorithm complexity, CSP dichotomy

⋅ 21st-B-10:50 − 11:10 $\Gamma$-graphic delta-matroids and its applications (Donggyu Kim, Duksang Lee, Sang-il Oum)

김동규*((카이스트)), 이덕상((카이스트)), 엄상일((기초과학연구원 이산수학그룹))
Donggyu Kim*, KAIST, Duksang Lee, KAIST, Sang-il Oum, IBS Discrete Mathematics Group

For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose vertices are labelled by elements of $\Gamma$. We prove that a certain collection of edge sets of a $\Gamma$-labelled graph forms a delta-matroid, which we call a $\Gamma$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $\Gamma$-graphic delta-matroids.

2010 Mathematics Subject Classification: 05C85
Key Words and Phrases: Delta-matroid, group-labelled graph, greedy algorithm, tree packing

⋅ 21st-B-11:20 − 11:40 On independent domination of regular graphs (Eun-Kyung Cho, Ilkyoo Choi, Boram Park)

조은경*((한국외대)), 최일규((한국외대)), 박보람((아주대))
Eun-Kyung Cho*, Hankuk University of Foreign Studies, Ilkyoo Choi, Hankuk University of Foreign Studies, Boram Park, Ajou University

Given a graph $G$, a {\it dominating set} of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The {\it domination number} of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The {\it independent domination number} of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Note that every graph has an independent dominating set, as a maximal independent set is equivalent to an independent dominating set. Let $G$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 4$. Generalizing a result by Lam, Shiu, and Sun, we prove that $i(G)\le \frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by Goddard et al. in the affirmative. We also show that $\frac{i(G)}{\gamma(G)} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, \v Skrekovski, and Tepeh. In addition, we prove that a graph $G'$ with maximum degree at most $4$ satisfies $i(G') \le \frac{5}{9}|V(G')|$, which is also tight.

2010 Mathematics Subject Classification: 05C69
Key Words and Phrases: Domination number, independent domination number, regular graph, bounded maximum degree

⋅ 21st-B-11:40 − 12:00 Towards constant-factor approximation for chordal/distance-hereditary vertex deletion (Jungho Ahn, Eun Jung Kim, Euiwoong Lee)

안정호*((카이스트)), 김은정((CNRS, LAMSADE, Unversit\'e Paris-Dauphine)), 이의웅((University of Michigan))
Jungho Ahn*, KAIST, Eun Jung Kim, CNRS, LAMSADE, Unversité Paris-Dauphine, Euiwoong Lee, University of Michigan

For a family of graphs $F$, Weighted $F$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G-S$ is in $F$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $F$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

2010 Mathematics Subject Classification: 05C85, 68R10
Key Words and Phrases: Chordal graphs, distance-hereditary graphs, Ptolemaic graphs, approximation algorithm, linear programming, feedback vertex set

⋅ 21st-C-14:50 − 16:00   Chair: Seunghyun Seo (Kangwon National University)

⋅ 21st-C-14:50 − 15:10 On strong Sidon sets of integers (Yoshiharu Kohayakawa, Sang June Lee, Carlos Gustavo Moreira, Vojtech Rodl)

Yoshiharu Kohayakawa((University of Sao Paulo)), 이상준*((경희대)), Carlos Gustavo Moreira((IMPA and Nankai University)), Vojtech Rodl((Emory University))
Yoshiharu Kohayakawa, University of Sao Paulo, Sang June Lee*, Kyung Hee University, Carlos Gustavo Moreira, IMPA and Nankai University, Vojtech Rodl, Emory University

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 a joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojt\v ech R\"{o}dl.

2010 Mathematics Subject Classification: 05D40
Key Words and Phrases: Strong Sidon set, Sidon set

⋅ 21st-C-15:10 − 15:30 Leray numbers of tolerance complexes (Minki Kim, Alan Lew)

김민기*((기초과학연구원 이산수학그룹)), Alan Lew((Technion-Israel Institute of Technology))
Minki Kim*, IBS Discrete Mathematics Group, Alan Lew, Technion-Israel Institute of Technology

Let $K$ be a simplicial complex $K$ on vertex set $V$. $K$ is $d$-Leray if the homology group in dimension $d$ or greater is trivial for every induced subcomplex. $K$ is $d$-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most $d$ that is contained in a unique maximal face. Motivated by results of Montejano and Oliveros on ``tolerant" versions of Helly's theorem, we define the $t$-tolerance complex of $K$ as the simplicial complex on $V$ whose simplices are formed as the union of a simplex in $K$ and a vertex subset of size at most $t$. We prove that, for every $d$ and $t$, there exists a positive integer $h(t,d)$ such that the $t$-tolerance complex of a $d$-collapsible complex is always $h(t,d)$-Leray.

2010 Mathematics Subject Classification: 05E45
Key Words and Phrases: $d$-collapsible complex, $d$-Leary complex, $t$-tolerance complex

⋅ 21st-C-15:40 − 16:00 Lattice path interpretations of self-conjugate simultaneous core partitions (Hyunsoo Cho, JiSun Huh, Jaebum Sohn)

조현수*((이화여대)), 허지선((아주대)), 손재범((연세대))
Hyunsoo Cho*, Ewha Womans University, JiSun Huh, Ajou University, Jaebum Sohn, Yonsei University

In this talk, we introduce recent results on self-conjugate simultaneous core partitions. It contains the symmetric path interpretation for self-conjugate $(s,s+1,\dots,s+p)$-core partitions, an explicit formula for the number of self-conjugate $(s,s+1,\dots,s+p)$-core partitions, the free Motzkin path interpretation for self-conjugate $(s,s+d,\dots,s+pd)$-core partitions, and formulas for the number of self-conjugate $(s,s+d,s+2d)$-core partitions and self-conjugate $(s,s+d,s+2d,s+3d)$-core partitions.

2010 Mathematics Subject Classification: 05A17, 05A19
Key Words and Phrases: Simultaneous core partition, self-conjugate partition, Motzkin path
Cryptography

⋅ 21st-C-15:10 − 16:20 Chair: Kyung-Ah Shim (NIMS)

⋅ 21st-C-15:10 − 15:30 Enhancing differential privacy for federated learning at scale (Chunghun Baek, Sungwook Kim, Dongkyun Nam, Jihoon Park)

백충훈((삼성전자 삼성리서치)), 김성욱*((서울여대)), 남동균((삼성전자 삼성리서치)), 박지훈((삼성전자 삼성리서치))
Chunghun Baek, Samsung Research, Samsung Electronics, Sungwook Kim*, Seoul Women's University, Dongkyun Nam, Samsung Research, Samsung Electronics, Jihoon Park, Samsung Research, Samsung Electronics

Federated learning (FL) is an emerging technique that trains machine learning models across multiple de-centralized systems. It enables local devices to collaboratively learn a model by aggregating locally computed updates via a server. Privacy is a core aspect of FL, and recent works in this area are advancing the privacy guarantee of an FL network. To ensure rigorous privacy guarantee for FL, prior works have focused on methods to securely aggregate local updates and provide differential privacy (DP). In this talk, we investigate a new privacy risk for FL. Specifically, FL may frequently encounter unexpected user dropouts because it is implemented over a large-scale network. We first observe that user dropouts of an FL network may lead to failure in achieving the desired level of privacy protection, i.e., over-consumption of the privacy budget. Subsequently, we develop a DP mechanism robust to user dropouts by dynamically calibrating noise to consider the dropout rate. We evaluate the proposed technique to train convolutional neural network models on MNIST and FEMNIST datasets over a simulated FL network. Our results show that our approach significantly improves privacy guarantee for user dropouts compared to existing DP algorithms on FL networks.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Differential privacy, federated learning, user dropouts, noise calibration

⋅ 21st-C-15:40 − 16:00 Privacy-preserving median selection and secure aggregation in federated learning (Junghee Cheon, Keewoo Lee, Jaehyun Nam)

천정희((서울대)), 이기우((서울대)), 남재현*((서울대))
Junghee Cheon, Seoul National University, Keewoo Lee, Seoul National University, Jaehyun Nam*, Seoul National University

Federated learning (FL) is a distributed machine learning (ML) paradigm which enhances privacy by coordinating multiple data owners to train a shared ML model without needs to send their raw data to the server. However, FL alone does not provide full security. (i) Full Privacy: There still remains a room for so-called inference attacks by the server from the communicated local updates. (ii) Byzantine-tolerance: Plain FL is vulnerable to faulty or malicious clients, allowing them to compromise the model to be trained. In this paper, we propose the first practical FL solution which addresses both privacy and Byzantine-tolerance. Our solution is a non-trivial combination of two approaches: (i) Secure Multi-party Computation (MPC) for privacy and (ii) Coordinate-wise Median Aggregator for Byzantine-tolerance. The main technical contribution is on how to efficiently compute (approximate) medians in MPC setting. We propose using median-of-medians approach which is found to be MPC-friendly.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Federated learning, multi-party computation, median

⋅ 21st-C-16:00 − 16:20 New differentially 4-uniform permutations from modifications of the inverse function (Jaeseong Jeong, Namhun Koo, Soonhak Kwon)

정재성((성균관대)), 구남훈*((이화여대)), 권순학((성균관대))
Jaeseong Jeong, Sungkyunkwan University, Namhun Koo*, Ewha Womans University, Soonhak Kwon, Sungkyunkwan University

Finding permutations with good cryptographic parameters is a good research topic about constructing a secure S-box in substitution-permutation networks. In particular constructing differentially 4-uniform permutations has made considerable progress in recent years. In this paper, we present new differentially 4-uniform permutations from the inverse function composed by disjoint cycles. Our new differentially 4-uniform permutations have high nonlinearity and low differential-linear uniformity. We give the differential spectrum and the extended Walsh spectrum of some of our differentially 4-uniform permutations, and then we can see that they are CCZ-inequivalent to some permutations whose differential spectrum and extended Walsh spectrum are known.

2010 Mathematics Subject Classification: 94A60, 06E30
Key Words and Phrases: Differential uniformity, nonlinearity, differential-linear uniformity, differentially 4-uniform permutations

⋅ 21st-D-16:40 − 17:20 Chair: Woo-Hwan Kim (NSRI)

⋅ 21st-D-16:40 − 17:00 ZLR: a fast online authenticated encryption achieving full security (Wonseok Choi, Seongha Hwang, ByeongHak Lee, Jooyoung Lee)

최원석((카이스트)), 황성하*((카이스트)), 이병학((카이스트)), 이주영((카이스트))
Wonseok Choi, KAIST, Seongha Hwang*, KAIST, ByeongHak Lee, KAIST, Jooyoung Lee, KAIST

Online authenticated encryptions have been considered in many environments, especially in lightweight cryptography, due to its nature of online property: low latancy and constant memory usage. In this paper, we propose a new tweakable block cipher-based online authenticated encryption scheme, dubbed the ZHash-Luby-Rackoff ($\mathsf{ZLR}$) mode. $\mathsf{ZLR}$ follows the Encrypt-Mix-Encrypt paradigm. However, on the contrast to the other schemes using Encrypt-Mix-Encrypt paradigm such that $\mathsf{ELmE}$ or $\mathsf{CoLM}$, $\mathsf{ZLR}$ enjoys $n$-bit security by introducing larger internal state and maintaining it by efficient $\mathsf{ZHash}$ algorithm. In this way, $\mathsf{ZLR}$ can process $2n$-bit block calculations with only one primitive call for hashing and two primitive calls for encryption/decryption. As a result, $\mathsf{ZLR}$ has a rate $2/3$ and efficiently supports parallel computation with online nonce-misuse resistant property.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Online authenticated encryption, nonce-misuse resistance, tweakable block ciphers

⋅ 21st-D-17:00 − 17:20 Parallelizable message authentication codes with length independent security (ByeongHak Lee, Jincheol Ha, Hwigyeom Kim, Hyojun Kim, Jooyoung Lee)

이병학*((카이스트)), 하진철((카이스트)), 김휘겸((카이스트)), 김효준((카이스트)), 이주영((카이스트))
ByeongHak Lee*, KAIST, Jincheol Ha, KAIST, Hwigyeom Kim, KAIST, Hyojun Kim, KAIST, Jooyoung Lee, KAIST

In this paper, we propose new constructions for parallelizable MACs having length independent security. Most block cipher based parallelizable MACs have either length dependent security or expansive masking functions that limit parallelizability. Adopting the structure of chaining based MACs, we propose CPMAC and CLMAC, variants of PMAC and LightMAC having length independent security and highly parallelizable structures. In CPMAC and CLMAC, the input message is divided into segments of $w$ blocks for some fixed $w$. Each segment outputs its hash value through PHash and LightHash, which are the hash parts of PMAC and LightMAC, respectively, and the hash value is fed to the next segment. The final tag is computed by encrypting the sum of the all hash values using a different key from the hash. As we compute only $w$ blocks in parallel, all masks in CPMAC can be pre-computed and CLMAC uses only $\log w$ bits counter in each block. We prove the security of CPMAC and CLMAC by constructing an input collision graph, which is similar to the structured graph used in the proof of EMAC. Bounding the collision property of the input collision graph, we achieve the security of CPMAC and CLMAC up to $2^{n/2}$ queries assuming that $l \leq 2^{n/4}$ and $w$ is a constant.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Message authentication codes, length independent security, PMAC, lightMAC

Trends in Arithmetic Geometry

⋅ 21st-A-08:40 − 10:10 Chair: Sungmun Cho (POSTECH)

⋅ 21st-A-08:40 − 09:10 Bounds on the torsion subgroups of N\'eron–Severi groups (Hyuk Jun Kweon)

권혁준((MIT))
Hyuk Jun Kweon, MIT

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron--Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_\mathrm{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements in various situations.

2010 Mathematics Subject Classification: 14C05, 14C20, 14C22
Key Words and Phrases: N\'eron–Severi group, second cohomology

⋅ 21st-A-09:10 − 09:40 On a comparison between Dwork and rigid cohomologies of projective complements (Junyeong Park)

박준영((포항공대))
Junyeong Park, POSTECH

For homogeneous polynomials $G_1,\ldots,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork's theory. In this talk, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots,G_k$ to the Monsky-Washnitzer complex associated to some affine bundle over the complement $\mathbb{P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots,G_k$, which computes the rigid cohomology of $\mathbb{P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb{P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes respectively. Consequently, we extend Katz's comparison results for projective hypersurface complements to arbitrary projective complements.

2010 Mathematics Subject Classification: 14F30, 11G25, 14F40, 14G20
Key Words and Phrases: Dwork cohomology, rigid cohomology, the Cayley trick, twisted de Rham complexes

⋅ 21st-A-09:40 − 10:10 Diophantine study of Stokes matrices (Jun Ho Whang)

황준호((서울대))
Jun Ho Whang, Seoul National University

The space of $N$-by-$N$ Stokes matrices (i.e., unipotent upper triangular matrices) with fixed Coxeter invariant carries a natural nonlinear action of the braid group on $N$ strands. We will motivate the Diophantine study of this action on integral points. After reviewing the classical theory for $N = 3$ going back to Markoff, we describe recent joint work with Yu-Wei Fan for $N = 4$ and, time permitting, finish by presenting new results (and problems) for $N \ge 5$.

2010 Mathematics Subject Classification: 11D72
Key Words and Phrases: Diophantine equations, Stokes matrices

⋅ 21st-C-14:50 − 16:20 Chair: Yeansu Kim (Chonnam National University)

⋅ 21st-C-14:50 − 15:20 Prismatic crystals and crystalline representations in the relative case (Heng Du, Tong Liu, Yong Suk Moon, Koji Shimizu)

Heng Du((Tsinghua University)), Tong Liu((Purdue University)), 문용석*((University of Arizona)), Koji Shimizu((UC Berkeley))
Heng Du, Tsinghua University, Tong Liu, Purdue University, Yong Suk Moon*, University of Arizona, Koji Shimizu, UC Berkeley

Let $K$ be a complete discretely valued field of mixed characteristic $(0, p)$ with perfect residue field. Recently, Bhatt-Scholze proved that the category of lattices in crystalline $\mathrm{Gal}(\overline{K}/K)$-representations is equivalent to the category of prismatic $F$-crystals on $\mathcal{O}_K$. We study the analogous question in some special relative settings, and give a characterization of crystalline representations up to isogeny via ``completed" prismatic $F$-crystals.

2010 Mathematics Subject Classification: 11S99
Key Words and Phrases: Crystalline representation, prismatic crystal

⋅ 21st-C-15:20 − 15:50 Brauer groups in derived/spectral algebraic geometry (Chang-Yeon Chough)

조창연((모듈과 공간의 양자구조 연구센터))
Chang-Yeon Chough, QSMS

To\"en gave an affirmative answer to Grothendieck's question of comparing the Brauer group and the cohomological Brauer group of a scheme for all quasi-compact and quasi-separated (derived) schemes by introducing the notion of derived Azumaya algebras. I'll give a glimpse of the extension of this result to algebraic stacks in the setting of derived/spectral algebraic geometry. If time permits, my latest work on twisted derived equivalences in the derived/spectral setting, which is based on the aforementioned extension, will be presented.

2010 Mathematics Subject Classification: 14F22, 14F05, 14A20
Key Words and Phrases: Brauer groups, derived categories, algebraic stacks

⋅ 21st-C-15:50 − 16:20 Arithmetic of the moduli of fibrations (Jun Yong Park, Johannes Schmitt)

박준용*((Max Planck Institute for Mathematics)), Johannes Schmitt((Max Planck Institute for Mathematics))
Jun Yong Park*, Max Planck Institute for Mathematics, Johannes Schmitt, Max Planck Institute for Mathematics

We will first consider the explicit formulation of the moduli stacks of fibered algebraic surfaces as the moduli stacks of rational curves on $\overline{\mathcal{M}}_{g,n}$. This will lead us to exact arithmetic invariants on those moduli stacks via motives in the Grothendieck ring of stacks introduced by the late Torsten Ekedahl that will, in turn, render their point counts over finite fields. We then enumerate elliptic \& hyperelliptic curves over $\mathbb{P}_{\mathbb{F}_q}^{1}$ with precise lower order terms ordered by bounded discriminant height. Along the way, we will glance at 2 important analogies in number theory \& geometry that are 1. Global fields analogy, 2. Rational points \& Rational curves.

2010 Mathematics Subject Classification: 14D23, 14C35, 14J27
Key Words and Phrases: Arithmetic, moduli, Weierstrass fibrations, motive
Recent Developments in Automorphic Forms and q-Series

⋅ 21st-B-10:30 − 12:00 Chair: Sungmun Cho (POSTECH)

⋅ 21st-B-10:30 − 11:00 The weak test vector problems and local periods (Yeongseong Jo)

조영성((The University of Maine))
Yeongseong Jo, The University of Maine

Let $F$ be a non-archimedean local field of characteristic zero and $\pi$ an irreducible admissible generic representation of $GL_m(F)$. By definition, the local $L$-function associated to $\pi$ is a priori given by a finite sum of Rankin-Selberg type integrals. The purpose of this talk is to finds a pair of Whittaker functions and Schwartz-Bruhant functions so that so-called the formal $L$-function can be expressed as a single integral. This is referred to the weak test problem and in general the formal $L$-function divides the original one. Time permitting, we explain how one can extend it to archimedean local fields. The last part is a joint work with Peter Humphries.

2010 Mathematics Subject Classification: 11F70
Key Words and Phrases: Test vector problems, local Rankin-Selberg L-functions, local period integrals, newforms

⋅ 21st-B-11:00 − 11:30 Classification theorems for vector bundles on the Fargues-Fontaine curve (Serin Hong)

홍세린((University of Michigan))
Serin Hong, University of Michigan

A recent work of Fargues-Scholze reveals a remarkable connection between the Langlands program and its geometric counter part. Their main result is that the local Langlands correspondence can be formulated and constructed as the geometric Langlands correspondence for the Fargues-Fontaine curve. In this talk, we discuss several classification theorems about vector bundles on the curve, and discuss their connection to the work of Fargues-Scholze.

2010 Mathematics Subject Classification: 11F85, 14H60, 14G20
Key Words and Phrases: Local Langlands correspondence, Fargues-Fontaine curve, vector bundles

⋅ 21st-B-11:30 − 12:00 Special cycles on the unitary Shimura variety with minuscule parahoric level structure (Sungyoon Cho)

조성윤((University of Arizona))
Sungyoon Cho, University of Arizona

The Kudla-Rapoport conjecture predicts a relation between the arithmetic intersection numbers of special cycles on the unitary Shimura variety with hyperspecial level structure and the derivative of representation densities for hermitian forms. In this talk, we will give conjectural formulas that relate the arithmetic intersection numbers of special cycles on the unitary Shimura varieties with minuscule parahoric level structure and weighted representation densities. We will also discuss a reformulation of these conjectures in terms of weighted lattice counting.

2010 Mathematics Subject Classification: 14G35
Key Words and Phrases: Special cycles

⋅ 21st-D-16:40 − 18:10 Chair: Yeansu Kim (Chonnam National University)

⋅ 21st-D-16:40 − 17:10 Automorphic spectrum of adjacency operator on a non-uniform quotient of $PGL_3$ (Sanghoon Kwon, Soonki Hong)

권상훈*((가톨릭관동대)), 홍순기((가톨릭관동대))
Sanghoon Kwon*, Catholic Kwandong University, Soonki Hong, Catholic Kwandong University

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb F_q[t])$ quotient of affine building. We prove that the set of non-trivial approximate eigenvalues $(\lambda^+,\lambda^-)$ of the weighted adjacency operators on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb F_q[t])\setminus PGL(3,\mathbb F_q((t^{-1})))/PGL(3,\mathbb F_q[[t^{-1}]])$ is not a Ramanujan complex, from a combinatorial aspect. Based on the joint work with Soonki Hong.

2010 Mathematics Subject Classification: 20G25, 20E42, 47A25
Key Words and Phrases: Building, spectrum, adjacency operators, Ramanujan complex

⋅ 21st-D-17:10 − 17:40 On a converse theorem for finite $G_2$ (Qing Zhang)

21st-D-17:10 − 17:40
Qing Zhang, KAIST

For irreducible generic representations of classical groups over $p$-adic fields, one can define local gamma factors twisted by irreducible generic representations of $GL(n)$. These are important invariants of representations. The local converse problem asks whether these invariants can determine the irreducible generic representations uniquely. For classical groups, such local converse theorems are proved recently. In this talk, I will report a converse theorem for exceptional group $G_2$ over finite fields of odd characteristic. This is a joint work with Baiying Liu.

2010 Mathematics Subject Classification: 20C33
Key Words and Phrases: Converse theorem, exceptional group, multiplicity one

⋅ 21st-D-17:40 − 18:10 On the non-vanishing of central values of certain automorphic L-functions of ${\rm GL}(2n)$ (Jaeho Haan)

한재호((연세대))
Jaeho Haan, Yonsei University

The global Gan-Gross-Prasad (GGP) conjecture predicts that the non-vanishing of certain automorphic periods is equivalent to that of their central L-values. In this talk, we discuss the GGP conjecture for symplectic-metaplectic groups and the proof of one direction of the conjecture. This has an application on the non-vanishing of the quadratic twists of automorphic L-functions. That is, combined with the theory of the twisted automorphic descent, we can prove that for a cuspidal automorphic representation of ${\rm GL}_{2n}$, there are infinitely many its quadratic twists whose central L-values are non-zero.

2010 Mathematics Subject Classification: 11F67
Key Words and Phrases: Automorphic form, periods, L-values, Gan-Gross-Prasad conjecture
Representation Theory and Related Topics

⋅ 21st-A-08:40 − 09:40   Chair: Uhi Rinn Suh (Seoul National University)

⋅ 21st-A-08:40 − 09:10 Chiral homology and classical series identities (Jethro van Ekeren, Reimundo Heluani, George Andrews)

Jethro van Ekeren*((Universidade Federal Fluminense)), Reimundo Heluani((IMPA)), George Andrews((Penn State University))
Jethro van Ekeren*, Universidade Federal Fluminense, Reimundo Heluani, IMPA, George Andrews, Penn State University

I will discuss results of an ongoing project on the chiral homology of elliptic curves with coefficients in conformal vertex algebras. We find interesting links between this structure and classical number theoretic identities of Rogers-Ramanujan type (joint work with George Andrews and Reimundo Heluani).

2010 Mathematics Subject Classification: 17B69
Key Words and Phrases: Chiral homology, elliptic curve, conformal vertex algebra

⋅ 21st-A-09:10 − 09:40 Feigin-Semikhatov duality in W-superalgebras (Thomas Creutzig, Naoki Genra, Shigenori Nakatsuka, Ryo Sato)

Thomas Creutzig((University of Alberta)), Naoki Genra((University of Alberta)), Shigenori Nakatsuka((The University of Tokyo)), Ryo Sato*((Kyoto University))
Thomas Creutzig, University of Alberta, Naoki Genra, University of Alberta, Shigenori Nakatsuka, The University of Tokyo, Ryo Sato*, Kyoto university

W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. It has been known that princial W-algebras satisfy a certain duality relation (Feigin-Frenkel duality) which can be regarded as a quantization of the geometric Langlands correspondence. Recently, D. Gaiotto and M. Rapčák found dualities between more general W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of their conjecture is proved by T. Creutzig and A. Linshaw, and a more specific subclass (Feigin-Semikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will explain how to upgrade the latter case to the level of representation theory by using relative semi-infinite cohomology. This is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka.

2010 Mathematics Subject Classification: 17B69
Key Words and Phrases: W-superalgebra, Feigin-Seikhatov duality

⋅ 21st-A-09:50 − 10:20   Chair: Jae-Hoon Kwon (Seoul National University)

⋅ 21st-A-09:50 − 10:20 Representation theory via quantum field theory (Philsang Yoo)

유필상((Tsinghua University))
Philsang Yoo, Tsinghua University

It is known that some subjects in mathematics may be enriched by finding their context in physics. In this talk, we argue that representation theory is no exception. After explaining the physical context for representation theory of a finite group as the most basic example, we discuss a research program on how to use ideas of quantum field theory to study certain objects of interest in geometric representation theory.

2010 Mathematics Subject Classification: 57K16
Key Words and Phrases: Representation theory, topological quantum field theory

⋅ 21st-B-10:20 − 10:50   Chair: Jae-Hoon Kwon (Seoul National University)

⋅ 21st-B-10:20 − 10:50 A combinatorial formula for Kazhdan-Lusztig polynomials of sparse paving matroids and its connections to representation theory (George D. Nasr)

21st-B-10:20 − 10:50
George D. Nasr, University of Oregon

In 2016, Elias, Proudfoot, and Wakefield introduced Kazhdan-Lusztig polynomials for a class of combinatorial objects called matroids. Later, they presented the equivariant (representation-theoretic) version of these polynomials. We will introduce both these topics and discuss results in the case of sparse paving matroids. For the ordinary Kazhdan-Lusztig polynomials, we present a combinatorial formula using skew Young tableaux for the coefficients of these polynomials for sparse paving matroids. In the case of uniform matroids (a special case of sparse paving matroids), this formula results in a nice combinatorial interpretation that arises in the equivariant version of these polynomials.

2010 Mathematics Subject Classification: 17B10
Key Words and Phrases: Kazhdan-Lusztig polynomials, skew Young tableaux

⋅ 21st-B-11:00 − 12:00   Chair: Young-Tak Oh (Sogang University)

⋅ 21st-B-11:00 − 11:30 A tugging symmetry conjecture for the modified Macdonald polynomials (Seung Jin Lee, Jaeseong Oh)

이승진((서울대)), 오재성*((고등과학원))
Seung Jin Lee, Seoul National University, Jaeseong Oh*, KIAS

In this talk, we propose a conjecture which is a symmetry relation for the modified Macdonald polynomials of stretched partitions, $\widetilde{H}_{k\mu}[X;q,q^k]=\widetilde{H}_{\mu^k}[X;q^k,q]$. Using the LLT-expansion of the modified Macdonald polynomials and linear relations of the LLT polynomials, we prove the conjecture for one column shape partition $\mu=(1^l)$. This is based on the joint work with Seung Jin Lee.

2010 Mathematics Subject Classification: 05E10, 05E05, 05A30
Key Words and Phrases: Macdonald polynomials, LLT polynomials

⋅ 21st-B-11:30 − 12:00 Extensions of 0-Hecke modules for dual immaculate quasisymmetric functions by simple modules (Seung-Il Choi, Young-Hun Kim, Sun-Young Nam, Young-Tak Oh)

최승일((서울대)), 김영훈*((서울대)), 남선영((서강대)), 오영탁((서강대))
Seung-Il Choi, Seoul National University, Young-Hun Kim*, Seoul National University, Sun-Young Nam, Sogang University, Young-Tak Oh, Sogang University

For each composition $\alpha$, Berg {\it et al.} introduced an indecomposable $0$-Hecke module $\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the quasisymmetric characteristic image. In this talk, we study extensions of $\mathcal{V}_\alpha$ by simple modules. To do this, we construct a minimal projective presentation of $\mathcal{V}_\alpha$ and calculate $\mathrm{Ext}^1$-group between $\mathcal{V}_\alpha$ and simple modules. Then we describe all non-split extensions of $\mathcal{V}_\alpha$ by simple modules in a combinatorial manner. As a corollary, it is shown that $\mathcal{V}_\alpha$ is rigid. This is a joint work with S.-I. Choi, S.-Y. Nam, and Y.-T. Oh.

2010 Mathematics Subject Classification: 20C08, 16E30, 05E10
Key Words and Phrases: 0-Hecke algebra, dual immaculate functions, extensions, Ext-group
Mathematical Logic and Its Applications

⋅ 21st-B-10:30 − 11:10   Chair: Hyeungjoon Kim (UNIST)

⋅ 21st-B-10:30 − 10:50 One-variable theorem for the localized ATP (JinHoo Ahn, Joonhee Kim, Junguk Lee)

안진후((고등과학원)), 김준희*((연세대)), 이정욱((카이스트))
JinHoo Ahn, KIAS, Joonhee Kim*, Yonsei University, Junguk Lee, KAIST

We introduce a notion of a localized version of antichain tree property and discuss its combinatorial property. Unlike the localized version of SOP2 in the paper of J. Dobrowolski and H. Kim, it is not known whether the localized ATP is witnessed by a strongly indiscernible tree in general. But we observed that if ATP is localized by a finite type, then we can apply the strong modeling property and obtain a witness with strong indiscernibility. Additionally, if the finite type is a set of formulas of a single free variable, then we can find a witness of a single free variable that witness the localized ATP. As an application, we show that for a complete geometric theory, being NATP is preserved by taking dense/codense expansion. This is a joint work with JinHoo Ahn and Junguk Lee.

2010 Mathematics Subject Classification: 03C45, 03C60, 05C25, 12J10
Key Words and Phrases: Tree property, classification theory, antichain tree property

⋅ 21st-B-10:50 − 11:10 The AKE principle meets classification theory (JinHoo Ahn, Joonhee Kim, Junguk Lee)

안진후((고등과학원)), 김준희((연세대)), 이정욱*((카이스트))
JinHoo Ahn, KIAS, Joonhee Kim, Yonsei University, Junguk Lee*, KAIST

In model theory of valued fields, the Ax-Kochen-Ershov principle, in short, the AKE principle, says that two unramified henselian valued fields of characteristic zero are elementary equivalent if and only if their residue fields and their value groups are elementary equivalent. In Classification Theory for valued fields, the AKE principle still works. It is well-known that given unramified henselian valued field of equicharacteristic zero is NIP if and only if the residue field is NIP (by Delon), and it also works for NTP2 (by Chernikov). We will show that it works for NATP. As a corollary, a Hahn field over a omega-free PAC field of characteristic zero is NATP, which is SOP1 (by the value group) and TP2 (by the residue field, known to be NSOP1-TP2 by the results of Chatzidakis, and of Chernikov-Ramsey). This is a joint work with Jinhoo Ahn and Joonhee Kim.

2010 Mathematics Subject Classification: 03C45, 03C60, 12J10
Key Words and Phrases: The AKE principle, classification theory, antichan tree property

⋅ 21st-B-11:20 − 12:00   Chair: Martin A Ziegler (KAIST)

⋅ 21st-B-11:20 − 11:40 Bit-complexity analysis for the heat and wave equations and provably optimal algorithms (Svetlana Selivanova)

21st-B-11:20 − 11:40
Svetlana Selivanova, KAIST

In this talk we overview our recent results about bit-complexity of computing solutions to equations of mathematical physics, emphasizing the examples of the Heat and Wave equations. We show the solutions belong to PSPACE; for the periodic case in $\#P$; provably $\#P_1$-optimal for the Heat equation in general; PTIME computable (even uniformly!) in the case of analytic initial data. The underlying frameworks of Computable Analysis and Real-valued Bit-complexity provide a bridge between classical fields: ``discrete" computability and complexity theories, from one side; and ``continuous" analysis (including functional spaces, linear algebra, manifolds, differential equations etc.) from the other, to mutual benefit of both. Relating different problems of analysis to various complexity classes, such as $P$, $NP$, $\#P$, PSPACE, etc. gives valuable insights and helps create reliable and efficient algorithms for their solution. The talk is based on joint works with Ivan Koswara, Martin Ziegler (KAIST), Gleb Pogudin (\'Ecole Politechnique), Florian Steinberg (Inria), Holger Thies (Kyoto University).

2010 Mathematics Subject Classification: 03D15
Key Words and Phrases: Provably optimal algorithms, bit-complexity, computations with real numbers

⋅ 21st-B-11:40 − 12:00 Nondeterminism in constructive metric completeness (Michal Konečný, Sewon Park, Holger Thies)

Michal Konečný((Aston University)), 박세원*((카이스트)), Holger Thies((Kyoto University))
Michal Konečný, Aston University, Sewon Park*, KAIST, Holger Thies, Kyoto University

The computational content of constructive metric completeness is the operator that computes limits of Cauchy sequences. Using it, we can construct certified programs that compute interesting transcendental real numbers from sequences of approximations. The desired nondeterministic version of it would be to nondeterministically compute real numbers from nondeterministic approximations. An example is to compute a square root (nondeterministically picked amongst the multiple square roots) of a complex number by nondeterministic approximations. However, it is not obvious how this nondeterministic metric completeness can be and should be formalized. In this talk, we specify some primitive properties of the nondeterminism monad in the category of assemblies over Kleene's second algebra. It can be used together with ordinary metric completeness to realize nondeterministic metric completeness. We suggest that there is no need to extend the axiomatic structure of constructive real numbers in order to have nondeterministic metric completeness.

2010 Mathematics Subject Classification: 03F60
Key Words and Phrases: Computable analysis, axiomatic real numbers, nondeterminism, categorical interpretation, realizability

⋅ 21st-C-14:50 − 15:30   Chair: Joonhee Kim (Yonsei University)

⋅ 21st-C-14:50 − 15:10 Generic linear-order structures and NATP theories (JinHoo Ahn)

안진후((고등과학원))
JinHoo Ahn, KIAS

Let an $\mathcal(L)$-theory $T$ and $\mathcal(L)'\supset\mathcal(L)$ be given. For an $\mathcal(L)'$-theory $T'$ obtained by adding reasonable axioms to $T$, we can find a model companion of $T'$, which is often called a generic expansion of $T$. Generic expansion often gives proper examples in neo-stability theories such as simple unstable structures (by Chatzidakis and Pillay) and strict NTP$_2$ structures (by Chernikov). In this talk, we focus on the generic expansion by adding linear orders and show that if a theory $T$ is geometric and has quantifier elimination, then the non-existence of antichain tree property is preserved under the expansion of $T$.

2010 Mathematics Subject Classification: 03C45
Key Words and Phrases: Antichain tree property, Generic expansion, Dense linear ordering

⋅ 21st-C-15:10 − 15:30 Independence notions in NTP$_2$, NSOP$_1$ and NATP theories (Hyoyoon Lee)

이효윤((연세대))
Hyoyoon Lee, Yonsei University

By A. Chernikov and I. Kaplan, it is shown that in an NTP$_2$ theory, over a model, forking and dividing coincide and there's an analogue of the Kim's Lemma, called the existence of universal Morley sequence. Further, in an NSOP$_1$ theory, Kim's Lemma holds for Kim-dividing, shown by I. Kaplan and N. Ramsey. The proofs of these will be briefly examined in this talk, and then discuss the most natural candidate for pre-independence notion in an NATP theory, which contains NTP$_2$ and NSOP$_1$ theories.

2010 Mathematics Subject Classification: 03C45
Key Words and Phrases: Independence relation

⋅ 21st-C-15:40 − 16:00   Chair: Sewon Park (KAIST)

⋅ 21st-C-15:40 − 16:00 Coloring sets in $\mathbb{R}^n$, focusing on drawing the Mandelbrot set (Hyunwoo Lee, Jihoon Hyun, Jiman Hwang, Martin Ziegler)

이현우((카이스트)), 현지훈*((카이스트)), 황지만((카이스트)), Martin Ziegler((카이스트))
Hyunwoo Lee, KAIST, Jihoon Hyun*, KAIST, Jiman Hwang, KAIST, Martin Ziegler, KAIST

It is well known that no proper nonempty subsets of $\mathbb{R}^n$ can give a clear answer to the query which asks if the point is in the set or not, within a finite time. Also, most previous works on drawing a set mainly focuses on open or closed sets. Jiman and Sewon had also worked on computing a connected path in Euclidean space. In this session, we give a new way of drawing a set in $\mathbb{R}^n$. For each cell of desired resolution, we color the cell as positive only if the cell intersects with the set, negative only if the cell doesn't intersect with the set, and 0 only if the cell is close enough to the boundary of the set. We will discuss more about properties of such method, discuss about an algorithm to draw a set with certain constraints, and finally apply it to draw a Mandelbrot set in $\mathbb{R}^2$.

2010 Mathematics Subject Classification: 37F46
Key Words and Phrases: Computability, Mandelbrot set
Commutative Algebra and Related Fields

⋅ 21st-C-14:50 − 16:40 Chair: Sunsook Noh (Ewha Womans University)

⋅ 21st-C-14:50 − 15:10 Uniformly $S$-Noetherian rings (Wei Qi, Hwankoo Kim, Fanggui Wang, Mingzhao Chen, Wei Zhao)

Wei Qi((Sichuan Normal University)), 김환구*((호서대)), Fanggui Wang((Sich\-uan Normal University)), Mingzhao Chen((Leshan Normal University)), Wei Zhao ((ABa Teachers University))
Wei Qi, Sichuan Normal University, Hwankoo Kim*, Hoseo University, Fanggui Wang, Sichuan Normal University, Mingzhao Chen, Leshan Normal University, Wei Zhao, ABa Teachers University

Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ring provided that there exists $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated subideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for uniformly $S$-Noetherian rings. Besides the uniformly $S$-Noetherian properties on several ring constructions are given. The notion of $S$-injective modules is also introduced and studied. Finally we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetherian rings.

2010 Mathematics Subject Classification: 13E05, 13A15
Key Words and Phrases: Uniformly $S$-Noetherian ring, $S$-Noetherian ring, $S$-injective module, ring construction

⋅ 21st-C-15:10 − 15:30 Remarks on Castelnuovo theory (Wan Seok Lee, Euisung Park)

이완석*((부경대)), 박의성((고려대))
Wan Seok Lee*, Pukyong National University, Euisung Park, Korea University

For a nondegenerate projective curve $C \subset \mathbb{P}^r$ of degree $d$ and arithmetic genus $g$, G. Castelnuovo obtained an upper bound $\pi_0(d,r)$ on genus $g$. He further characterized the extremal curves, say Castelnuovo curves, showing in particular that the curves of degree $d \geq 2r+1$ attain the maximally possible value $\pi_0(d,r)$ lie on a surface of minimal degree. Castelnuovo's theorem has been extended recently by Eisenbud-Harris and I. Petrakiev. Their main idea is to investigate the Hilbert function and the configuration of a set of finite points which is a general hyperplane section of $C$. This idea was reinterpreted as the relations between the number of quadratic generators of the defining ideal of $C$ and the degree of surfaces which contain $C$. Along this line, I and Euisung Park study the relations among following three expectations: $\textbf{A}$ (Eisenbud-Harris): Let $C \subset \mathbb{P}^r$ be an irreducible nondegenrate curve of degree $d \geq 2r+2k-1$ for $1 \leq k \leq r-2$ and of arithmetic genus $g$. If $g >\pi_k(d,r)$, then $C$ lies on a surface of degree at most $r+k-2$. $\textbf{B}$: Let $\Gamma \subset \mathbb{P}^{r-1}$ be a set of $d \geq 2r+2k-1$ points in uniform position for $1 \leq k \leq r-2$. Suppose that $h_{\Gamma}(2)=2r+k-2$. Then $\Gamma$ lies on a curve $D$ of degree at most $r+k-2$. $\textbf{C}$: Let $C \subset \mathbb{P}^r$ be an irreducible nondegenrate curve of degree $d \geq 2r+2k-1$ for $1 \leq k \leq r-2$. Then ${\rm dim}_{\mathbb{K}}I(C)= {{r-1}\choose{2}}+1-k$ if and only if $C$ lies on a surface of degree at most $r+k-2$.

2010 Mathematics Subject Classification: 14N05
Key Words and Phrases: Castelnuovo theory, Hilbert function, quadratic generators

⋅ 21st-C-15:40 − 16:00 Radicals in Ore extensions (Nam Kyun Kim)

김남균((한밭대))
Nam Kyun Kim, Hanbat National University

In this talk, we first characterize the prime radical and Levitzki radical of the Ore extension. We next provide formulas for the strongly prime radical and the uniformly strongly prime radical of the Ore extensions.

2010 Mathematics Subject Classification: 16N40, 16S36
Key Words and Phrases: Ore extension, prime radical, Levitzki radical, strongly prime radical, uniformly strongly prime radical

⋅ 21st-C-16:00 − 16:20 Some generalizations of reversible and Armendariz rings (Juncheol Han, Tai Keun Kwak, Chang Ik Lee, Yang Lee)

한준철((부산대)), 곽태근((대진대)), 이창익*((부산대)), 이양((Yanbian University))
Juncheol Han, Pusan National University, Tai Keun Kwak, Daejin University, Chang Ik Lee*, Pusan National University, Yang Lee, Yanbian University

In this talk we concerns several ring theoretic properties related to matrices and polynomials. The basic properties of $\pi$-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing $\pi$-reversibility and power-Armendariz condition.

2010 Mathematics Subject Classification: 16U80, 16S36
Key Words and Phrases: Matrix ring, polynomial ring, $\pi$-reversible ring, power-Armendariz ring

⋅ 21st-C-16:20 − 16:40 Non-isomorphic $n\times n$ structural matrix rings (Sangmin Chun, Gangyong Lee, Mauricio Medina-B\'arcenas, Khanh Tung Nguyen)

천상민((중앙대)), 이강용*((충남대)), Mauricio Medina-B\'arcenas((Universidad \\ Aut\'onoma de Puebla)), Khanh Tung Nguyen((Vietnam National University))
Sangmin Chun, Chung-Ang University, Gangyong Lee*, Chungnam National University, Mauricio Medina-B\'arcenas, Universidad Aut\'onoma de Puebla, Khanh Tung Nguyen, Vietnam National University

In 1988, Wyk [2] defined the structural matrix ring as a subring of a full $n\times n$ matrix ring over a ring using a Boolean matrix and a binary relation (preorder). Recently, Lee, Roman, and Zhang defined a partial matrix ring [1]. We call an $n\times n$ \emph{partial matrix ring over a ring $\mathfrak{a}$}, denoted by $\mathsf{PM}_n(\mathfrak{a})$, a subring of a full $n\times n$ matrix ring over $\mathfrak{a}$, with elements matrices whose entries are either elements of $\mathfrak{a}$ or $0$, such that nonzero entries are independent of each other, i.e., $\mathsf{PM}_n(\mathfrak{a})=\sum_{(i, j)\in\mathcal{U}}e_{ij}\mathfrak{a}$ where $e_{ij}$ are matrix units and $\mathcal{U}$ is a subset of the index set $\mathcal{I}\times \mathcal{I}$, $\mathcal{I}=\{1, 2, \dots, n\}$. Note that in a partial matrix ring $R=\mathsf{PM}_n(\mathfrak{a})$, $\sum_{i=1}^ne_{ii}\mathfrak{a}\subseteq R$ (because $R$ has the unity) and not every choice of an index-pair set $\mathcal U$ will generate a structure closed under multiplication of matrices. In this talk, we provide that the structural matrix ring and the partial matrix ring coincide. Also, we discuss the set of all $n\times n$ partial matrix rings over a ring and related concepts. We provide the numbers of $n\times n$ partial matrix rings and isomorphism classes in $n\times n$ partial matrix rings over a ring using the number of all topologies on $\{1, 2, \dots, n\}$. Note that it is known that the number of all topologies on $\{1, 2, \dots, n\}$ is the same as that of all preorders on $\{1, 2, \dots, n\}$. This talk is based on a joint work with Sangmin Chun, Mauricio Medina-B\'arcenas, and Khanh Tung Nguyen. \begin{thebibliography}{9} \bibitem{1} G. Lee, C. S. Roman, and X. Zhang, {\it Modules whose endomorphism rings are division rings}, Comm. Algebra {\bf 42} (2014), no. 12, 5205--5223. \bibitem{2} L. van Wyk, {\it Maximal left ideals in structural matrix rings}, Comm. Algebra {\bf 16} (1988), no. 2, 399--419. \end{thebibliography}

2010 Mathematics Subject Classification: 15B34, 06A06, 06E20
Key Words and Phrases: Structural matrix ring, partial matrix ring, topologies, preorders, Boolean matrix
Operator Theory on Reproducing Kernel Hilbert Spaces

⋅ 21st-C-14:40 − 16:30 Chair: In Sung Hwang (Sungkyunkwan University)

⋅ 21st-C-14:40 − 15:00 The Beurling degree of an inner matrix function (In Sung Hwang, Woo Young Lee, Jaehui Park)

황인성((성균관대)), 이우영((서울대)), 박재휘*((서울대))
In Sung Hwang, Sungkyunkwan University, Woo Young Lee, Seoul National University, Jaehui Park*, Seoul National University

The Beurling degree of an inner matrix function $\Delta\in H^{\infty}_{M_{N\times r}}$ is defined to be the smallest number $m$ for which $\ker H_{\breve{\Phi}}^{*}=\Delta H^2_{\mathbb{C}^r}$ for some $\Phi\in L^{2}_{M_{N\times m}}$. In this talk, we describe the Beurling degree of inner matrix function in terms of its entries.

2010 Mathematics Subject Classification: 47A56, 15A15, 30H10, 30J05, 47B35
Key Words and Phrases: Inner matrix functions, the Beurling degree, the spectral multiplicity

⋅ 21st-C-15:00 − 15:20 On rank one perturbations of normal operators and their local spectral properties (Ji Eun Lee, Eungil Ko)

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

In this paper, we focus on local spectral properties of rank one perturbations of normal operators $T=N+ u\otimes v$. In particular, we show that if $\mathbb{C}\setminus\overline{\sigma_p(D_{\Lambda})}$ is a connected set in $\mathbb{C}$ where $D_{\Lambda}$ is a diagonal normal operator, then rank one perturbation of a diagonal normal operator $D_{\Lambda}$ is subscalar of order $2$. As an application of the main result, if the spectrum of rank one perturbation of a diagonal normal operator $D_{\Lambda}$ has a nonempty interior in $\mathbb{C}$, then it has a nontrivial invariant subspace.

2010 Mathematics Subject Classification: 47A11, 47B15
Key Words and Phrases: Rank one perturbations of normal operators, subscalar, nontrivial invariant subspace

⋅ 21st-C-15:20 − 15:40 Adjoint of strong $H^p$-functions (Sumin Kim)

김수민((한양대))
Sumin Kim, Hanyang University

A function $\Phi: \mathbb T \to \mathcal B(X, Y)$ is called a strong $H^p$-function if $\Phi(\cdot)x\in H^p(\mathbb T, Y)$. In this talk, we consider the adjoint of strong $H^p$-functions with values in $\mathcal B(X, Y)$. Let $H^p_{s}(\mathbb T, \mathcal B(X, Y))$ be the set of all (equivalence classes of) strong $H^p$-functions with values in $\mathcal B(X, Y)$. We show that if $X$ and $Y$ are reflexive, then the mapping $\Phi \to \Phi^*$ is an isometric isomorphism from $H^p_{s}(\mathbb T, \mathcal B(X, Y))$ onto $H^p_{s}(\mathbb T,\mathcal B(Y^*, X^*))$. This talk is based on a joint work with In Sung Hwang.

2010 Mathematics Subject Classification: 47A05, 46E40
Key Words and Phrases: Operator-valued functions, adjoint operator

⋅ 21st-C-15:50 − 16:10 The extremal vector of seminilpotent operators (Hyoung Joon Kim)

김형준((서울대))
Hyoung Joon Kim, Seoul National University

In this talk, we introduce the notion of extremal vector and study its properties. And we apply them to seminilpotent operators.

2010 Mathematics Subject Classification: 47A15
Key Words and Phrases: Seminilpotent operators

⋅ 21st-C-16:10 − 16:30 H-Toeplitz operators on the Bergman spaces (Jongrak Lee)

이종락((제주대))
Jongrak Lee, Jeju National University

In this talk, we introduce the H-Toeplitz operators on the Bergman space and we study the properties of H-Toeplitz operators. In particular, we consider the expansive and contractive Toeplitz operators with special polynomial symbols.

2010 Mathematics Subject Classification: 47B35
Key Words and Phrases: H-Toeplitz operator, Bergman space

⋅ 21st-D-16:40 − 18:10 Chair: In Hyoun Kim (Incheon National University)

⋅ 21st-D-16:40 − 17:00 On properties of $C$-normal operators (Mee-Jung Lee, Eungil Ko, Ji Eun Lee)

이미정*((이화여대)), 고응일((이화여대)), 이지은((세종대))
Mee-Jung Lee*, Ewha Womans University, Eungil Ko, Ewha Womans University, Ji Eun Lee, Sejong University

A bounded linear operator $T: {\mathcal H}\rightarrow {\mathcal H}$ is a {\it $C$-normal operator} if there exists a conjugation $C$ on ${\mathcal H}$ such that $[CT, (CT)^{\ast}]=0$ where $[R,S]:=RS-SR.$ In this talk, we study properties of $C$-normal operators. In particular, we prove that $T-\lambda$ is $C$-normal for all $\lambda\in{\mathbb C}$ if and only if $T$ is a complex symmetric operator with the conjugation $C$. Moreover, we show that if $T$ is $C$-normal, then the following statements are equivalent; (i) $T$ is normal, (ii) $T$ is quasinormal, (iii) $T$ is hyponormal, (iv) $T$ is $p$-hyponormal for $0<p\leq 1$. Finally, we consider operator transforms of $C$-normal operators.

2010 Mathematics Subject Classification: 47A05, 47B15, 47B20
Key Words and Phrases: $C$-normal operator, complex symmetric operator, operator transforms

⋅ 21st-D-17:00 − 17:20 Matrix representations of bounded linear operators on $L^2$ space (Young-Bok Chung)

정영복((전남대))
Young-Bok Chung, Chonnam National University

We find matrix representations of two bounded linear operators, Hankel and composition operators on Hilbert subspaces of $L^2$ functions on the unit circle with respect to a special orthonormal basis which is a generalization of monomials.

2010 Mathematics Subject Classification: 47B33, 47B02, 47B91, 30H10, 30C40, 47B35
Key Words and Phrases: Composition operator, matrix representation, Riemann map, Szeg\H o kernel, Henkel operator, Hankel matrix, Hardy space

⋅ 21st-D-17:30 − 17:50 Properties of spherical p-hyponormal and log-hyponormal commuting pairs of operators (Jaewoong Kim)

김재웅((육군사관학교))
Jaewoong Kim, Korea Military Academy

In this talk, we introduce spherical log-hyponormality and spherical p hyponormality for commuting pairs of operators and develop their several properties. In particular, we study whether spherical p-hyponormality and spherical log-hyponormality are invariant under spherical Aluthge transform and powers of operators.

2010 Mathematics Subject Classification: 47A13, 47B20
Key Words and Phrases: Spherical Aluthge transform, spherical p-hyponormality, spherical log-hyponormality, Taylor invertibility

⋅ 21st-D-17:50 − 18:10 A characterization of rational inner functions (Woo Young Lee)

이우영((서울대))
Woo Young Lee, Seoul National University

In this talk we show that every inner divisor of the operator-valued coordinate function, $zI_E$, is a Blaschke-Potapov factor. We also introduce a notion of operator-valued ``rational" function and then show that $\Delta$ is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product; this extends to operator-valued functions the well-known result proved by V. P. Potapov for matrix-valued functions. This work was collaborated with In Sung Hwang.

2010 Mathematics Subject Classification: 46E40, 30H10, 30J05, 47B35, 47B20
Key Words and Phrases: Inner functions, inner divisors, rational functions, finite Blaschke-Potapov product
Mathematical Analysis of PDEs Arising from Fluid Dynamics

⋅ 21st-B-10:30 − 11:15 Chair: Hantaek Bae (UNIST)

⋅ 21st-B-10:30 − 10:45 $L^{r}$--theory of the stationary Navier-Stokes equations with nonzero velocity at infinity (Dugyu Kim)

김두규((연세대))
Dugyu Kim, Yonsei University

In this talk, we consider the stationary motion of an incompressible Navier-Stokes fluid past obstacles in $\mathbb{R}^{3}$, subject to the given boundary velocity $u_{b}$, external force $f = {\rm div} F$ and nonzero constant vector $k_{e_1}$ at infinity. We first prove that the existence of at least one very weak solution $u$ in $L^{3}(\Omega) + L^{4}(\Omega)$ for arbitrary large $F \in L^{3/2}(\Omega) + L^{2}(\Omega)$ provided that the flux of $u_{b}$ on the boundary of each body is sufficiently small with respect to the viscosity $\nu$. Moreover, we establish weak-- and strong--regularity results for very weak solutions. As a consequence, our existence and regularity results enable us to prove the existence of a weak solution satisfying $\nabla u \in L^{r}(\Omega)$ for a given $F \in L^{r}(\Omega)$, $3/2 \leq r \leq 2$, and a strong solution satisfying $\nabla^{2} u \in L^{s}(\Omega)$ for a given $f \in L^{s}(\Omega)$, $1 < s \leq 6/5$, respectively.

2010 Mathematics Subject Classification: 35D30, 35D35, 76D05
Key Words and Phrases: Navier-Stokes equations, weak solution, very weak solution, exterior domain

⋅ 21st-B-10:45 − 11:00 On a shallow-water model with the Coriolis effect (Byungsoo Moon)

문병수((인천대))
Byungsoo Moon, Incheon National University

In this talk an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. The transport equation theory is then applied to investigate the local well-posedness and wave breaking phenomena for this model. The nonexistence of the Camassa-Holm-type peaked solution and classification of various traveling-wave solutions to the new system are also established. Moreover it is shown that all the symmetric waves to this model are traveling waves.

2010 Mathematics Subject Classification: 35B10, 35B65, 35Q35, 34Q85
Key Words and Phrases: Shallow water, asymptotic model, Coriolis force, Green-Naghdi equations, wave breaking, traveling wave

⋅ 21st-B-11:00 − 11:15 Global solutions to 3D incompressible rotational fluid systems (Jaewook Ahn, Jihoon Lee, Junha Kim)

안재욱*((동국대)), 이지훈((중앙대)), 김준하((중앙대))
Jaewook Ahn*, Dongguk University, Jihoon Lee, Chung-Ang University, Junha Kim, Chung-Ang University

In this talk, we discuss the global classical solvability in three-dimensional fluid systems with a strong Coriolis force. After reviewing some previous results, we consider its extensions to the fractional Navier-Stokes-Coriolis equations. If time allows, related works on the MHD-Coriolis equations would be also discussed. The talk is based on a joint work with Jihoon Lee and Junha Kim of Chung-Ang University.

2010 Mathematics Subject Classification: 76D03, 76U05
Key Words and Phrases: Coriolis force, global solvability

⋅ 21st-B-11:25 − 12:10 Chair: Minsuk Yang (Yonsei University)

⋅ 21st-B-11:25 − 11:40 Low regularity well-posedness of Hartree type Dirac equations in 2,3-dimensions (Yonggeun Cho, Kiyeon Lee, Tohru Ozawa)

조용근((전북대)), 이기연*((이화여대)), Tohru Ozawa((Waseda University))
Yonggeun Cho, Jeonbuk National University, Kiyeon Lee*, Ewha Womans University, Tohru Ozawa, Waseda University

We give a survey of small data scattering of 2,3 dimensional Dirac equation with Hartree type nonlinearity $c(|x|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$ and $c(\langle D \rangle^{-2} * \langle \psi, \beta \psi\rangle)\beta\psi$. Since the nonlinear potentials of our equations are Coulomb potential, these equations can be a simplified physical model of Chern-Simons-Dirac equations and coupled Dirac-Klein-Gordon equation. The results about global well-posedness and small data scattering of Dirac equations can be proven to control singularities of the frequency of potential. To handle the frequencies of potential we need the Up-Vp space argument and localized bilinear estimates arising from the null structure. In this talk, we will show the small initial data scattering for a solution to Dirac equations in the case $1 < \gamma <2$ and nonexistence result for scattering in the case $0<\gamma \le 1$ and describe the details of estimates. Also, we give small data scattering result with the Yukawa potential. This talk is based on co-work with Yonggeun Cho and Tohru Ozawa.

2010 Mathematics Subject Classification: 35Q55, 35Q40
Key Words and Phrases: Dirac equations, small data scattering, null-structure

⋅ 21st-B-11:40 − 11:55 Suitable weak solutions of the incompressible magnetohydrodynamic equations over the moving boundary domains (Yunsoo Jang, Dugyu Kim)

장윤수*((강원대)), 김두규((연세대))
Yunsoo Jang*, Kangwon National University, Dugyu Kim, Yonsei University

The purpose of this paper is to study the three-dimensional system of magnetohydrodynamic (MHD equations) for a viscous incompressible resistive fluid. We are interested in the existence of suitable weak solutions to the system in time varying domains. To do this, we consider the approximate equations related to the MHD equations and we apply the Leray -Schauder fixed point theorem to the solutions of the equations over the moving boundary domains. Existence of suitable weak solutions is established by the energy estimates and the compactness results in Lebesgue and Sobolev spaces.

2010 Mathematics Subject Classification: 35Q10, 35D35, 76D07
Key Words and Phrases: Suitable weak solution, MHD equations, localized energy inequality, moving boundary, Schauder theory

⋅ 21st-B-11:55 − 12:10 Gradient estimates for Stokes systems arising from composite materials (Jongkeun Choi)

최종근((부산대))
Jongkeun Choi, Pusan National University

The regularity theory for Stokes systems with irregular coefficients has some applications in mathematical fluid dynamics, for instance, Stokes flow over composite materials. In this talk, I will present recent work on partial regularity of weak solutions to the stationary Stokes systems with piecewise DMO coefficients in a domain which consists of a finite number of disjoint subdomains. This talk is based on a joint work with Hongjie Dong (Brown University) and Longjuan Xu (National University of Singapore).

2010 Mathematics Subject Classification: 76D07, 35B65, 35J47
Key Words and Phrases: Stokes system, piecewise DMO, gradient estimate
Nonlinear Differential Equations from Physics and Collective Behavior

⋅ 21st-A-08:40 − 10:10 Chair: Dongnam Ko (Catholic University)

⋅ 21st-A-08:40 − 09:00 Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics (Hansol Park)

박한솔((PIMS))
Hansol Park, Pacific Institute for the Mathematical Sciences

We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death (COD) state from the exponential $\ell^1$-stability and the existence of the equilibrium solution.

2010 Mathematics Subject Classification: 70G60, 34D06, 70F10
Key Words and Phrases: Winfree model, aggregation, complete synchronization, high-dimensional sphere

⋅ 21st-A-09:00 − 09:20 Entropy production estimate for the ES-BGK model with the correct Prandtl number (Doheon Kim, Myeong-Su Lee, Seok-Bae Yun)

김도헌*((고등과학원)), 이명수((성균관대)), 윤석배((성균관대))
Doheon Kim*, KIAS, Myeong-Su Lee, Sungkyunkwan University, Seok-Bae Yun, Sungkyunkwan University

In this paper, we establish the entropy-entropy production estimate for the ES-BGK model, a generalized version of the BGK model of the Boltzmann equation introduced for better approximation in the Navier-Stokes limit. Our result improves the previous entropy production estimates in that (1) the full range of Prandtl parameters $-1/2\leq\nu <1$ including the critical case $\nu=-1/2$ is covered, and (2) a sharper entropy production bound is obtained. An explicit characterization of the coefficient of the entropy-entropy production estimate is also presented.

2010 Mathematics Subject Classification: 35Q20
Key Words and Phrases: BGK model, ES-BGK model, entropy production estimate, Boltzmann equation, kinetic theory of gases

⋅ 21st-A-09:30 − 09:50 Curvature flows with obstacles (Taehun Lee, Ki-Ahm Lee, Hyunsuk Kang)

이태훈*((고등과학원)), 이기암((서울대)), 강현석((광주과학기술원))
Taehun Lee*, KIAS, Ki-Ahm Lee, Seoul National University, Hyunsuk Kang, GIST

Curvature flow are geometric evolutions of a hypersurface moved by curvature quantities such as the mean curvature and the Gaussian curvature, which have been applied in material science and image processing. The main difficulty to treat curvature flows is a development of singularities in finite time which arises in many cases. In this talk, we consider the evolution of hypersurface in the presence of obstacles, where the hypersurface cannot pass, so that the flows exist for a long time. We will discuss how a hypersurface with obstacles evolves under the mean curvature flow or the Gaussian curvature flow, which is based on joint works with Ki-Ahm Lee.

2010 Mathematics Subject Classification: 35R35, 53E10
Key Words and Phrases: Curvature flows, obstacle problems

⋅ 21st-A-09:50 − 10:10 Marcinkiewicz regularity for singular parabolic $p$-Laplace type equations with measure data (Jung-Tae Park)

박정태((고등과학원))
Jung-Tae Park, KIAS

In this talk, we consider a parabolic $p$-Laplace type equation when the right-hand side is a signed Radon measure with finite total mass, whose model is $$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}.$$ In the singular range $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$, we discuss integrability results for the spatial gradient of a solution in the Marcinkiewicz space, under a suitable density condition of the right-hand side measure $\mu$.

2010 Mathematics Subject Classification: 35K92, 35R06, 35B65
Key Words and Phrases: Singular parabolic equation, measure data, Marcinkiewicz space

⋅ 21st-B-10:30 − 12:00 Chair: Dohyun Kim (Sungshin Women's University)

⋅ 21st-B-10:30 − 10:50 Hydrodynamic limits of the nonlinear Schrodinger equation coupled with the Chern-Simons gauge fields (Jeongho Kim, Bora Moon)

김정호((한양대)), 문보라*((한양대))
Jeongho Kim, Hanyang University, Bora Moon*, Hanyang University

In this talk, we present the hydrodynamic limit problem for the Chern-Simons-Schro\-dinger (CSS) system. We consider the Madelung transformation and two different scalings of the CSS system to derive the compressible and incompressible Euler systems coupled with the Chern-Simons equations and Poisson equation, respectively. Both cases are based on modulated energy estimates for rigorous derivation. Here we focus on the case of compressible limit to which the classical theory of relative entropy method can be applied.

2010 Mathematics Subject Classification: 35Q55, 35B40
Key Words and Phrases: Chern-Simons-Schrodinger system, hydrodynamic limit, modulated energy, relative entropy

⋅ 21st-B-10:50 − 11:10 A relativistic generalization of the bgk model for gas mixtures (Byung-Hoon Hwang, Myeong-Su Lee, Seok-Bae Yun)

황병훈*((성균관대)), 이명수((성균관대)), 윤석배((성균관대))
Byung-Hoon Hwang*, Sungkyunkwan University, Myeong-Su Lee, Sungkyunkwan University, Seok-Bae Yun, Sungkyunkwan University

The BGK model is the best-known model equation of the Boltzmann equation, which satisfies the essential features of the Boltzmann equation at a much lower numerical cost. In this talk, we introduce a relativistic generalization of the BGK model for gas mixtures derived based on Marle's formulation of the relativistic BGK model. Here we consider an inert gas mixture satisfying the conservation laws for species number densities, global momentum, and total energy. We first present several properties that our model satisfies and discuss the determination problem of the auxiliary temperature provided by the nonlinear relations. Then, we show that our model can recover the classical BGK model proposed by Bisi et al. [J. Phys. A: Math. Theor., 51 (2018)] in the Newtonian limit. This is based on the joint work with Myeong-Su Lee and Seok-Bae Yun (Sungkyunkwan Univ.).

2010 Mathematics Subject Classification: 35Q20, 35Q75, 83A05
Key Words and Phrases: Kinetic theory of gases, relativistic Boltzmann equation, relativistic BGK model, gas mixtures

⋅ 21st-B-11:20 − 11:40 Local velocity grid conservative semi-Lagrangian schemes for the BGK model (Sebastiano Boscarino, Seung-Yeon Cho, Giovanni Russo)

Sebastiano Boscarino((University of Catania)), 조승연*((경상국립대)), Giovanni Russo((Uni\-versity of Catania))
Sebastiano Boscarino, University of Catania, Seung-Yeon Cho*, Gyeongsang National University, Giovanni Russo, University of Catania

Most numerical schemes proposed for solving kinetic equations for rarefied gas dynamics, such as Boltzmann equation and BGK model, are based on the discrete velocity approximation. Recently developed conservative semi-Lagrangian schemes allow accurate solutions on a wide range of Knudsen numbers, with very mild restrictions on the time step. In general, such approaches use fixed velocity grids, and one must secure a sufficient number of grid points in phase space to resolve the structure of the distribution function. When dealing with high Mach number problems, where large variation of mean velocity and temperature are present in the domain under consideration, the computational cost and memory allocation requirements become prohibitively large. Local velocity grid methods have been developed to overcome such difficulty in the context of Eulerian based schemes. In this talk, we introduce a velocity adaption technique for the semi-Lagrangian scheme applied to the BGK model. The velocity grids will be set locally in time and space. We apply a weighted minimization approach to impose global conservation, generalizing the $L^2$-minimization technique introduced by I. M. Gamba. We demonstrate the efficiency of the proposed scheme in several numerical examples.

2010 Mathematics Subject Classification: 65L06, 65M25, 76P05
Key Words and Phrases: Local velocity grid, semi-Lagrangian scheme, BGK model

⋅ 21st-B-11:40 − 12:00 Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schr\"odinger equation (Sangdon Jin, Younghun Hong)

진상돈*((중앙대)), 홍영훈((중앙대))
Sangdon Jin*, Chung-Ang University, Younghun Hong, Chung-Ang University

For the one-dimensional mass-critical/supercritical pseudo-relativistic nonlinear Schr\"od\-inger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.

2010 Mathematics Subject Classification: 35J50, 35J20
Key Words and Phrases: Orbital stability, uniqueness, non-relativistic limit
Functional Analysis and Mathematical Understanding of Quantum Phenomena

⋅ 21st-C-14:50 − 16:20 Chair: Un Cig Ji (Chungbuk National University)

⋅ 21st-C-14:50 − 15:20 The SIC existence problem and POVMs (Jaeseong Heo)

허재성((한양대))
Jaeseong Heo, Hanyang University

The purpose of this talk is to introduce the SIC existence problem and its equivalent problems. The SIC existence problem asks if there are $n^2$ equiangular lines in an $n$-dimensional complex Hilbert space. Such maximal equiangular lines are related to the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. If the SIC existence problem is true in every finite dimension, they would constitute a naturally distinguished class of POVMs which might be expected to have many interesting applications to quantum tomography, quantum cryptography, and quantum information theory generally.

2010 Mathematics Subject Classification: 81P15, 81P70
Key Words and Phrases: Symmetric informationally complete, POVM, equiangular lines

⋅ 21st-C-15:30 − 15:50 Quantum Markov semigroup of Fibonacci type oscillators (Franco Fagnola, Chul Ki Ko, Hyun Jae Yoo)

Franco Fagnola((Politecnico di Milano)), 고철기((연세대)), 유현재*((한경대))
Franco Fagnola, Politecnico di Milano, Chul Ki Ko, Yonsei University, Hyun Jae Yoo*, Hankyong National University

We discuss the quantum Markov semigroup that arises in the weak coupling limit of a system with a boson reservoir. Particularly, what we are interested in is the $(q,r)$-deformed oscillators, where the parameters $(q,r)$ generate a Fibonacci type sequences. The weak coupling limit type dynamics is constructed through GKSL generators given for each Bohr frequency resulting from the eigen values of the system Hamiltonian. For the specific parameter regions we discuss the dynamical properties such as the existence of the QMS, invariant states, and spectral gaps. This is a joint work with Franco Fagnola and Chul Ki Ko.

2010 Mathematics Subject Classification: 47D06
Key Words and Phrases: Quantum Markov semigroup, weak coupling limit, GKSL generator, spectral gap

⋅ 21st-C-15:50 − 16:20 Polytope structures for Greenberger-Horne-Zeilinger diagonal states (Kyung Hoon Han, Seung-Hyeok Kye)

한경훈*((수원대)), 계승혁((서울대))
Kyung Hoon Han*, University of Suwon, Seung-Hyeok Kye, Seoul National University

The notion of entanglement arising from quantum mechanics is now recognized as one of the most important resources in the current quantum information and computation theory. The Greenberger-Horne-Zeilinger states are key examples of genuine entanglement in multi-qubit systems, and have many applications in various fields of quantum information theory. They also play important roles in the classification of entanglement in multi-qubit systems. The Greenberger-Horne-Zeilinger diagonal states are mixtures of Greenberger-Horne-Zeilinger states. We discuss the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit Greenberger-Horne-Zeilinger diagonal states.

2010 Mathematics Subject Classification: 81P15, 15A30, 52B11, 46L05, 46L07
Key Words and Phrases: Greenberger-Horne-Zeilinger diagonal state, polytope, bi-separable, fully bi-separable, Mermin inequality, volume

⋅ 21st-D-16:40 − 18:10 Chair: Hun Hee Lee (Seoul National University)

⋅ 21st-D-16:40 − 17:00 The orthogonality on the space of bilinear forms (Sun Kwang Kim, Geunsu Choi)

김선광*((충북대)), 최근수((충북대))
Sun Kwang Kim*, Chungbuk National University, Geunsu Choi, Chungbuk National University

We deal with a certain concept of the orthogonality on the space of bilinear forms which is called Birkhoff-James orthononality. For a norm attaining multilinear map on a Banach space, we study the maps which are orthogonal to the original one. Specially, using this concept, we consider the set of maps with Bhatia-\v{S}emrl property and see whether it is dense or not. We not only introduce cases that it is norm dense, but also introduce cases that it is not dense.

2010 Mathematics Subject Classification: 46B04
Key Words and Phrases: Banach space, Birkhoff-James orthogonality, Bhatia-Semrl property

⋅ 21st-D-17:00 − 17:20 Convex feasibility problems in ${\rm CAT}(\kappa)$ spaces (Byoung Jin Choi)

최병진((제주대))
Byoung Jin Choi, Jeju National University

The Projection algorithm is one of the most simple and important algorithms for computing a point in the intersection of some convex sets, which is called the convex feasibility problem. In this talk, we study some projection methods to solving the convex feasibility problem in ${\rm CAT}(\kappa)$ spaces. Indeed, we study the convergence of the weighted averaged projection method and the Mann's alternating projection method in ${\rm CAT}(\kappa)$ spaces.

2010 Mathematics Subject Classification: 47J25
Key Words and Phrases: Weighted average sequence, Mann's iterative method, ${\rm CAT}(\kappa)$ spaces

⋅ 21st-D-17:20 − 17:40 On convex combinations of random quantum states and their partial transposes (Sang-Gyun Youn)

윤상균((서울대))
Sang-Gyun Youn, Seoul National University

One of the most important quantum phenomena is the existence of quantum entanglement, which implies that partial transposes of quantum states are not positive in general. However, their convex combinations still have possibilities to be positive, and the main aim of this talk is to explain how free probability theory can be applied to solve this question for random quantum states.

2010 Mathematics Subject Classification: 46L54
Key Words and Phrases: Quantum entanglement, partial transpose, random quantum state, free probability

⋅ 21st-D-17:40 − 17:55 Boundaries for Gelfand transform images of Banach algebras of holomorphic functions (Yun Sung Choi, Mingu Jung)

최윤성((포항공대)), 정민구*((포항공대))
Yun Sung Choi, POSTECH, Mingu Jung*, POSTECH

Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$, which is a uniform algebra on the spectrum of $\mathcal{A}$, we obtain an explicit description of the Shilov boundary for $\widehat{\mathcal{A}}$ for classical Banach spaces $X$ in the case where $\mathcal{A}$ is a certain Banach algebra, for instance, $\mathcal{A}_\infty (B_X)$, $\mathcal{A}_u (B_X)$ or $\mathcal{A}_{wu} (B_X)$. Some possible applications of our result to the famous Corona theorem are also briefly discussed.

2010 Mathematics Subject Classification: 46E50
Key Words and Phrases: Shilov boundary, holomorphic functions, peak points

⋅ 21st-D-17:55 − 18:10 Symplectic matrices and representations of the canonical commutation relations (Hyunmoon Kim)

김현문((서울대))
Hyunmoon Kim, Seoul National University

The canonical commutation relations are a fundamental tool in describing quantum phenomena. In this talk, I will explain how Grossman understood the canonical commutation relations geometrically from translations on a symplectic vector space. Then I will construct a representation of the canonical commutation relations for every complex symplectic matrix. Finally, I will discuss some analytic problems related to intertwining operators.

2010 Mathematics Subject Classification: 15A23, 22D10, 81R05
Key Words and Phrases: Riemann theta function, unitary representations of the Heisenberg group
Several Complex Variables and Related Topics

⋅ 21st-A-08:40 − 10:05   Chair: Sung Yeon Kim (Center for Complex Geometry, IBS)

⋅ 21st-A-08:40 − 09:05 Plurisubharmonic singularities in algebraic geometry (Dano Kim)

김다노((서울대))
Dano Kim, Seoul National University

Plurisubharmonic functions are fundamental objects in several complex variables. Their singularities naturally appear as transcendental objects in several contexts in algebraic geometry. We will discuss our recent results in some of such contexts: log canonical thresholds, jumping numbers of multiplier ideals and Demailly's strong continuity on higher Lelong numbers. These include joint works with Alexander Rashkovskii and with Hoseob Seo.

2010 Mathematics Subject Classification: 32U05, 14F18, 32U25
Key Words and Phrases: Plurisubharmonic functions, multiplier ideals, Lelong numbers

⋅ 21st-A-09:10 − 09:35 Holder continuous solutions to the complex Hessian equation (Slawomir Kolodziej, Ngoc Cuong Nguyen)

Slawomir Kolodziej((Jagiellonian University)), Ngoc Cuong Nguyen*((카이스트))
Slawomir Kolodziej, Jagiellonian University, Ngoc Cuong Nguyen*, KAIST

This is a joint work with Slawomir Kolodziej. We show that the complex Hessian measure of a Holder continuous $m$-subharmonic function is well dominated by the corresponding capacity. As consequence we obtain the Holder continuous subsolution theorem for the complex Hessian equation.

2010 Mathematics Subject Classification: 53C55, 35J96, 32U40
Key Words and Phrases: H\"older continuity, weak solutions, complex Hessian equations

⋅ 21st-A-09:40 − 10:05 Intersection of currents and the Monge-Amp\`ere operator (Lucas Kaufmann, Dinh Tuan Huynh, Duc-Viet Vu)

Lucas Kaufmann*((Institute for Basic Science)), Dinh Tuan Huynh((Chinese Academy of Sciences)), Duc-Viet Vu((University of Cologne))
Lucas Kaufmann*, Institute for Basic Science, Dinh Tuan Huynh, Chinese Academy of Sciences, Duc-Viet Vu, University of Cologne

Given positive closed currents on a complex manifold, the question of whether one can define a meaningful notion of intersection is of fundamental importance in complex analysis and its application in geometry and dynamics. In this talk I will recall recent notions of intersection and their relations with the study of the Monge-Amp\`ere operator acting on singular plurisubharmonic functions. This is based on joint works with D.-T. Huynh (Chinese Academy of Sciences) and D.-V. Vu (Cologne).

2010 Mathematics Subject Classification: 32U40, 32H50, 37F05
Key Words and Phrases: Monge-Ampere operator, positive closed currents, intersection theory

⋅ 21st-B-10:30 − 11:55   Chair: Kang-Hyurk Lee (Gyeongsang National University)

⋅ 21st-B-10:30 − 10:55 Equidistribution of positive closed currents for certain birational maps on $\mathbb{P}^k$ (Taeyong Ahn)

안태용((인하대))
Taeyong Ahn, Inha University

It is known that for a holomorphic endomorphism of $\mathbb{P}^k$ or a regular polynomial automorphism of $\mathbb{C}^k$, under certain conditions given in terms of the mapping, the normalized pull-backs of a given positive closed current $(p, p)$ of unit mass converges to the Green $(p, p)$-current associated with the given map where $1\leq p\leq k$. In this talk, we study the inverse images of positive closed currents of bidegree $(p, p)$ under regular birational maps. More precisely, we discuss a sufficient condition that a sequence of normalized pull-backs of a given current converges to the Green $(p, p)$-current.

2010 Mathematics Subject Classification: 37F80, 32U40
Key Words and Phrases: Equidistribution, birational maps, positive closed current, Green current, superpotential

⋅ 21st-B-11:00 − 11:25 Fiberwise K\"ahler-Einstein metric and K\"ahler-Ricci flow on a family of strongly pseudoconvex domains (Young-Jun Choi)

최영준((부산대))
Young-Jun Choi, Pusan National University

In 2012, Schumacher proved that the fiberwise K\"ahler-Einstein metric on a family of canonically polarized compact K\"ahler manifolds is positive definite on the total space. He showed that the geodesic curvature, which measures the positivity of the horizontal direction, satisfies a certain elliptic PDE. The positivity is obtained by applying the maximum principle. In 2013, Berman proved the parabolic version of the Schumahcer's result. More precisely, he proved that the geodesic curvature of the fiberwise K\"ahler-Ricci flow satisfies a parabolic equation. A parabolic maximum principle implies that the positivity of the geodesic curvature is preserved along the K\"ahler-Ricci flow. In this talk, we will introduce the results of Schumacher and Berman and how to generalize those results to a family of strongly pseudoconvex domains.

2010 Mathematics Subject Classification: 53G55, 32G05, 32T15
Key Words and Phrases: K\"ahler-Einstein metric, K\"ahler-Ricci flow, family of strongly pseudoconvex domains

⋅ 21st-B-11:30 − 11:55 Totally geodesic Kobayashi isometric embedding between bounded symmetric domains (Aeryeong Seo)

서애령((경북대))
Aeryeong Seo, Kyungpook National University

In this talk, we study totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. We show that any $C^1$ totally geodesic rank one Kobayashi isometric disc which extends continuously to the boundary in a bounded symmetric domain is either holomorphic or anti-holomorphic. In particular, it is either a minimal disc or a conjugate of a minimal disc. Furthermore, for bounded symmetric domains $\Omega,\,\Omega'$ any $C^1$ totally geodesic Kobayashi isometric embedding from $\Omega$ to $\Omega'$ which extends continuously to the boundary is either holomorphic or anti-holomorphic provided that $\Omega$ is irreducible and $\text{rank}(\Omega)\geq \text{rank}(\Omega')$.

2010 Mathematics Subject Classification: 32F45, 32M15, 53C35
Key Words and Phrases: Kobayashi pseudometric, bounded symmetric domain, totally geodesic isometric embedding, holomorphicity

⋅ 21st-C-14:50 − 16:15   Chair: Jong-Do Park (Kyung Hee University)

⋅ 21st-C-14:50 − 15:15 Compact difference of composition operators on the Hardy spaces (Boo Rim Choe, Koeun Choi, Hyungwoon Koo, Inyoung Park)

최부림((고려대)), 최고은((고려대)), 구형운((고려대)), 박인영*((한양대))
Boo Rim Choe, Korea University, Koeun Choi, Korea University, Hyungwoon Koo, Korea University, Inyoung Park*, Hanyang University

Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this talk, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

2010 Mathematics Subject Classification: 47B33, 30H20, 30H10
Key Words and Phrases: Difference of composition operator, compactness, Hardy space

⋅ 21st-C-15:20 − 15:45 Weighted sub-mean-value inequality on a non-isotropic ball (Hong Rae Cho, Han-Wool Lee, Soohyun Park)

조홍래((부산대)), 이한울((부산대)), 박수현*((부산대))
Hong Rae Cho, Pusan National University, Han-Wool Lee, Pusan National University, Soohyun Park*, Pusan National University

This talk concerns the Bergman space with an exponential weight on the unit ball. In this talk, we present a sub-mean-value inequality on a non-isotropic ball with an exponential weight for investigation of the weighted Bergman space. The inequality allows a certain type of Carleson embedding theorem and some applications to integral operators. We also discuss the relationship between the sub-mean-value inequality and pointwise Bergman kernel estimates. It is based on the joint work with Hong Rae Cho and Han-Wool Lee.

2010 Mathematics Subject Classification: 32A36
Key Words and Phrases: Weighted Bergman spaces, Carleson measure, integral operators

⋅ 21st-C-15:50 − 16:15 Difference of weighted composition operators II (Koeun Choi, Boo Rim Choe, Hyungwoon Koo, Jongho Yang)

최고은*((고려대)), 최부림((고려대)), 구형운((고려대)), 양종호((한국교원대))
Koeun Choi*, Korea University, Boo Rim Choe, Korea University, Hyungwoon Koo, Korea University, Jongho Yang, Korea National University of Education

In the setting of the unit disk we have recently obtained characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting from a standard weighted Bergman space into the corresponding weighted Lebesgue space. In this talk, we extend those results to the case when the exponents of the domain space and the target space are different.

2010 Mathematics Subject Classification: 47B33, 30H20, 30H10
Key Words and Phrases: Difference, weighted composition operator, Carleson measure, boundedness, compactness

⋅ 21st-D-16:40 − 18:05   Chair: Jaehyun Hong (Center for Complex Geometry, IBS)

⋅ 21st-D-16:40 − 17:05 Infinitesimal symmetries of weakly pseudoconvex manifolds (Martin Kolar, Shin-Young Kim)

Martin Kolar((Masaryk University)), 김신영*((기초과학연구원))
Martin Kolar, Masaryk University, Shin-Young Kim*, IBS

We consider weakly pseudoconvex hypersurfaces with polynomial models in $\mathbb{C}^N$ and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds.

2010 Mathematics Subject Classification: 32V35
Key Words and Phrases: Infinitesimal symmetries, weakly pseudoconvex hypersurfaces

⋅ 21st-D-17:10 − 17:35 Characterization of Diederich-Fornaess index and Steinness index (Jihun Yum)

염지훈((기초과학연구원))
Jihun Yum, IBS

Pseudoconvex domains are fundamental objects in Several Complex Variables. Every pseudoconvex domain can be exhausted by strongly pseudoconvex domains from inside, and Diederich-Fornaess index (roughly speaking) measures how fast it is approximated. Also, if a pseudoconvex domain can be exhausted by strongly pseudoconvex domains from outside(not always possible), Steinness index (roughly speaking) measures how fast it is approximated. We will completely characterize the two indices in terms of D'Angelo 1-forms on the boundary of the domain. Then we see some application of it.

2010 Mathematics Subject Classification: 32T20, 32T35
Key Words and Phrases: Pseudoconvex domain, Diederich-Fornaess index, D'Angelo 1-form

⋅ 21st-D-17:40 − 18:05 Singular K\"{a}hler-Einstein metrics on Gorenstein Fano compactifications with rank two (Jae-Hyouk Lee, Kyeong-Dong Park, Sungmin Yoo)

이재혁((이화여대)), 박경동((고등과학원)), 유성민*((기초과학연구원))
Jae-Hyouk Lee, Ewha Womans University, Kyeong-Dong Park, KIAS, Sungmin Yoo*, IBS

By the works of Chen-Donaldson-Sun and Li-Tian-Wang, the Q-Fano variety admits a singular K\"{a}hler-Einstein metric if and only if it is K-stable. However, in general, it is difficult to check the K-stability condition since we should consider all possible test configurations. Despite these difficulties, we can check the K-stability if the variety has large symmetries. In this talk, we give a classification of Gorenstein Fano bi-equivariant compactifcations of complex semisimple Lie groups with rank two. By computing the barycenter of each moment polytope, we determine which of them admit singular K\"{a}hler-Einstein metrics. This is based on the joint work with Jae-Hyouk Lee and Kyeong-Dong Park.

2010 Mathematics Subject Classification: 32Q20
Key Words and Phrases: Kaehler-Einstein metrics, group compactification, Gorenstein Fano
Toric Topology

⋅ 21st-B-10:30 − 12:00 Chair: Dong Youp Suh (KAIST)

⋅ 21st-B-10:30 − 11:00 Regular semisimple Hessenberg varieties whose cohomology rings are generated by degree two elements (Mikiya Masuda)

21st-B-10:30 − 11:00
Mikiya Masuda, Osaka City University Advanced Mathematical Institute (OCAMI)

Schubert varieties are a family of subvarieties of the flag variety $\mathrm{Fl}(\mathbb{C}^n)$ and their topology, geometry and combinatorics are well studied (Schubert calculus). Hessenberg varieties are a rather new family of subvarieties of $\mathrm{Fl}(\mathbb{C}^n)$ including Springer varieties (or fibers), permutohedral varieties and Peterson varieties. Among Hessenberg varieties, what are called \emph{regular semisimple} Hessenberg varieties are generic and play a central role. The cohomology of a regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$, which is determined by a function $h\colon \{1,\dots,n\}\to \{1,\dots,n\}$ satisfying $h(i)\ge i$ and $h(i)\le h(i+1)$, becomes a module over the symmetric group $\mathfrak{S}_n$ on $n$ letters. Interestingly, its analysis leads to a solution of the Stanley-Stembridge conjecture in graph theory. Recently, Cho-Hong-Lee [1] presented a basis of the $\mathfrak{S}_n$-module $H^2(\mathrm{Hess}(S,h))$ using the Bialynicki-Birula cell decomposition (i.e., geometrically). In this talk, I will present a basis of the $\mathfrak{S}_n$-module $H^2(\mathrm{Hess}(S,h))$ using the GKM theory (i.e., combinatorially) and also discuss when $H^\ast(\mathrm{Hess}(S,h))$ is generated by $H^2(\mathrm{Hess}(S,h))$ as a ring. This is a joint work with Takashi Sato in progress. \begin{thebibliography}{9} \bibitem{1} S. Cho, J. Hong, and E. Lee, {\it Permutation module decomposition of the second cohomology of a regular semisimple Hessenberg variety}, arXiv:2107.00863 \end{thebibliography}

2010 Mathematics Subject Classification: 14M15
Key Words and Phrases: Hessenberg varieties

⋅ 21st-B-11:10 − 11:35 Cubic Bruhat interval polytopes (Eunjeong Lee, Mikiya Masuda, Seonjeong Park)

이은정*((IBS 기하학 수리물리 연구단)), Mikiya Masuda((Osaka City University Advanced Mathematical Institute ((OCAMI)))), 박선정((Jeonju University))
Eunjeong Lee*, IBS Center for Geometry and Physics, Mikiya Masuda, Osaka City University Advanced Mathematical Institute (OCAMI), Seonjeong Park, Jeonju University

Schubert varieties and Richardson varieties are some of the most interesting subvarieties of the full flag varieties. A maximal torus acts on the full flag variety and these subvarieties are stable under the action. Considering the restriction of the moment map on Richardson varieties, we obtain Bruhat interval polytopes. In this talk, we consider an interesting family of Bruhat interval polytopes each element of which is associated with a smooth projective Fano toric variety, called a toric variety of Catalan type. In fact, they are Fano Bott manifolds, which are interesting objects in toric topology. Moreover, we study correspondences between toric varieties of Catalan type, polygon triangulations, and cubic Bruhat interval polytopes. This talk is based on a joint work with Mikiya Masuda and Seonjeong Park.

2010 Mathematics Subject Classification: Primary 14M25, 14M15; Secondary 05A05
Key Words and Phrases: Richardson varieties, Bruhat interval polytopes, Bott manifolds, Catalan numbers, Wedderburn--Etherington numbers

⋅ 21st-B-11:35 − 12:00 On the homotopy type and the cohomology of toric spaces (Xin Fu)

21st-B-11:35 − 12:00
Xin Fu, Ajou University

In this talk, I will talk about how methods from various fields can be applied to study the topology of toric spaces. For instance, use the theory of twisting cochains and combinatorics of fans to find twisting terms on Tor algebras of partial quotients and compute their cohomology rings. Another example is to use the perspectives of homotopy theory and q-CW-complexes to prove the homotopy rigidity property of toric orbifolds in four dimensions.

2010 Mathematics Subject Classification: 57R18, 14M25, 57S12
Key Words and Phrases: Toric spaces, toric orbifolds

⋅ 21st-C-14:50 − 16:15 Chair: Seonjeong Park (Jeonju University)

⋅ 21st-C-14:50 − 15:15 Circle actions on oriented manifolds with 3 fixed points and non-existence in dimension 12 (Donghoon Jang)

장동훈((부산대))
Donghoon Jang, Pusan National University

The study of rotations of geometric objects, such as rotations of spheres, is a natural and classical topic in geometry and topology. Let the circle act on a compact oriented manifold M with a non-empty finite fixed point set. The following are well-known. (1) The dimension of M is even. (2) If there is exactly one fixed point, M is the point. (3) If there are exactly two fixed points, a rotation of an even dimensional sphere provides an example in any even dimension. (4) On any even dimension (greater than 2), a disjoint union (a connected sum) of even spheres provides an (connected, respectively) example with any even number of fixed points. (5) If the number of fixed points is odd, the dimension of M is a multiple of 4. (6) Circle actions on $\mathbb{CP}^2$, $\mathbb{HP}^2$, and $\mathbb{OP}^2$ provide examples with 3 fixed points in real dimensions 4, 8, and 16, respectively. Therefore, both from small numbers of fixed points and from low dimensions, the first case for which an answer is not known is whether there exists a 12-dimensional manifold with exactly three fixed points. During this talk, we prove that there does not exist a circle action on a 12-dimensional compact oriented manifold with exactly three fixed points.

2010 Mathematics Subject Classification: 58C30
Key Words and Phrases: Oriented manifold, circle action, fixed point

⋅ 21st-C-15:20 − 15:45 Hodge structures on tropical varieties (Hoil Kim)

김호일((경북대 \& 대구경북과학기술원))
Hoil Kim, Kyungpook National Univerisity \& DGIST

Tropical varieties are combinatorial geometries which share much with toric varieties. These relate complex geometry with p-adic geometry. They are also closely related to mirror symmetry in extended way. I this talk we want to discuss on the Hodge structures on tropical abelian varieties, surfaces and three-folds extending the results on tropical curves.

2010 Mathematics Subject Classification: 14T15, 14T20, 14J28, 14J32, 14J33, 14K05
Key Words and Phrases: Tropical variety, p-adic geometry, abelian varieties, mirror symmetry, surfaces, threefolds

⋅ 21st-C-15:50 − 16:15 Conic decomposition of a toric variety (Jongbaek Song, Seonjeong Park)

송종백*((고등과학원)), 박선정((전주대))
Jongbaek Song*, KIAS, Seonjeong Park, Jeonju University

We introduce the notion of a conic decomposition of a complete toric variety. It is a certain sequence of cofibrations, which is parametrized by a combinatorial sequence of the associated polytope. This gives us several vanishing results in the rational cohomology of a toric variety. We also discuss the Poincar\'e polynomials for a large class of singular toric varieties.

2010 Mathematics Subject Classification: 14M25, 55N10, 52B05, 52B11
Key Words and Phrases: Toric variety, singular cohomology, Betti number
Stochastic Partial Differential Equations

⋅ 21st-A-08:40 − 10:10 Chair: Ildoo Kim (Korea University)

⋅ 21st-A-08:40 − 09:00 Solvability of the stochastic modified Burgers' equation driven by multiplicative space-time white noise (Beom-Seok Han)

한범석((포항공대))
Beom-Seok Han, POSTECH

In this talk, we consider a stochastic modified Burgers' equation driven by multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + \bar bu^\lambda u_{x} + \sigma(u)\dot W_t,\quad (t,x)\in(0,\infty)\times\mathbb R;\quad u(0,\cdot) = u_0, $$ where $\lambda >0$ and $\dot W$ is a space-time white noise. The initial data $u_0$ is nonnegative, the coefficients $a,b$ and $c$ depend on $(\omega,t,x)$, and $\bar b$ depend on $(\omega,t)$. The coefficient $\sigma(u)$ depends on $(\omega,t,x,u)$. According to the types of $\sigma(u)$ and noise $\dot W$, two different cases are considered. (i) The diffusion coefficient $\sigma(u)$ is a bounded Lipschitz function in $u$, $\lambda\in (0,1]$. (ii) The diffusion coefficient $\sigma(u) = u^{1+\lambda_0}$ and $\lambda,\lambda_0\in(0,1)$. We discuss the uniqueness, existence, $L_p$-regularity, and maximal H\"older regularity of a strong solution to the above equation. The maximal H\"older regularity of the solution depends on $\bar b u^\lambda u_x$ and $\sigma(u)$. In other words, if $\lambda \in (0,1]$ and $\sigma(u)$ is a bounded Lipschitz function in $u$, we have (a.s.) $$u\in C_{t,x}^{1/4-\varepsilon,1/2-\varepsilon}([0,T]\times\mathbb R). $$ Otherwise, if $\sigma(u) = u^{1+\lambda_0}$ and $\lambda,\lambda_0\in(0,1)$, then (a.s.) $$u\in C_{t,x}^{\frac{1/2-(\lambda - 1/2)\vee \lambda_0}{2}-\varepsilon,\frac{1}{2}-(\lambda - \frac{1}{2})\vee\lambda_0-\varepsilon}([0,T]\times\mathbb R). $$

2010 Mathematics Subject Classification: 60H15, 35R60
Key Words and Phrases: Stochastic Burgers' equations, space-time white noise

⋅ 21st-A-09:00 − 09:20 Regularity of solutions to Nonlinear SPDE: focusing on random noise (Jae-Hwan Choi, Beom-Seok Han)

최재환*((고려대)), 한범석((포항공대))
Jae-Hwan Choi*, Korea University, Beom-Seok Han, POSTECH

Consider the following nonlinear SPDE ; $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d,\quad u(0,\cdot)=u_0, $$ where $\lambda \geq 0$, the coefficients depend on $(\omega,t,x)$ and $F$ is a spatially homogeneous gaussian random noise. The main strategy for dealing with $F$ is an analysis of the Reproducing Kernel Hilbert Space (RKHS) generated by $F$. In this talk, the regularity of a solution focusing on the effect of $F$ is presented. This talk is based on joint works with Beom-Seok Han.

2010 Mathematics Subject Classification: 60H15, 35R60
Key Words and Phrases: Stochastic partial differential equation, nonlinear, spatially homogeneous Gaussian noise

⋅ 21st-A-09:30 − 09:50 Deterministic and stochastic parabolic equations with space-time non-local operators (Kyeong-Hun Kim, Daehan Park, Junhee Ryu)

김경훈((고려대)), 박대한((카이스트)), 유준희*((고려대))
Kyeong-Hun Kim, Korea University, Daehan Park, KAIST, Junhee Ryu*, Korea University

In this talk, we introduce a Sobolev space theory for deterministic and stochastic PDEs with space-time non-local operators. We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. This talk is based on joint works with Kyeong-Hun Kim and Daehan Park.

2010 Mathematics Subject Classification: 60H15, 35R60, 26A33, 47G20
Key Words and Phrases: Stochastic partial differential equations, space-time nonlocal equations, maximal $L_p$-regularity, multi-dimensional space-time white noise

⋅ 21st-A-09:50 − 10:10 A Sobolev space theory for the time-fractional stochastic partial differential equations driven by Levy processes (Kyeong-Hun Kim, Daehan Park)

김경훈((고려대)), 박대한*((카이스트))
Kyeong-Hun Kim, Korea University, Daehan Park*, KAIST

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\Delta u +\sum_{k=1}^{\infty}\partial^{\beta}_{t}\int_{0}^{t} g^{k} dZ^k_{s} $$ Here $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ are the Caputo fractional derivatives, $$ 0<\alpha<2, \qquad \beta<\alpha+1/p, $$ and $\{Z^k_t:k=1,2,\ldots\}$ is a sequence of independent L\'evy processes. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity of the solution.

2010 Mathematics Subject Classification: 60H15, 35R60, 45D05
Key Words and Phrases: Stochastic partial differential equations, time-fractional derivatives, Levy processes, maximal $L_p$-regularity
Special on Applied Algebra and Optimization

⋅ 21st-B-10:30 − 12:00 Chair: Dae Yeol Jeon (Kongju National University)

⋅ 21st-B-10:30 − 11:00 Fast mixing random walks and almost spanning expanders (Debsoumya Chakraborti, Jaehoon Kim, Jinha Kim, Minki Kim, Liu Hong)

Debsoumya Chakraborti((기초과학연구원)), 김재훈*((카이스트)), 김진하((기초과학연구\\원)), 김민기((기초과학연구원)), Liu Hong((University of Warwick))
Debsoumya Chakraborti, IBS, Jaehoon Kim*, KAIST, Jinha Kim, IBS, Minki Kim, IBS, Liu Hong, University of Warwick

We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).

2010 Mathematics Subject Classification: 05C48, 05C81
Key Words and Phrases: Random walks, expander

⋅ 21st-B-11:00 − 11:30 On the computation of $r$-th roots in finite fields (Gook Hwa Cho)

조국화((이화여대))
Gook Hwa Cho, Ewha Womans University

Let $q$ be a power of a prime such that $q \equiv 1 \pmod r$. Let $c$ be an $r$-th power residue over $\mathbb F_q$. In this paper, we present a new $r$-th root formula which generalizes G. H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. H. C. Williams gave an $r$-th root algorithm in $O(r^3 \log q)$ when $r$ is odd prime. This was then improved by K. S. Williams and Hardy. So their algorithm requires $O(r^2 \log q +r^4)$ multiplications. Our algorithm requires also $O(r^2 \log q +r^4)$ multiplications, but our algorithm has half of Williams-Hardy's multiplications in main exponential computation. Also, we briefly introduce the recent results how to solve generalized NTRU equation.

2010 Mathematics Subject Classification: 11T06, 11Y16, 68W40
Key Words and Phrases: Finite field, $r$-th root, Cipolla-Lehmer algorithm

⋅ 21st-B-11:30 − 12:00 The dynamical system for Lucas polynomials over the ring of 2-adic integers (Myunghyun Jung)

정명현((성균관대))
Myunghyun Jung, Sungkyunkwan University

Like the relation between Fibonacci numbers and Fibonacci polynomials, Lucas polynomials are a polynomial sequence which can be considered as a generalization of the Lucas numbers. \[ L_n(x)=\sum_{k=0}^{\lfloor n/2 \rfloor}\frac{n}{n-k}\binom{n-k}{k}x^{n-2k} \] We show the dynamical systems for Lucas polynomials over the ring of 2-adic integers by investigating the minimal decomposition which consists of minimal subsets and the attracting basin.

2010 Mathematics Subject Classification: 11S82, 37P05, 37P35
Key Words and Phrases: p-adic ring, minimal decomposition theory, Lucas polynomial

⋅ 21st-C-14:50 − 16:20 Chair: Yoon Mo Jung (Sungkyunkwan University)

⋅ 21st-C-14:50 − 15:20 Lattice paths and negatively indexed weight-dependent binomial coefficients (Meesue Yoo, Josef K\"stner, Michael Schlosser)

류미수*((충북대)), Josef K\"ustner((University of Vienna)), Michael Schlosser((University of Vienna))
Meesue Yoo*, Chungbuk National University, Josef K\"ustner, University of Vienna, Michael Schlosser, University of Vienna

In 1992, Daniel Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Sam Formichella and Armin Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In our work, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser, to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. In this talk I will also examine some important special cases of the weight-dependent binomial coefficients, including ordinary, $q$- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions. This is a joint work with Josef K\"ustner and Michael Schlosser.

2010 Mathematics Subject Classification: 05A30, 11B65, 33E05, 33E20
Key Words and Phrases: Binomial coefficients, weight-dependent pattice paths, elliptic functions

⋅ 21st-C-15:20 − 15:50 A unified framework for distributed optimization algorithms over time-varying directed graphs (Woocheol Choi, Doheon Kim, Seok-Bae Yun)

최우철((성균관대)), 김도헌*((고등과학원)), 윤석배((성균관대))
Woocheol Choi, Sungkyunkwan University, Doheon Kim*, KIAS, Seok-Bae Yun, Sungkyunkwan University

In this paper, we propose a framework under which several decentralized optimization algorithms can be treated in a unified manner. More precisely, we show that the distributed subgradient descent algorithms, the subgradient-push algorithm, and the distributed algorithm with row-stochastic matrix can be derived by making suitable choices of consensus matrices, step-size and subgradient from a decentralized subgradient descent algorithm. As a result of such unified understanding, we provide a convergence proof that covers several algorithms under a novel algebraic condition that is strictly weaker than the conventional graph-theoretic condition. This unification also enables us to derive a new distributed optimization scheme.

2010 Mathematics Subject Classification: 90C25, 68Q25
Key Words and Phrases: Distributed gradient methods, Unified framework, gradient-push algorithm

⋅ 21st-C-15:50 − 16:20 Bipartite consensus for multi-agent systems with Markov switching topologies and gain variations (Arumugam Parivallal)

21st-C-15:50 − 16:20
Arumugam Parivallal, Sungkyunkwan University

In this talk, the problem of output feedback control design for bipartite consensus of multi-agent systems will be addressed in the presence of Markov switching topologies. The primary objective of this talk is to discuss the output feedback controller through which the considered multi-agent system achieves bipartite consensus under controller gain variations. Precisely, the communication between agents of the considered multi-agent system is described with the aid of signed undirected Markovian switching graphs. By using Lyapunov stability theory and algebraic graph theory, required conditions for bipartite consensus of the considered multi-agent system will be derived in the form of linear matrix inequalities. Finally, the effectiveness of the proposed controller and developed theoretical results are verified using numerical results.

2010 Mathematics Subject Classification: 93C95
Key Words and Phrases: Multi-agent systems, bipartite consensus
GeoMath

⋅ 21st-B-10:30 − 12:00 Chair: Sangil Kim (Pusan National University)

⋅ 21st-B-10:30 − 10:50 Sea level rise estimation near the Korean peninsula using CEEMDAN with tidal data (Young Jin Kim, Okyu Kwon, Harksoo Song, Jongho Kim, Hyuk Kang)

김영진*((국가수리과학연구소)), 권오규 ((국가수리과학연구소)), 송학수((국가수리과학연구소)), 김종호((국가수리과학연구소)), 강혁((국가수리과학연구소))
Young Jin Kim*, NIMS, Okyu Kwon, NIMS, Harksoo Song, NIMS, Jongho Kim, NIMS, Hyuk Kang, NIMS

The sea level rise (SLR) due to global warming is an issue at a global level and its causes already have been studied clearly in early times. IPCC creates scenarios for greenhouse gas emissions and predicts global average sea level rise accordingly. Ice Sheet System Model is a numerical model of ice sheet dynamics process caused by the loss of ice sheets in polar regions. In particular, the global SLR prediction through the Glacial Isostatic Adjustment provides seawater flows from the ice sheet at a macroscopic scale. However, this global scale modeling has limitations in insufficient mesh size due to computation speed and makes it difficult to predict local differences in SLR in microscopic scale and complex terrain. In particular, in the Korean Peninsula, high SLR rate differences between tidal stations have been reported in microscopic areas ($\sim$ 100 km). In this study, we study regression and empirical mode decomposition for SLR prediction using tidal data near Korean Peninsula. And we also analyze correlation, causality, and volatility to understand the difference between tidal stations.

2010 Mathematics Subject Classification: 62R07
Key Words and Phrases: CEEMDAN, empirical mode decomposition, tidal level, data analysis, sea level rise

⋅ 21st-B-10:50 − 11:10 Projected the spatial distribution of common squid (Todarodes pacificus) in Korean waters using modeling approaches (Dongwha Sohn, Minkyoung Bang, Changsin Kim, Jung Jin Kim, Sangil Kim, Chan Joo Jang)

손동화*((부산대)), 방민경((한국해양과학기술원)), 김창신((국립수산과학원)), 김중진((국립수산과학원)), 김상일((부산대)), 장찬주((한국해양과학기술원))
Dongwha Sohn*, Pusan National University, Minkyoung Bang, Korea Institute of Ocean Science ＆ Technology, Changsin Kim, National Institute of Fisheries Science, Jung Jin Kim, National Institute of Fisheries Science, Sangil Kim, Pusan National University, Chan Joo Jang, Korea Institute of Ocean Science ＆ Technology

Changes in environmental conditions in marine ecosystems due to global warming could directly or indirectly affect the spatial distribution of fishery resources. Understanding species-specific responses to environmental variability is crucial for managing commercially important species. Common squid (Todarodes pacificus) is a commercially and ecologically important species in Korean waters. Their commercial catches have been decreased since the mid-2000s. In this study, to examine effects of biotic and abiotic factors on the spatial dynamics of the common squid in Korean waters, we applied both machine learning and statistical approaches (i.e., maximum entropy model, random forest, generalized additive model etc.). We constructed species distribution models using species occurrence data and environmental data. The full model included abiotic (i.e., seawater temperature, salinity, water velocity, sea surface height, mixed layer depth) and biotic (i.e., chlorophyll-a concentration) factors as independent variables. Our results showed that the range of common squid distribution was expanded to northward during summer and fall. Seasonal differences in their spatial distribution were related with variability of seawater temperature and mixed layer depth. Furthermore, using the best fitted model, we projected their monthly spatial distribution in the 2050s under two different representative concentration pathways (RCP 2.6 and RCP 8.5) from CMIP5 (Coupled Model Intercomparison Project Phase 5).

2010 Mathematics Subject Classification: 92-10
Key Words and Phrases: Common squid, Korean waters, spatial distribution, species distribution model, future projection

⋅ 21st-B-11:20 − 11:40 Impact of late 1980s climate regime shift on the spawning area of walleye pollock (Gadus chalcogramma) in the western East Sea (Japan Sea) (Yong-Yub Kim, Yang-Ki Cho, Yu-Kyeong Kang, Seung-Tae Lee, Hae Kun Jung, Chung il Lee, Sangil Kim)

김용엽*((서울대)), 조양기((서울대)), 강유경((서울대)), 이승태((서울대)), 정해근((국립수산과학원)), 이충일((강릉원주대)), 김상일((부산대))
Yong-Yub Kim*, Seoul National University, Yang-Ki Cho, Seoul National University, Yu-Kyeong Kang, Seoul National University, Seung-Tae Lee, Seoul National University, Hae Kun Jung, National Institute of Fisheries Science, Chung il Lee, Gangneung-Wonju National University, Sangil Kim, Pusan National University

Walleye pollock (Gadus chalcogramma) fishery stock in the Korean fishing area dramatically decreased in the late 1980s. To investigate the impact of climate regime shift on the collapse of pollock in the late 1980s, we conducted a three-dimension hydrodynamic model with data assimilation and particle tracking model. Spawned eggs of walleye pollock were tracked by particle tracking model for 30 days in each day of January and February from 1983 to 1992. The individuals transported further north during 1988-1992 than 1983-1987. Changes in the transport of individuals are associated with ocean currents due to climate regime shift in the late 1980s. Moreover, water temperature increases due to global warming and climate regime shift provide unfavorable conditions in the spawning and nursery of Pollock.

2010 Mathematics Subject Classification: 92-10
Key Words and Phrases: Walleye pollock, East Sea (Japan Sea), larval transport, numerical model

⋅ 21st-B-11:40 − 12:00 Optimal harvest strategies of sandfish in eastern coastal waters of Korea (Giphil Cho)

조기필((부산대))
Giphil Cho, Pusan National University

We propose optimal harvest strategies using a stage-structured fishery model with impulsive system. The parameters of reproduction rate and growth equation of the sandfish in the model were derived from data of otolith and gonad analyses of female sandfish collected from 2005 to 2008. And we estimated the age-specific natural mortality of the sandfish assuming natural mortality as an inverse function of total length. 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 sandfish 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 sandfish with and without considering monthly price. We expect that maximum sustainable yield and profit of the sandfish can be increased by approximately 13.5\% and 23\% due to the optimal harvest strategies.

2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: Sandfish, optimal harvest, impulsive system
Algebraic and Combinatorial Coding Theory with Its Applications

⋅ 21st-A-09:00 − 10:10 Chair: Jon-Lark Kim (Sogang University)

⋅ 21st-A-09:00 − 09:20 Locally recoverable codes on algebraic geometry curves (Boran Kim)

김보란((경북대))
Boran Kim, Kyungpook National University

Locally recoverable codes have been significantly studied because they are motivated by the reliability and efficiency of distributed storage systems. In this talk, we present locally recoverable codes C on algebraic geometry curves. In particular, we improve the lower bound of the minimum distance for C by using certain methods.

2010 Mathematics Subject Classification: 11T71, 11G20
Key Words and Phrases: Locally recoverable codes, algebraic geometry codes

⋅ 21st-A-09:20 − 09:40 Some structures of weighted posets and digraphs admitting the extended Hamming code to be a perfect code (Hyun Kwang Kim, Jieun Kwon)

김현광((포항공대)), 권지은*((포항공대))
Hyun Kwang Kim, POSTECH, Jieun Kwon*, POSTECH

The weighted posets and directed graphs admitting the extended Hamming code $\widetilde{\mathcal{H}}_3$ to be a $2$-perfect code are all classified. In this research, we continued the classification problem further. An element of a weighted poset or digraph of level one and weight one is called a root and let $s$ be the number of roots. We will classify structure vectors of weighted posets (resp. directed graphs) which admit the extended Hamming code to be a $2$-perfect code when $s=3,4,5$ (resp. $s=3,4$).

2010 Mathematics Subject Classification: 94B99
Key Words and Phrases: Extended Hamming code, directed graphs, posets, metric

⋅ 21st-A-09:50 − 10:10 Codes over a non-commutative non-unital ring E with four elements and its applications (Jon-Lark Kim)

김종락((서강대))
Jon-Lark Kim, Sogang University

We introduce codes over a non-commutative non-unital ring $E$ with four elements. The ring $E$ is defined by $E = \langle a, b \,|\, 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle$. We describe the structures of codes over $E$ and introduce how to construct quasi self-dual codes, and LCD codes over $E$. We show how to apply it to the DNA codes and Cryptography.

2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: Codes, LCD codes, DNA codes, rings, self-dual codes

⋅ 21st-D-16:40 − 18:10 Chair: Whan-Hyuk Choi (UNIST)

⋅ 21st-D-16:40 − 17:00 Projection and lifting of codes over finite chain rings with respect to MDS self-dual codes (Sunghyu Han)

한성휴((한국기술교육대))
Sunghyu Han, Koreatech

In this research, we study the projection and lifting of codes over finite chain rings with respect to MDS self-dual codes. If the characteristic of a finite field is odd, then we can lift MDS self-dual codes over the field to MDS self-dual codes over the corresponding finite chain rings. On the other hand, if the characteristic of a finite field is even, then there is no lifting like odd characteristic case. For even characteristic case, we study various aspects of the existence of MDS self-dual codes.

2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: Finite chain ring, lifting, MDS code, projection, self-dual code

⋅ 21st-D-17:00 − 17:20 Optimal linear codes from simplicial complexes (Jong Yoon Hyun, Yoonjin Lee, Jungun Lee)

현종윤*((건국대)), 이윤진((이화여대)), 이정연((강원대))
Jong Yoon Hyun*, Konkuk University, Yoonjin Lee, Ewha Womans University, Jungun Lee, Kangwon University

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal linear codes $C_{\Delta^c}$ constructed from simplicial complexes in $\mathbb{F}^n_2$, where $\Delta$ is a simplicial complex in $\mathbb{F}^n_2$ and $\Delta^c$ the complement of $\Delta$. We first find an explicit computable criterion for $C_{\Delta^c}$ to be optimal; this criterion is given in terms of the 2-adic valuation of $\sum_{i=1}^s 2^{|A_i|-1},$ where the $A_i$'s are maximal elements of $\Delta$. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of $\Delta$. In particular, we find that $C_{\Delta^c}$ is a Griesmer code. if and only if the maximal elements of $\Delta$ are pairwise disjoint and their sizes are all distinct. Specially, when $\mathcal{F}$ has exactly two maximal elements, we explicitly determine the weight distribution of $C_{\Delta^c}$. Tables 2 through 4 present optimal linear codes.

2010 Mathematics Subject Classification: 94B05, 94A60
Key Words and Phrases: Optimal codes, simplicial complex, weight distributions

⋅ 21st-D-17:30 − 17:50 Base field extension of AG codes for decoding (Kwankyu Lee)

이관규((조선대))
Kwankyu Lee, Chosun University

As previous decoding algorithms for AG codes all assumed existence of an extra rational place on the base algebraic curve, the AG codes supported by all rational places of the curve were excluded from the domain of applicability of the decoding algorithms. We presents a decoding algorithm for those AG codes using an extra place of higher degree and by constant field extension. Hence finally all AG codes, as Goppa defined 40 years ago, are equipped with a fast decoding algorithm.

2010 Mathematics Subject Classification: 94B35
Key Words and Phrases: AG codes, decoding

⋅ 21st-D-17:50 − 18:10 Self-orthogonality matrix and Reed-Muller codes (Whan-Hyuk Choi, Jon-Lark Kim)

최환혁*((UNIST)), 김종락((서강대))
Whan-Hyuk Choi*, UNIST, Jon-Lark Kim, Sogang University

We obtain a new method for checking self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Kim et al.~(2021). We introduce a self-orthogonality matrix, which is used to check the self-orthogonality of a linear code. We show that self-orthogonality matrix is obtained by puncturing a Reed-Muller code.

2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: Self-orthogonality matrix, Reed-Muller codes, optimal codes