Program and Abstracts

Algebra

⋅ 19th-C-15:00 − 16:30 Chair: Dohyeong Kim (Seoul National University)

⋅ 19th-C-15:00 − 15:20 Some properties of the standard non-uniform arithmetic quotient of $PGL_3$ (Sanghoon Kwon)

권상훈((가톨릭관동대))
Sanghoon Kwon, Catholic Kwandong University

There are a lot of formal similarities between two kinds of fields: algebraic number fields and global function fields. In many cases, it is relatively easier to work on function fields than on number fields. Motivated by some correspondence between number theoretical aspects and dynamical properties on $PGL(n,\mathbb{Z})\backslash PGL(n,\mathbb{R})$ ($n=2,3$), we discuss certain properties of the standard non-uniform arithmetic quotient of $PGL_n$, $n=2,3$ over a field of formal series.

2010 Mathematics Subject Classification: 20G25, 11F06
Key Words and Phrases: Arithmetic groups, global function fields, Bruhat-Tits buildings

⋅ 19th-C-15:20 − 15:40 Weak Bruhat interval modules of the $0$-Hecke algebras for genomic Schur functions (Young-Hun Kim, Semin Yoo)

김영훈*((서울대)), 유세민((고등과학원))
Young-Hun Kim*, Seoul National University, Semin Yoo, KIAS

In 2017, Pechenik and Yong introduced a deformation of Schur functions, called the genomic Schur functions, in the context of the K-theory of Grassmannians. Recently, Pechenik verified that every genomic Schur function is a nonnegative linear combination of fundamental quasisymmetric functions. We construct modules of the $0$-Hecke algebra whose quasisymmetric characteristics are the homogeneous components of a genomic Schur function. We then decompose these modules into weak Bruhat interval modules recently defined by Jung--Kim--Lee--Oh. Furthermore, we find the projective cover of each weak Bruhat interval module we construct. This is joint work with Semin Yoo.

2010 Mathematics Subject Classification: 20C08, 05E10, 05E05
Key Words and Phrases: $0$-Hecke algebra, weak Bruhat order, genomic Schur function, quasisymmetric characteristic, projective cover

⋅ 19th-C-15:50 − 16:10 Supersymmetric classical W-algebras (Eric Ragoucy, Arim Song, Uhi Rinn Suh)

Eric Ragoucy((Laboratoire de Physique Th\'eeorique LAPTh, CNRS, Universit \'e Savoie Mont Blanc and U.G.A.)), 송아림*((서울대)), 서의린((서울대))
Eric Ragoucy, Laboratoire de Physique Th\'eeorique LAPTh, CNRS, Universit \'ee Savoie Mont Blanc and U.G.A., Arim Song*, Seoul National University, Uhi Rinn Suh, Seoul National University

For a Lie algebra, one associates to it a $\mathbb{C}$-algebra called W-algebra, which is a main algebraic object in 2-dimensional conformal field theory. When given a Lie superalgebra, one gets a supersymmetric (SUSY) analogue of it, called SUSY W-algebra. In this talk, we mainly deal with the classical limits of SUSY W-algebras called SUSY classical W-algebras. We first introduce the notion of (SUSY) Poisson vertex algebra, which is the algebra structure of (SUSY) classical W-algebra and explain how SUSY and nonSUSY Poisson structures are related. Next, we introduce SUSY W-algebras associated with Lie superalgebras and show how to find their generators. This talk is based on the joint work with Ragoucy and Suh.

2010 Mathematics Subject Classification: 17B69
Key Words and Phrases: Supersymmetric W-algebra, Lie superalgebra, Poisson vertex algebra

⋅ 19th-C-16:10 − 16:30 Essential dimension of seimsimple groups (Yeongjong Kim, Sanghoon Baek)

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

We provide a simple method to compute an upper bound of the essential dimension of a semisimple group and its strict reductive envelope through its generically free representation. Combining our upper bound with previously known lower bound, the exact value of the essential dimension is calculated for some types of semisimple groups. As an application, we determine the essential dimension of a semisimple group of types A, B, C, D, and E under certain conditions on its center. This extends previous works on simple simply connected groups of type B or D by Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, and semisimple groups of type A by Cernele-Reichstein and type B by the authors to any semisimple groups in a uniform way.

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

⋅ 19th-D-16:50 − 17:10 Chair: Young-Hoon Kiem (Seoul National University)

⋅ 19th-D-16:50 − 17:10 Deformations of weighted homogeneous surface singularities and Koll\'ar conjecture (Jaekwan Jeon, Dongsoo Shin)

전재관*((충남대)), 신동수((충남대))
Jaekwan Jeon*, Chungnam National University, Dongsoo Shin, Chungnam National University

In 1988, J\'anos Koll\'ar conjectured that every irreducible component of deformation spaces of rational singularities corresponds to some partial resolutions of the singularities called `P-modification'. On the other hand, T. de Jong and D. van Straten prove that deformations of sandwiched singularities (containing some weighted homogeneous surface singularities) can be described by ``picture deformations" which are deformations of plane curves. We prove that every irreducible component of deformation spaces of weighted homogeneous surface singularities with certain conditions corresponds to `P-resolution' (included in P-modifications) using picture deformations and certain MMP-algorithm. Moreover, if there is enough time, we discuss more broader cases and the relation with symplectic fillings.

2010 Mathematics Subject Classification: 14D15
Key Words and Phrases: Weighted homogeneous surface singularities, sandwiched singularity, deformation, picture deformation, P-resolution
Analysis

⋅ 19th-C-15:00 − 16:30 Chair: Jinwan Park (Kongju National University)

⋅ 19th-C-15:00 − 15:20 Global well-posedness of fifth-order modified KdV equations on periodic domain (Chulkwang Kwak, Kiyeon Lee)

곽철광((이화여대)), 이기연*((카이스트))
Chulkwang Kwak, Ewha Womans University, Kiyeon Lee*, KAIST

In this talk, we will discuss the global well-posedness of fifth-order modified KdV equations (5-mKdV) on energy space $H^2$. For the derivation of our main equation, we introduce a concept of KdV hierarchy and Hamiltonian with some conservation laws. To prove the global well-posedness of 5-mKdV, we exploit the short time $X^{s,b}$ space. Especially, we impose the weighted into the function spaces to show GWP on the energy space. This talk is based on the joint work with Chulkwang Kwak.

2010 Mathematics Subject Classification: 35Q53, 37K10
Key Words and Phrases: KdV equation, short time $X^{s,b}$ space, energy estimates, Kdv hierarchy

⋅ 19th-C-15:20 − 15:40 A new necessary and sufficient condition for the existence of global solutions to semilinear parabolic equations on bounded domains (Jaeho Hwang, Soon-Yeong Chung)

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

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove:\\ \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if }\\ &\mbox{the function } f \mbox{ satisfies}\\ &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\epsilon \,\lVert S(t)u_{0}\rVert_{\infty}\right)}{\lVert S(t)u_{0}\rVert_{\infty}}dt=\infty\\ &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.

2010 Mathematics Subject Classification: 35B44, 35K57, 35K61
Key Words and Phrases: Semilinear parabolic equation, Fujita blow-up, critical exponent

⋅ 19th-C-15:50 − 16:10 Nonlocal Schr\"odinger-Kirchhoff equations of convex--concave type with the external magnetic field (Yun-Ho Kim, Seol Vin Kim)

김연호((상명대)), 김설빈*((상명대))
Yun-Ho Kim, Sangmyung University, Seol Vin Kim*, Sangmyung University

In the present talk we are to ensure the existence of infinitely many large- or small-energy solutions weak solutions to nonlocal Schr\"{o}dinger--Kirchhoff equations of convex--concave type with the external magnetic field when the nonlinear growth $f$ does not satisfy the condition of Ambrosetti-Rabinowitz type. The strategy of the proof for these results is to approach the problem variationally by applying the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition.

2010 Mathematics Subject Classification: 35A15, 35J60, 35R11, 47G20
Key Words and Phrases: Schr\"odinger-Kirchhoff equation, fractional magnetic operators, variational methods

⋅ 19th-C-16:10 − 16:30 Properties of the free boundary near the fixed boundary of the double obstacle problems (Jinwan Park)

박진완((공주대))
Jinwan Park, Kongju National University

In this talk, I will introduce the tangential touch and $C^1$ regularity of the free boundary near the fixed boundary of the double obstacle problem. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle. In this talk, a new type of difficulty that comes from the second type of upper obstacle will be discussed. The obstacle problems are typical examples of the free boundary problem and arise in porous media, elasto-plasticity, optimal control, and financial mathematics. In the last decades, many properties of the obstacle problems have been studied by L. Caffarelli, K.-A. Lee, A. Figalli, H. Shahgholian, and various researchers. The double obstacle problem which is the problem with two obstacle functions arises in the study of option pricing with transaction costs, the game of tug-of-war, and semiconductor devices.

2010 Mathematics Subject Classification: 35R35, 35B65
Key Words and Phrases: Free boundary problem, obstacle problem, double obstacle problem, regularity of free boundary

⋅ 19th-D-16:50 − 18:00 Chair: Dong Hyun Cho (Kyonggi University)

⋅ 19th-D-16:50 − 17:10 Solutions and stability of the $p$-radical additive and Jensen functional equations related to the $p$-power function (Gwang Hui Kim)

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

The aim of this paper is to investigate a solution and the stability for the $p$-radical additive functional equation: $$f\left(\sqrt[p]{x+y}\right) =f\left(\sqrt[p]{x}\right)+ f\left(\sqrt[p]{y}\right),$$ and stability for the $p$-radical Jensen and an alternative Jensen functional equations: $$f\left(\sqrt[p]{x+y}\right)+f\left(\sqrt[p]{x-y}\right)=2f(\sqrt[p]{x)},$$ $$f\left(\sqrt[p]{x+y}\right)-f\left(\sqrt[p]{x-y}\right)=2f(\sqrt[p]{y)},$$ related to the $p$-power function, where $p$ is an odd positive integer and $f$ is a real valued function. Furthermore, the obtained results can be extended onto Banach space.

2010 Mathematics Subject Classification: 39B82, 39A30, 39B52
Key Words and Phrases: Stability, superstability, additive functional equation, Jensen functional equation, $p$-power function, trigonometric functional equation

⋅ 19th-D-17:10 − 17:30 On norm-attainment in symmetric tensor products of Banach spaces (Sheldon Dantas, Luis C. Garcia-Lirola, Mingu Jung, Abraham Rueda Zoca)

Sheldon Dantas((Universitat Jaume I)), Luis C. Garcia-Lirola((Universidad de Zaragoza)), 정민구*((고등과학원)), Abraham Rueda Zoca((Universidad de Murcia))
Sheldon Dantas, Universitat Jaume I, Luis C. Garcia-Lirola, Universidad de Zaragoza, Mingu Jung*, KIAS, Abraham Rueda Zoca, Universidad de Murcia

In this talk, we study a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{\pi, s, N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of $N$-homogeneous polynomials on $X$. Due to this relation, we can observe that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in $\widehat{\otimes}_{\pi, s, N} X$ is dense. We show that such a set is dense in $\widehat{\otimes}_{\pi, s, N} X$ for a large class of Banach spaces such as $L_p$-spaces, isometric $L_1$-predual spaces or Banach spaces with monotone Schauder basis, among others. This is based on joint work with Sheldon Dantas, Luis C. Garcia-Lirola and Abraham Rueda Zoca.

2010 Mathematics Subject Classification: 46B04
Key Words and Phrases: Symmetric tensor product, norm attaining polynomials

⋅ 19th-D-17:40 − 18:00 Properties of a generalized Brownian motion over paths in abstract Wiener space (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 $\mathbb B$, the space of $\mathbb B$-valued continuous functions on $[a,b]$. In this talk, we introduce a positive finite measure with a scale on $C^{\mathbb B}[a,b]$ which is a generalized analogue of Wiener measure. Then we investigate various properties of a generalized Brownian motion on the space including the Fourier-transforms of the motion. As an application of the results, we derive a simple formula evaluating the Radon-Nikodym derivatives of integrable functions on $C^{\mathbb B}[a,b]$ with an infinite dimensional conditioning function.

2010 Mathematics Subject Classification: 28C20, 42B10, 46G12, 46T12, 60G15
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
Geometry

⋅ 19th-C-15:00 − 16:30 Chair: Juncheol Pyo (Pusan National University)

⋅ 19th-C-15:00 − 15:20 Symplectic geometry and the Ma\~n\'e critical value (Seongchan Kim, Will Merry)

김성찬*((공주대)), Will Merry((None))
Seongchan Kim*, Kongju National University, Will Merry, None

A subset $U$ of a symplectic manifold $(M, \omega)$ is said to be displaceable if there is a Hamiltonian diffeomorphism $\phi$ satisfying $\phi(U) \cap U = \emptyset.$ In this talk I will briefly introduce the Ma\~n\'e critical value of a Hamiltonian system defined on a magnetic cotangent bundle of a closed manifold and its relation to displaceability of energy levels.

2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Floer homology, Ma\~n\'e critical value, displaceability

⋅ 19th-C-15:20 − 15:40 Uniqueness results for the critical catenoid (Dong Hwi Seo)

서동휘((한양대))
Dong Hwi Seo, Hanyang University

A free boundary minimal surface in the three-dimensional unit ball is a properly immersed minimal surface in the unit ball that meets the unit sphere orthogonally along the boundary of the surface. The topic was initiated by Nitsche in 1985, derived from studies by Gergonne, Schwarz, Courant, and Lewy. Basic examples are the equatorial disk and the critical catenoid. The equatorial disk is the only immersed free boundary minimal disk in the ball up to congruence. The critical catenoid is claimed to be the only embedded free boundary minimal annulus in the ball up to congruence. Recently, the problem has been attempted using a relationship with the Steklov eigenvalue problem. In this talk, I will describe previous studies in this direction and explain my uniqueness results for the critical catenoid as the embedded free boundary minimal annuli in the ball under symmetry conditions on the boundaries.

2010 Mathematics Subject Classification: 53A10, 58C40, 52A10
Key Words and Phrases: Minimal surface, Steklov eigenvalue problem, free boundary problem

⋅ 19th-C-15:50 − 16:10 Design and connection of parametric surfaces through given regular curves (Hyun Chol Lee, Dae Won Yoon)

이현철*((경상대)), 윤대원((경상대))
Hyun Chol Lee*, Gyeongsang National University, Dae Won Yoon, Gyeongsang National University

This talk focused on the connections between two parametric surfaces through the given regular curves. Accordingly, we analyzed a $C^0$-continuous connected surface in terms of the marching-scale functions of these surfaces. It should be noted that, in general, differentiability along a common curve for a $C^0$-continuously connected surface is not guaranteed. To solve this problem, we introduced a $C^1$-continuous connection and proved its existence for such continuous connections. These connections are improved versions of the $G^1$-connection of developable surfaces introduced by Paluszny. Moreover, we suggested applications to illustrate the $C^1$-continuous connection using B\'{e}zier curves and some marching-scale functions of the parametric surfaces.

2010 Mathematics Subject Classification: 53A05, 65D17, 68U05
Key Words and Phrases: B\'{e}zier curve, developable surface, marching-scale function, continuous connection

⋅ 19th-C-16:10 − 16:30 Catenoids in the three-dimensional light cone (Won Joo Lee)

이원주((고려대))
Won Joo Lee, Korea University

We introduce three-dimensional light cone in four-dimensional Lorentz space with using several models. In addition, we construct examples of catenoids and trinoids with using holomorphic spinor representation formula for surfaces of zero mean curvature in three-dimensional light cone and monodromy conditions.

2010 Mathematics Subject Classification: 53A35
Key Words and Phrases: Three-dimensional light cone, monodromy condition, catenoids, trinoids

⋅ 19th-D-16:50 − 18:00 Chair: Seungsu Hwang (Chung-Ang University)

⋅ 19th-D-16:50 − 17:10 A classification on parallel Ricci tensor for real hypersurfaces in the complex quadric (Hyunjin Lee, Young Jin Suh)

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

First we introduce the notion of parallel Ricci tensor $\nabla \mathrm{Ric} =0 $ for real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2} / SO_{m} SO_{2}$ and show that the unit normal vector field $N$ is singular. Next we give a new classification of real hypersurfaces in the complex quadric $Q^{m}$ with parallel Ricci tensor.

2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: Parallel Ricci tensor, singular tangent vector field, Kahler structure, complex conjugation, complex quadric

⋅ 19th-D-17:10 − 17:30 Semi-symmetric hypersurfaces in complex hyperbolic two-plane Grassmannians (Chang Hwa Woo, Doo Hyun Hwang)

우창화*((부경대)), 황두현((경북대))
Chang Hwa Woo*, Pukyong National University, Doo Hyun Hwang, Kyungpook National University

In this talk, we introduce new notions of symmetric operators such as semi-symmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians with such notions.

2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C15
Key Words and Phrases: Real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, semi-symmetric shape operator, semi-symmetric structure Jacobi operator

⋅ 19th-D-17:40 − 18:00 Monotone property of the first eigenvalue of Laplacian by IMCF (Juncheol Pyo)

표준철((부산대))
Juncheol Pyo, Pusan National University

In this talk, we discuss the first nonzero eigenvalue $\lambda_{p,1}$ of the $p$-Laplacian on free boundary hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that $\lambda_{p,1}$ is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of $\lambda_{p,1}$ of a free boundary disk type hypersurface in the unit ball. This is joint work with Pak Tung Ho.

2010 Mathematics Subject Classification: 58C40
Key Words and Phrases: Inverse mean curvature flow, first eigenvalue, $p$-Laplacian
Topology

⋅ 19th-A-09:00 − 10:30   Chair: Hyoungjun Kim (Kookmin University)

⋅ 19th-A-09:00 − 09:20 The integral cohomology ring of toric surfaces (Xin Fu, Tseleung So, Jongbaek Song)

Xin Fu((아주대)), Tseleung So((University of Western Ontario)), 송종백*((고등과학원))
Xin Fu, Ajou University, Tseleung So, University of Western Ontario, Jongbaek Song*, KIAS

It is well-known that the rational cohomology ring of toric varieties with orbifold singularities behaves similarly to the integral cohomology ring of smooth toric varieties. What is known for the integral cohomology of an arbitrary toric variety is somewhat restrictive and complicated for computational purposes. In this talk, we consider toric surfaces, namely the toric varieties of complex dimension 2, which have at worst orbifold singularities in general. The main result determines the integral cohomology ring structure of toric surfaces in terms of basis and relations.

2010 Mathematics Subject Classification: 14M25, 57S12
Key Words and Phrases: Toric variety, toric orbifold, cohomology ring

⋅ 19th-A-09:20 − 09:40 The Betti numbers of real toric varieties associated to Weyl chambers of type E7 (Suyoung Choi, Seonghyeon Yu, Younghan Yoon)

최수영((아주대)), 유성현((아주대)), 윤영한*((아주대))
Suyoung Choi, Ajou University, Seonghyeon Yu, Ajou University, Younghan Yoon*, Ajou University

In this talk, we compute the Betti numbers of real toric varieties associated to Weyl chambers of type E7.

2010 Mathematics Subject Classification: 57N65, 57S12
Key Words and Phrases: Real toric variety, real toric manifold, Betti number, root system, Weyl chambers, type E7

⋅ 19th-A-09:50 − 10:10 The enumeration of toric colorable pl-spheres with picard number 4 (Suyoung Choi, Hyeontae Jang, Mathieu Vall\'ee)

최수영((아주대)), 장현태*((아주대)), Mathieu Vall\'ee((Universit\'e Paris 13 Nord))
Suyoung Choi, Ajou University, Hyeontae Jang*, Ajou University, Mathieu Vall\'ee, Universit\'e Paris 13 Nord

Toric colorable PL-sphere is a PL-sphere which supports a non-singular characteristic map. To classify quasitoric or toric manifolds, it can be a first step to classify toric colorable PL-spheres. The wedge operation is a classical operation of simplicial complexes. We call a simplicial complex which can not be obtained by wedge operation as a seed. It is known that the number of seeds with a fixed picard number is finite. In this talk, we provide new combinatorial method which allows us to construct a powerful strategy for obtaining the complete list of toric colorable seed PL-spheres with Picard number 4.

2010 Mathematics Subject Classification: 57S12
Key Words and Phrases: Toric topology, pl-sphere

⋅ 19th-A-10:10 − 10:30 Hodge theory for tropical varieties (Hoil Kim)

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

Tropical variety is the variety expressed combinatorially for the complex geometry or p-adic geometry. Tropical varieties have been studied for curves, abelian varieties and some surfaces so far. In this talk we discuss the Hodge structures mostly for K-3 surfaces, abelian varieties and Calabi-Yau varieties.

2010 Mathematics Subject Classification: 14T15, 14T20, 14T25
Key Words and Phrases: Tropical varieties, K3 surfaces, abelian varieties, Calaby-Yau varieties, Hodge structures

⋅ 19th-B-10:50 − 12:20   Chair: Suyoung Choi (Ajou University)

⋅ 19th-B-10:50 − 11:10 Closed string mirror symmetry for punctured spheres (Hansol Hong, Hyeongjun Jin, Sangwook Lee)

홍한솔((연세대)), 진형준*((연세대)), 이상욱((숭실대))
Hansol Hong, Yonsei University, Hyeongjun Jin*, Yonsei University, Sangwook Lee, Soongsil University

In this talk, we construct a mirror of a punctured sphere as an orbifold Landau-Ginzburg model using the group action symmetry. We will see that the mirror is given as the quotient of that of the pair-of-pants, whose mirror picture is the cyclic quotient of the punctured sphere to the pair-of-pants. We then relate the symplectic cohomology of the punctured sphere and the Koszul cohomology of its LG mirror via the closed-open (Kodaira-Spencer) map adapted to the equivariant setting.

2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Symplectic geometry, Floer theory, Kodaira-Spencer map, mirror symmetry

⋅ 19th-B-11:10 − 11:30 Mirror symmetry for log Calabi-Yau surfaces (Hyunbin Kim, Hansol Hong)

김현빈*((연세대)), 홍한솔((연세대))
Hyunbin Kim*, Yonsei University, Hansol Hong, Yonsei University

In this talk, we review the construction of Landau-Ginzburg mirror of Log Calabi-Yau surfaces. We prove closed-string mirror symmetry assuming certain positivity conditions. Using energy induction and the combinatorics of Newton polytopes, we compare the number of critical points of potential functions with the rank of quantum cohomology.

2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Symplectic geometry, Lagrangian Floer theory, mirror symmetry

⋅ 19th-B-11:40 − 12:00 The generalized Harer conjecture for the homology triviality (Wonjun Chang, Byung Chun Kim, Yongjin Song)

장원준*((인하대)), 김병천((인하대)), 송용진((인하대))
Wonjun Chang*, Inha University, Byung Chun Kim, Inha University, Yongjin Song, Inha University

The classical Harer conjecture is about the stable homology triviality of the obvious embedding $\phi : B_{2g+2} \hookrightarrow \Gamma_{g}$, which was proved by Song and Tillmann. The main part of the proof is to show that $\operatorname{B}\phi^{+} : \operatorname{B}B_{\infty}^{+} \rightarrow \operatorname{B}\Gamma_{\infty}^{+}$ induced from $\phi$ is a double loop space map. In this paper, we give a proof of the generalized Harer conjecture which is about the homology triviality for an $arbitrary$ embedding $\phi : B_{n} \hookrightarrow \Gamma_{g,k}$. We first show that it suffices to prove it for a $regular$ embedding in which all atomic surfaces are regarded as identical and each atomic twist is a $simple$ $twist$ interchanging two identical sub-parts of atomic surfaces. The main strategy of the proof is to show that the map $\Phi : \mathcal{C} \rightarrow \mathcal{S}$ induced by $\operatorname{B}\phi:\operatorname{Conf}_n(D)\rightarrow\mathcal{M}_{g,k}$ preserves the actions of the framed little 2-disks operad.

2010 Mathematics Subject Classification: 57M12, 57M50
Key Words and Phrases: Braid group embedding, mapping class groups, branched covering

⋅ 19th-B-12:00 − 12:20 Comultiplication structures on the localization of a wedge of spheres and Moore spaces (Dae-Woong Lee)

이대웅((전북대))
Dae-Woong Lee, Jeonbuk National University

In this talk, we study the set of comultiplications on a wedge sum of CW-spaces. More specifically, we provide the methods of how to calculate the homotopy comultiplications on CW-spaces. Our methods involve basic Whitehead products in a wedge sum of spheres and Hopf-Hilton invariants. In particular, we are interested in the structure of homotopy comultiplications on the localization $L_{(p)}$ of a wedge sum $L: = \mathbb S^m \vee M(G,n)$ of the homotopy spheres and the Moore spaces localized at a prime $p$ for $2 \leq m < n$.

2010 Mathematics Subject Classification: 55Q05, 13B30, 55Q15, 55Q52, 55P60
Key Words and Phrases: Comultiplication, localization, basic Whitehead product, Hilton-Milnor formula, Hopf-Hilton invariant

⋅ 19th-C-15:00 − 16:30   Chair: Seung Yeop Yang (Kyungpook National University)

⋅ 19th-C-15:00 − 15:20 Self-pair homotopy equivalences related to contra variant functors (Hyung Seok Oh, Ho Won Choi)

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

The category of pairs is the category whose objects are continuous maps between two based spaces and morphisms are pair-maps from one object to another object. In general, self-pair homotopy equivalence is a difficult and complicated problem in the category of pairs. There are few results using algebraic tools. In this paper, we introduce certain subgroups and exact sequences of self-pair homotopy equivalence related to contra variant functors. Using a contra variant functor, these are interpreted as group's objects and pair homomorphisms. Using this fact, we study about the self-pair homotopy equivalences related contra variant functors. We defined some groups and it's subgroups and give to results of certain exact sequence. These results lead up to cohomotopy and cohomology theory. In cohomotoy and cohomology, we give some examples which are depending on objects even if we fixed two based spaces. we investigate two certain subgroups $\mathscr E_{\ast}^{n}(\alpha)$ and $\mathscr E_{n}^{\ast}(\alpha)$. For the certain condition, we have $\mathscr E_{\ast}^{n}(\alpha)=\mathscr E_{n}^{\ast}(\alpha)$.

2010 Mathematics Subject Classification: 55P10, 55Q05, 55Q55
Key Words and Phrases: Category of pairs, self-pair homotopy equivalence, contra variant functor

⋅ 19th-C-15:20 − 15:40 Ropelength of 2-bridge knots using superhelices (Hyoungjun Kim, Youngsik Huh, Seungsang Oh)

김형준*((국민대)), 허영식((한양대)), 오승상((고려대))
Hyoungjun Kim*, Kookmin University, Youngsik Huh, Hanyang University, Seungsang Oh, Korea University

The ropelength is a mathematical quantity that regulates the tightness of flexible strands in the three-dimensional space. The superhelical conformation of long twisted strands is known to be more efficient in terms of ropelength compared with the circular double helical conformation. In this talk, we present a conformation of 2-bridge knots by using ropelength-minimizing superhelical curves and derive an upper bound on the ropelength of 2-bridge knots. Our superhelical model of 2-bridge knots is shown to be more efficient than the standard double helical one if the iterative twisted parts are long enough.

2010 Mathematics Subject Classification: 57K10
Key Words and Phrases: Ropelength, superhelices, 2-bridge knot

⋅ 19th-C-15:50 − 16:10 Aztec bipyramid and dicube tilings (Sunmook Choi, Sangyop Lee, Seungsang Oh)

최선묵*((고려대)), 이상엽((중앙대)), 오승상((고려대))
Sunmook Choi*, Korea University, Sangyop Lee, Chung-Ang University, Seungsang Oh, Korea University

We consider the enumeration of dicube tilings, each of which is a three-dimensional tessellation of a polycube with dicubes. The enumeration of domino tilings of the Aztec diamond and the augmented Aztec diamond is well studied. As a three-dimensional analogue of the Aztec diamond, the Aztec bipyramid is the polycube consisting of unit cubes, which resembles a platonic octahedron. In this paper, we find a bijection between dicube tilings of the Aztec bipyramid and three-dimensional Delannoy paths, and use it to count the number of dicube tilings of the Aztec bipyramid.

2010 Mathematics Subject Classification: 05A15, 05B45, 05C70
Key Words and Phrases: Aztec bipyramid, dicube tiling, perfect matching

⋅ 19th-C-16:10 − 16:30 Linking numbers of 2-bridge links and Montesinos links (Sung Jong No, Hyoungjun Kim, Hyungkee Yoo)

노성종*((경기대)), 김형준((국민대)), 유형기((이화여대))
Sung Jong No*, Kyonggi University, Hyoungjun Kim, Kookmin University, Hyungkee Yoo, Ewha Womans University

The linking number of an oriented two-component link is an invariant indicating how intertwined the two components are. A $\frac{p}{q}$-rational link is a two-component link if and only if $p$ is even. Tuler proved that the linking number of $\frac{p}{q}$-rational link is $$\sum^{\frac{|p|}{2}}_{k=1} (-1)^{\big\lfloor (2k-1) \frac{q}{p} \big\rfloor }.$$ In this talk, we introduce the numerical algorithm to find the linking number of $\frac{p}{q}$-rational link. Furthermore we also introduce the linking number of two-component Montesinos link.

2010 Mathematics Subject Classification: 57K10
Key Words and Phrases: Linking number, 2-bridge link, rational link, Montesinos link

⋅ 19th-D-16:50 − 18:20   Chair: Sung Jong No (Kyonggi University)

⋅ 19th-D-16:50 − 17:10 A robust property of measure continuum-wise expansive systems (Manseob Lee, Seunghee Lee, Bomi Shin)

이만섭((목원대)), 이승희*((건양대의료원)), 신보미((성균관대))
Manseob Lee, Mokwon University, Seunghee Lee*, Konyang University Medical Center, Bomi Shin, Sungkyunkwan University

In this talk, we consider measure continuum-wise expansive systems which link the continuum theory to measurable dynamics. For the concept, we study the relationship with continuum-wise expansive measure and hyperbolic structure. More precisely, for a diffeomorphism $f:M \rightarrow M $ of a compact smooth manifold $M$, we prove that if $f$ is a robustly measure continuum-wise expansive, then it is quasi-Anosov. Moreover we show that a local behavior of such system.

2010 Mathematics Subject Classification: 37C40, 37D05
Key Words and Phrases: Continuum-wise expansive, quasi-Anosov, hyperbolicity

⋅ 19th-D-17:10 − 17:30 Mosaics and invariants for surface-links (Seonmi Choi, Sam Nelson)

최선미*((경북대)), Sam Nelson((Claremont McKenna College))
Seonmi Choi*, Kyungpook National University, Sam Nelson, Claremont McKenna College

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. In 2014, Kuriya and Shehab proved Lomonaco-Kauffman conjecture which means that knot mosaic type is a complete invariant of tame knots. The mosaic number of a knot $K$ is the smallest integer $n$ for which $K$ can be represented on an $n\times n$ mosaic board. In this talk, we define a mosaic system for surface-links using marked graph diagrams and discuss some invariants of surface-links via marked graph mosaics. This is joint work with Sam Nelson.

2010 Mathematics Subject Classification: 57K10, 57K12
Key Words and Phrases: Mosaic, marked graph, surface-knot, mosaic number

⋅ 19th-D-17:40 − 18:00 A note on the extreme Khovanov homology of a pretzel link (Hongdae Yun, Jinseok Oh, Seung Yeop Yang)

윤홍대*((경북대)), 오진석((경북대)), 양승엽((경북대))
Hongdae Yun*, Kyungpook National University, Jinseok Oh, Kyungpook National University, Seung Yeop Yang, Kyungpook National University

Khovanov (co)homology was introduced by Mikhail Khovanov in 2000, and Viro described it as an enhanced state of a link diagram. J. Gonz\'alez-Meneses, P. M. G. Manch\'on, M. Silvero showed the (potential) extreme Khovanov homology of a link is isomorphic to the independence simplicial complex of a Lando graph at the link. In this talk, we examine the extreme Khovanov homology of 3-strand pretzel links and their geometric realizations. This is joint work with Jinseok Oh and Seung Yeop Yang.

2010 Mathematics Subject Classification: 57K10, 57M15, 57K18
Key Words and Phrases: Khovanov homology, pretzel knot, geometric realization

⋅ 19th-D-18:00 − 18:20 Betti numbers of the Yang-Baxter homology of Alexander biquandles (Jinseok Oh, Seung Yeop Yang, Hongdae Yun)

오진석*((경북대)), 양승엽((경북대)), 윤홍대((경북대))
Jinseok Oh*, Kyungpook National University, Seung Yeop Yang, Kyungpook National University, Hongdae Yun, Kyungpook National University

A homology theory of set-theoretic Yang-Baxter operators was established by Carter, Elhamdadi, and Saito. Biquandles, a generalization of quandles, are special solutions of the set-theoretic Yang-Baxter equation. The free parts of quandle homology groups were completely determined, but only little is known about biquandles. In this talk, we first review the definition of set-theoretic Yang-Baxter (co)homology and determine the Betti numbers of some finite Alexander biquandles. This is joint work with Seung Yeop Yang and Hongdae Yun.

2010 Mathematics Subject Classification: 57K12, 57K10, 55N35
Key Words and Phrases: Alexander biquandle, Betti number
Probability and Statistics

⋅ 19th-C-15:00 − 16:30 Chair: Ildoo Kim (Korea University)

⋅ 19th-C-15:00 − 15:20 Potential theory of Dirichlet forms with jump kernels blowing up at the boundary (Panki Kim)

김판기((서울대))
Panki Kim, Seoul National University

In this talk, we study the potential theory of Dirichlet forms on the half-space defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha} B(x,y)$ and the killing potential $\kappa x_d^{-\alpha}$, where $\alpha\in (0, 2)$ and $B(x,y)$ can blow up to infinity at the boundary. The jump kernel and the killing potential depend on several parameters. For all admissible values of the parameters involved and all $d \ge 1$, we prove that the boundary Harnack principle holds, and establish sharp two-sided estimates on the Green functions of these processes.

2010 Mathematics Subject Classification: 60J45
Key Words and Phrases: Jump processes, jump kernel degenerate at the boundary, Carleson estimate, boundary Harnack principle, Green function

⋅ 19th-C-15:20 − 15:40 Local limit theorem of Brownian motion on metric trees (Soon Ki Hong)

홍순기((가톨릭관동대))
Soon Ki Hong, Catholic Kwandong University

Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group $\Gamma$ acts isometrically and geometrically on the tree $\mathcal{T}$. Assume that the length spectrum of $\Gamma$ is 2-Diophantine or 4-Diophantine. In this talk, using thermodynamic formalism for the measure on the space of geodesic lines, we show that there exists a continuous function $C$ on $\mathcal{T}\times \mathcal{T}$ such that the heat kernel $p(t,x,y)$ of $\mathcal{T}$ satisfies $$\lim_{t\rightarrow \infty}t^{3/2}e^{\lambda_0t}p(t,x,y)=C(x,y)$$ for any $x,y\in \mathcal{T}\times \mathcal{T}$. Here $\lambda_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$.

2010 Mathematics Subject Classification: 37A50, 37A25
Key Words and Phrases: Dynamical systems and their relations with probability theory and stochastic processes, rates of mixing

⋅ 19th-C-15:50 − 16:10 On the characteristic polynomial of the eigenvalue moduli of random normal matrices (Sung-Soo Byun, Christophe Charlier)

변성수*((고등과학원)), Christophe Charlier((Lund University))
Sung-Soo Byun*, KIAS, Christophe Charlier, Lund University

In this talk, I will discuss the characteristic polynomial of the eigenvalue moduli drawn from the Mittag-Leffler ensemble, a two-dimensional determinantal point process that generalises the Ginibre point process. I will present precise asymptotic behaviours of the moment generating function, which involves a large structured determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities. In particular, I will explain that such asymptotic behaviours surprisingly yield a new kind of ingredient called associated Hermite polynomials.

2010 Mathematics Subject Classification: 41A60, 60B20, 60G55
Key Words and Phrases: Jump- and root-type singularities along circles, moment generating functions, random matrix theory, asymptotic analysis

⋅ 19th-C-16:10 − 16:30 Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space (Haesung Lee, Gerald Trutnau)

이해성*((한국과학영재학교)), Gerald Trutnau((서울대))
Haesung Lee*, Korea Science Academy of KAIST, Gerald Trutnau, Seoul National University

We consider a locally uniformly strictly elliptic second order partial differential operator in $\mathbb{R}^d$, $d \geq 2$, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt process is known to solve a corresponding stochastic differential equation that is pathwise unique. Our main result is that recurrence implies uniqueness of infinitesimally invariant measures, as well as existence and uniqueness of invariant measures. We can hence make in particular use of various explicit analytic criteria for recurrence that have been previously developed in the context of (generalized) Dirichlet forms and present diverse examples and counterexamples for uniqueness of infinitesimally invariant measures, as well as invariant measures. Furthermore, we illustrate how our results can be applied to related work and vice versa. This is joint work with Gerald Trutnau (Seoul National University).

2010 Mathematics Subject Classification: 60J46, 31C25
Key Words and Phrases: (Infinitesimally) invariant measures, recurrence, conservativeness, $C_0$-semigroup, (generalized) Dirichlet forms

⋅ 19th-D-16:50 − 17:30 Chair: Ildoo Kim (Korea University)

⋅ 19th-D-16:50 − 17:10 Clinical time delay distributions of COVID-19 in 2020–2022 in the Republic of Korea: Inferences from a nationwide database analysis (Eunha Shim, Wongyeong Choi, Youngji Song)

심은하((숭실대)), 최원경*((숭실대)), 송영지((숭실대))
Eunha Shim, Soongsil University, Wongyeong Choi*, Soongsil University, Youngji Song, Soongsil University

Estimates in epidemiological distributions of the coronavirus disease 2019 (COVID-19) essentially characterize disease spread and thereby allow the implementation of effective containment strategies. Using the most complete individual-level data obtained from the Korea Disease Control and Prevention Agency and the Central Disease Control Headquarters, we estimated the following key epidemiological distributions: onset-to-diagnosis, onset-to-reporting, onset-to-death, reporting-to-death, and serial interval from before and during the Delta variant’s predominance. We adopted a Bayesian model comparison to determine the best model to fit the data, and a hierarchical Bayesian model with partial pooling to estimate the model parameters at the region level. We found that the COVID-19 pandemic in the Republic of Korea was characterized by relatively short onset-to-diagnosis and onset-to-report intervals but by long serial intervals. Gamma distribution showed the best fit for the onset-to-death interval (with heterogeneity in age, sex, and comorbidities) and the reporting-to-death interval, whereas Log-normal distribution was optimal for ascertaining the onset-to-diagnosis and onset-to-report intervals. Serial interval (days) was shorter before the Delta variant-induced outbreaks than during the Delta variant’s predominance (4.4 vs. 5.2 days), indicating the higher transmission potential of the Delta variant. The identified heterogeneity in region-, age-, sex-, and period-based distributions of the transmission dynamics of COVID-19 will facilitate the development of effective interventions and disease-control strategies.

2010 Mathematics Subject Classification: 62F15
Key Words and Phrases: COVID-19, epidemiological distribution, Delta variant, Republic of Korea, serial interval, SARS-CoV-2

⋅ 19th-D-17:10 − 17:30 Phase transition in the generalized stochastic block model (Sunmin Lee, Jioon Lee)

이선민*((카이스트)), 이지운((카이스트))
Sunmin Lee*, KAIST, Jioon Lee, KAIST

We study the problem of detecting the community structure from the generalized stochastic block model (GSBM). Based on the analysis of the Stieljtes transform of the empirical spectral distribution, we prove a BBP-type transition for the largest eigenvalue of the GSBM. For specific models such as a hidden community model and an unbalanced stochastic model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.

2010 Mathematics Subject Classification: 68Q87, 15B51
Key Words and Phrases: BBP-type transition, empirical spectral distribution, Wigner-type matrix, hidden community model, unbalanced stochastic model
Applied Mathematics(including AI, Data Science)

⋅ 19th-C-15:00 − 16:30 Chair: Kwang-Yeon Kim (Kangwon National University)

⋅ 19th-C-15:00 − 15:20 Construction of interpolatory Hermite subdivision schemes preserving polynomials (Byeongseon Jeong)

정병선((계명대))
Byeongseon Jeong, Keimyung University

We construct a new family of interpolatory Hermite subdivision schemes that act on a sequence of vectors whose components can be respectively regarded as approximations to values of a function and its first order derivative on a uniform grid. Our construction is based on the polynomial preservation property of a Hermite subdivision scheme, which is strongly connected to the scheme's order of approximation. The proposed Hermite schemes have free tension parameters to be utilized according to design circumstances. We show that the limit function generated by our scheme whose associated masks have the shortest support is four times continuously differentiable. Some numerical examples verifying the theoretical results are presented.

2010 Mathematics Subject Classification: 41A05, 65D15, 65D17
Key Words and Phrases: Hermite subdivision, interpolatory scheme, polynomial preservation, approximation order

⋅ 19th-C-15:20 − 15:40 Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with random initial data (Seungchan Ko, Guanglian Li, Yi Yu)

고승찬*((성균관대)), Guanglian Li((The University of Hong Kong)), Yi Yu((The University of Hong Kong))
Seungchan Ko*, Sungkyunkwan University, Guanglian Li, The University of Hong Kong, Yi Yu, The University of Hong Kong

In this paper, we analyze the numerical approximation of the Navier-Stokes problem in a polygonal domain in $\mathbb{R}^2$, where the initial condition is given by a lognormal random field. We aim to compute the expectation value of linear functionals of the solution of the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes finite element, fully-discrete space-time discretizations, truncated Karhunen-Loeve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expectation values over parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the high-dimensional domain, which guarantees the error decays with $\mathcal{O}(N^{-1+\delta})$ where $N$ is the number of sampling points, $\delta>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.

2010 Mathematics Subject Classification: 65D30, 65N30, 76D05
Key Words and Phrases: Quasi-Monte Carlo method, finite element method, uncertainty quantification, Navier-Stokes equations, random initial data, lognormal random field, Karhunen-Loeve expansion

⋅ 19th-C-15:50 − 16:10 Universal approximation theorem on metric spaces (Woochul Jung)

정우철((건양대병원))
Woochul Jung, Konyang University Hospital

Deep learning has widely attracted in many areas and recently developed mathematical studies for supervised learning which is a training process for seeking an approximation solution as a target function. In this talk, we discuss an overview of deep learning theory from an approximation perspective. We characterize a sort of extensor property for metric spaces and present the universal approximation property of certain function classes on metric spaces.

2010 Mathematics Subject Classification: 41A65
Key Words and Phrases: Deep learning, approximation theory

⋅ 19th-C-16:10 − 16:30 Proportion of pre-symptomatic transmission events associated with COVID-19 in South Korea (Youngji Song, Eunha Shim)

송영지*((숭실대)), 심은하((숭실대))
Youngji Song*, Soongsil University, Eunha Shim, Soongsil University

Pre-symptomatic transmission potentially reduces the effectiveness of symptom-onset-based containment and control strategies for the coronavirus disease (COVID-19). Despite evidence from multiple settings, the proportion of pre-symptomatic transmission varies among countries. To estimate the extent of pre-symptomatic transmission in South Korea, we used individual-level COVID-19 case records from the Korea Disease Control and Prevention Agency and Central Disease Control Headquarters. We inferred the probability of symptom onset per day since infection based on the density distribution of the incubation period to stratify the serial interval distribution in Period 1 (20 January–10 February 2020) and Period 2 (25 July–4 December 2021), without and with expanded testing or implementation of social distancing strategies, respectively. Assuming both no correlation as well as positive and negative correlations between the incubation period and the serial interval, we estimated the proportion of pre-symptomatic transmission in South Korea as 43.5\% (accounting for correlation, range: 9.9–45.4\%) and 60.0\% (56.2–64.1\%) without and with expanded testing, respectively, during the Delta variant's predominance. This study highlights the importance of considering pre-symptomatic transmission for COVID-19 containment and mitigation strategies because pre-symptomatic transmission may play a key role in the epidemiology of COVID-19.

2010 Mathematics Subject Classification: 92D30
Key Words and Phrases: COVID-19, Korea, pre-symptomatic, serial interval, incubation period, SARS-CoV-2, statistical, mathematical, expanded testing, Delta variant
Mathematical Education

⋅ 19th-C-15:00 − 15:20 Chair: Young Rock Kim (Hankuk University of Foreign Studies)

⋅ 19th-C-15:00 − 15:20 Case study on college calculus education with coding (Jae Hwa Lee, Sang-Gu Lee)

이재화*((성균관대)), 이상구((성균관대))
Jae Hwa Lee*, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University

In this talk, we introduced the case of college calculus course with coding. We suggest this case as an alternative to overcome mathematics anxiety. Contents, python/SageMath codes, and textbook for this course, which help students to easily and quickly review middle and high school mathematics, were newly developed by authors. Due to the use of codes and chat with classmates in learning management system, most of the students who took this course reported that they no longer felt anxious in complex mathematics problems, had a full understanding of calculus concepts, could solve almost problems in any calculus textbooks with or without codes, and could explain calculus concepts to other students in their own words. In this way if mathematics and coding is properly used in mathematics education, it helps students with weak mathematical backgrounds or mathematics anxiety to restore confidence in mathematics in college. This could be applicable in secondary mathematics education.

2010 Mathematics Subject Classification: 97U50, 97U70
Key Words and Phrases: Calculus, coding, college mathematics education
Discrete Mathematics

⋅ 19th-C-15:00 − 16:30 Chair: Soojin Cho (Ajou University)

⋅ 19th-C-15:00 − 15:20 The $\alpha$-analogue $r$-Whitney numbers via normal ordering (Hye Kyung Kim)

김혜경((대구가톨릭대))
Hye Kyung Kim, Daegu Catholic University

The normal ordering of an integral power of the number operator $a^{\dag}a$ in terms of boson annihilation $a$ and creation operators $a^{\dag}$ operators is expressed with the help of the Stirling numbers of the second kind. The normal ordering problems directly links the problems to combinatorics. In this paper, we consider new combinatorial numbers which are the $\alpha$-analogue $r$-Whitney of the first and second kind $(\alpha \in \mathbb{R}, r \in \mathbb{N})$, respectively. The combinatorial properties of the these new numbers are shown to follow from the algebraic properties of the boson operators. In detail, we derive some properties, recurrence relations and several identities on these new numbers arising from $\alpha$-analogue normal ordering.

2010 Mathematics Subject Classification: 05A17, 11B73, 05A18
Key Words and Phrases: $\alpha$-analogue $r$-Stirling numbers of the first kind, $\alpha$-analogue $r$-Stirling numbers of the second kind, $r$-Whitney numbers of the first kind, $r$-Whitney numbers of the second kind

⋅ 19th-C-15:20 − 15:40 The circuit-cocircuit intersection conjecture for intersection size $k\le 6$ (Jaeho Shin)

신재호((서울대))
Jaeho Shin, Seoul National University

Oxley conjectured (1992) that if a matroid has a circuit-cocircuit intersection of size $k\ge4$, then it has a circuit-cocircuit intersection of size $k-2$. We show that this conjecture holds for $k\le6$.

2010 Mathematics Subject Classification: 05B35
Key Words and Phrases: Matroid, circuit-cocircuit intersection

⋅ 19th-C-15:50 − 16:10 The importance of the empty set in Katona's circle method and implications (Stijn Cambie)

19th-C-15:50 − 16:10
Stijn Cambie, IBS Ecopro

The Erdos-Ko-Rado theorem from 1961 is one of the foundational results in extremal set theory. The uniform version says that a the trivial intersecting family is maximum among all possible intersecting families. An elementary proof uses Katana's circle method. This method can be extended to multiple family versions, provided these are non-empty. As an example of the power of this simple observation, we sketch how one could prove a weak form of Frankl's conjecture on cross-union families. A stronger form of the conjecture has been proven in a paper with Jaehoon Kim, Hong Liu and Tuan Tran, arxiv.org/abs/2202.10365.

2010 Mathematics Subject Classification: 05D05
Key Words and Phrases: Intersecting sets

⋅ 19th-C-16:10 − 16:30 Transversal numbers of stacked 2-spheres (Minho Cho, Jinha Kim, Minki Kim)

조민호*((카이스트)), 김진하((기초과학연구원 이산수학 그룹)), 김민기((광주과학기술원))
Minho Cho*, KAIST, Jinha Kim, IBS Discrete Mathematics Group (DIMAG), Minki Kim, GIST

A simplicial complex $B$ is called \emph{stacked $(d+1)$-ball} if it is constructed by repeatedly attaching one $(d+1)$-simplex at a time along a free $d$-face, starting with a $(d+1)$-simplex. A simplicial complex $S$ is \emph{stacked $d$-sphere} if it is the boundary of a stacked $(d+1)$-ball. It is natural but challenging problem to find the maximum transversal number of facet hypergraphs of $d$-polytopes having $n$ vertices. We ask the same question for the class of stacked 2-spheres instead of general polytopes, and provide both upper and lower bounds for the maximum transversal number. Both proofs are constructive and computer-assisted.

2010 Mathematics Subject Classification: 05D15, 52B12
Key Words and Phrases: Stacked sphere, stacked ball, transversal number, simplicial polytope

⋅ 19th-D-16:50 − 18:00 Chair: O Joung Kwon (Hanyang University)

⋅ 19th-D-16:50 − 17:10 Odd covers by complete bipartite graphs (Calum Buchanan, Eric Culver, Alexander Clifton, Jiaxi Nie, Jason O'Neill, Puck Rombach, Mei Yin)

19th-D-16:50 − 17:10
Calum Buchanan, University of Vermont, Eric Culver, Brigham Young University, Alexander Clifton*, Discrete Mathematics Group, Institute for Basic Science, Jiaxi Nie, Max Planck Institute for Mathematics in the Sciences, Jason O'Neill, California State University, Los Angeles, Puck Rombach, University of Vermont, Mei Yin, University of Denver

Given a finite simple graph $G$, an {\em odd cover of $G$} is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$ appears in an odd number of bicliques and each non-edge of $G$ appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of $G$ by $b_2(G)$ and prove that $b_2(G)$ is bounded below by half of the rank over $\mathbb{F}_2$ of the adjacency matrix of $G$. We show that this lower bound is tight in the case when $G$ is a bipartite graph and almost tight when $G$ is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from $b_2(G)$.

2010 Mathematics Subject Classification: 05C62, 05C75, 05C50
Key Words and Phrases: Odd cover problem, complete bipartite graph, Graham-Pollak, bipartite subgraph complementation

⋅ 19th-D-17:10 − 17:30 A proper $h$-conflict-free coloring of a graph (Eun-Kyung Cho, Ilkyoo Choi, Hyemin Kwon, Boram Park)

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

For a positive integer $h$, a proper $h$-conflict-free $c$-coloring of a graph $G$ is a proper $c$-coloring such that each vertex has at least $\min\{d_G(v),h\}$ colors appearing uniquely on its neighborhood. The proper $h$-conflict-free chromatic number of a graph $G$, denoted $\chi_{pcf}^h(G)$, is the minimum $c$ such that $G$ has a proper $h$-conflict-free $c$-coloring. We present recent results on $\chi_{pcf}^h(G)$ by focusing on Brook’s type results. We prove for a graph $G$ with $1\le h \le \Delta(G)-2$, $\chi_{pcf}^h(G)\le (h+1)\Delta-1$. This is based on joint work with Eun-Kyung Cho, Ilkyoo Choi, and Boram Park.

2010 Mathematics Subject Classification: 05C69
Key Words and Phrases: Proper $h$-conflict-free coloring

⋅ 19th-D-17:40 − 18:00 The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs (Jungho Ahn, Seonghyuk Im, Sang-il Oum)

안정호*((카이스트)), 임성혁((카이스트)), 엄상일((기초과학연구원, 이산수학그룹))
Jungho Ahn*, KAIST, Seonghyuk Im, KAIST, Sang-il Oum, Institute for Basic Science (IBS), Discrete Mathematics Group (DIMAG)

A proper coloring of a graph is proper conflict-free if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the PCF $k$-Coloring problem asks whether an input graph admits a proper conflict-free $k$-coloring and the Odd $k$-Coloring asks whether an input graph admits an odd $k$-coloring. We show that for every integer $k\geq3$, both problems are NP-complete, even if the input graph is bipartite. Furthermore, we show that the PCF $4$-Coloring problem is NP-complete when the input graph is planar.

2010 Mathematics Subject Classification: 05C15, 68Q17
Key Words and Phrases: Proper conflict-free coloring, odd coloring, planar graph, bipartite graph
Cryptography

⋅ 19th-C-15:00 − 16:30 Chair: Sungwook Kim (Seoul Women's University)

⋅ 19th-C-15:00 − 15:20 META-BTS: Bootstrapping precision beyond the limit (Wonhee Cho, Youngjin Bae, Jung Hee Cheon, Jaehyung Kim, Taekyung Kim)

조원희*((서울대)), 배영진((((주))크립토랩)), 천정희((((주))크립토랩)), 김재형((((주))크립토랩)), 김태경((((주))크립토랩))
Wonhee Cho*, Seoul National University, Youngjin Bae, CryptoLab. Inc., Jung Hee Cheon, CryptoLab. Inc., Jaehyung Kim, CryptoLab. Inc., Taekyung Kim, CryptoLab. Inc.

Bootstrapping, which enables the full homomorphic encryption scheme that can perform an infinite number of operations by restoring the modulus of the ciphertext with a small modulus, is an essential step in homomorphic encryption. However, bootstrapping is the most time and memory consuming of all homomorphic operations. As we increase the precision of bootstrapping, a large amount of computational resources is required. Specifically, for any of the previous bootstrap designs, the precision of bootstrapping is limited by rescaling precision. In this paper, we propose a new bootstrapping algorithm of the Cheon-Kim-Kim-Song (CKKS) scheme to use a known bootstrapping algorithm repeatedly, so called {\it Meta-BTS}. By repeating the original bootstrapping operation twice, one can obtain another bootstrapping with its precision essentially doubled; it can be generalized to be $k$-fold bootstrapping operations for some $k>1$ while the ciphertext size is large enough. Our algorithm overcomes the precision limitation given by the rescale operation.

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: Fully homomorphic encryption, bootstrapping, high precision

⋅ 19th-C-15:20 − 15:40 Attack on secure triplet loss (Bora Jeong, Sunpill Kim, Seunghun Paik, Jae Hong Seo)

정보라*((한양대)), 김선필((한양대)), 백승훈((한양대)), 서재홍((한양대))
Bora Jeong*, Hanyang University, Sunpill Kim, Hanyang University, Seunghun Paik, Hanyang University, Jae Hong Seo, Hanyang University

Major improvements in biometric authentication have been made in recent years due to the advancements in deep learning. Using a deep learning-based facial recognition model, more discriminative feature vectors can be extracted from faces. A large threat to user privacy could result from the disclosure of more discriminatory feature vectors. Among many biometric template protection (BTP) schemes, there have been studies that have attempted to protect feature vectors from a deep learning model, while considering security requirements. One of them is secure triplet loss (STL) based BTP, which is an end-to-end BTP scheme using deep learning model that merges an additional layer on a pre-trained facial recognition model. STL-based BTP takes a pre-defined key and an image as inputs, and it is designed to become closer only when both the identity and the key are matched simultaneously. In this paper, we propose efficient attacks on STL-based BTP that breaks irreversibility, which is one of the security requirements. Our attack is conducted in a black-box setting using only the similarity scores between a target template and the queried image and key pair. We succeeded in the attack using approximately 329.59 and 256.57 queries for the two types of black-box target systems.

2010 Mathematics Subject Classification: 68W25
Key Words and Phrases: Authentication, biometrics, face recognition, impersonation attack

⋅ 19th-C-15:50 − 16:10 Analysis on locality sensitive hashing-based biometric template protection schemes (Seunghun Paik, Sunpill Kim, Jae Hong Seo)

백승훈*((한양대)), 김선필((한양대)), 서재홍((한양대))
Seunghun Paik*, Hanyang University, Sunpill Kim, Hanyang University, Jae Hong Seo, Hanyang University

In recent decades, deep learning-based biometric authentication systems have been developed, and numerous applications have been deployed in practice. However, privacy issues such as reconstruction attacks that recover the original biometric information from the output features of the deep learning based model have been reported. Owing to the unchangeable nature of biometric information, a biometric template protection scheme (BTPS) is proposed. Locality-sensitive hashing (LSH) is a promising solution for designing BTPS in practice, and several constructions have already been proposed. In this paper, we propose an efficient attack algorithm to find the pre-image of a protected template from LSH-based BTPS, claiming that their schemes do not satisfy irreversibility, which is a standard security requirement for BTPS. We evaluated our attack algorithm using three types of LSH-based BTPSs and LFW datasets as target systems and datasets, respectively. According to the results, our algorithm achieves an attack success rate of 99.46\% against an LSH-based protection scheme that achieved an acceptance rate of 99.70\% for true pairs on the LFW dataset. Furthermore, we provide examples of reconstructed facial images corresponding to biometric information using the output of our attack algorithm as input to the existing reconstruction attack model.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Biometric authentication, inversion attack, locality senstive hashing, template protection

⋅ 19th-C-16:10 − 16:30 On subgaussian gadget decomposition in homomorphic encryption (Sohyun Jeon, Hyang-Sook Lee, Jeongeun Park)

전소현*((이화여대)), 이향숙((이화여대)), 박정은((imec-COSIC, KU Leuven, Leuven, Belgium))
Sohyun Jeon*, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University, Jeongeun Park, imec-COSIC, KU Leuven, Leuven, Belgium

Gadget decomposition is widely used in lattice based cryptography, especially homomorphic encryption (HE) to keep the noise growth slow. If it is randomized, it is called subgaussian (gadget) decomposition which guarantees that we can bound the noise contained in ciphertexts by its variance. This gives tighter and cleaner noise bound in average case, instead of the use of its norm. Even though there are few attempts to build efficient such algorithms, most of them are still not practical enough to be applied to homomorphic encryption schemes due to somewhat high overhead. Furthermore, there has been no detailed analysis of existing works. Therefore, HE schemes use the deterministic decomposition algorithm and rely on a Heuristic assumption that every output element follows a subgaussian distribution independently. In this work, we introduce a new practical subgaussian gadget decomposition algorithm which has the least time overhead over deterministic decomposition among existing works for certain parameter sets. We give a detailed comparison, even for other parameters, with all the competitive algorithms for applications based on homomorphic encryption to choose the best algorithm for their choice of parameters.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Gadget decomposition, subgaussian distribution, lattice

⋅ 19th-D-16:50 − 18:00 Chair: Changmin Lee (KIAS)

⋅ 19th-D-16:50 − 17:10 Efficient polynomial commitment scheme over groups of unknown order (Sungwook Kim, Sungju Kim, Jihye Kim, Hyunok Oh)

김성욱*((서울여대)), 김성주((Zkrypto)), 김지혜((국민대)), 오현옥((한양대))
Sungwook Kim*, Seoul Women's University, Sungju Kim, Zkrypto, Jihye Kim, Kookmin University, Hyunok Oh, Hanyang University

We propose a new efficient transparent polynomial commitment scheme. In a conventional polynomial commitment scheme, a prover commits a polynomial, and then a verifier chooses a random point and sends it to a prover. Then the polynomial is evaluated on the given random point with generating a proof that the evaluated value is correctly computed according to the committed function. The evaluated value is regarded as a representative to the polynomial. Our construction starts with a semi-polynomial commitment scheme using a different presentation on a polynomial that is called a plain polynomial commitment scheme. More precisely, while conventional polynomial commitment schemes evaluate $f(z)=w$, the proposed approach computes $w = f(q) \mod z$ where $f(X)$ is committed, $z$ is randomly chosen by a verifier, and $q$ is large enough so that $f(q)$ is bijective to $f(X)$. Our construction is based on the polynomial commitment scheme (the DARK compiler) proposed by B{\"{u}}nz, Fisch, and Szepieniec in EUROCRYPT 2020. By aggregating commitments from recursive steps in DARK, the proposed plain polynomial commitment reduces the proof size by half compared to DARK. By adopting the proposed polynomial commitment scheme, the efficiency of transparent SNARKs from polynomial IOPs can be significantly improved.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Polynomial commitment scheme, trustless setup, zk-SNARKs, unknown order group

⋅ 19th-D-17:10 − 17:30 Multi-user security of the sum of truncated random permutations (Wonseok Choi, Hwigyeom Kim, Jooyoung Lee, Yeongmin Lee)

최원석*((고등과학원)), 김휘겸((카이스트)), 이주영((카이스트)), 이영민((카이스트))
Wonseok Choi*, KIAS, Hwigyeom Kim, KAIST, Jooyoung Lee, KAIST, Yeongmin Lee, KAIST

For several decades, constructing pseudorandom functions from pseudorandom permutations, so-called Luby-Rackoff backward construction, has been a popular cryptographic problem. Two methods are well-known and comprehensively studied for this problem: summing two random permutations and truncating partial bits of the output from a random permutation. By combining both summation and truncation, we propose new Luby-Rackoff backward constructions, dubbed SaT1 and SaT2. SaT2 is obtained by partially truncating output bits from the sum of two independent random permutations, and SaT1 is its single permutation-based variant using domain separation. The distinguishing advantage against SaT1 and SaT2 is upper bounded by $$O({\sqrt{\mu q_{\max}}}/{2^{n-0.5m}})\text{ and } O({\sqrt{\mu }q_{\max}^{1.5}}/{2^{2n-0.5m}}),$$ respectively, in the multi-user setting, where $n$ is the size of the underlying permutation, $m$ is the output size of the construction, $\mu$ is the number of users, and $q_{\max}$ is the maximum number of queries per user. We also prove the distinguishing advantage against a variant of XORP[3] (studied by Bhattacharya and Nandi at Asiacrypt 2021) using independent permutations, dubbed SoP3-2, which is upper bounded by $O({\sqrt{\mu} q_{\max}^2}/{2^{2.5n}})$. For $\mu = O(2^{n-m})$, multi-user security of a truncated random permutation is bounded by the birthday bound, while SaT1 and SaT2 are fully secure, i.e., allowing $O(2^n)$ queries for each user. It is the same security level as XORP[3], which requires three permutation calls while SaT1 and SaT2 only need two permutation calls.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Pseudorandom function, Luby-Rackoff backward, sum of permutations, truncated random permutation, multi-user security

⋅ 19th-D-17:40 − 18:00 On the number of linearly independent equations for power functions (Mingyu Cho, Jincheol Ha, Jooyoung Lee)

조민규*((카이스트)), 하진철((카이스트)), 이주영((카이스트))
Mingyu Cho*, KAIST, Jincheol Ha, KAIST, Jooyoung Lee, KAIST

Resistance of an S-box against algebraic attacks heavily depends on the number of linearly independent multivariate equations obtained from its algebraic representation. When it comes to $n$-bit S-boxes based on power functions over $\mathbb{F}_{2^n}$, Nawaz \textit{et al.} proposed an algorithm that computes the number of linearly independent biaffine and quadratic equations in $O(n^2)$ time. In this paper, we further generalize this work, proposing a new algorithm that computes the exact number of linearly independent equations of degree $D$ in $O(n^D)$ time. We also implement our algorithm for $D = 3$, and experimentally show that a power function having a smaller number of quadratic equations might not necessarily have a smaller number of cubic equations.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Algebraic attack, linearly independent equations, power functions, S-box

Algebra and Number Theory

⋅ 20th-F-10:10 − 11:20   Chair: Young-Hoon Kiem (Seoul National University)

⋅ 20th-F-10:10 − 10:45 Numerical local Langlands correspondence for $G_2$ (Ju-Lee Kim, Melissa Emory, Maria Fox)

김주리*((MIT)), Melissa Emory((Oklahoma State University)), Maria Fox((Oklahoma State University))
Ju-Lee Kim*, MIT, Melissa Emory, Oklahoma State University, Maria Fox, Oklahoma State University

We discuss numerical local Langlands correspondence for $G_2$.

2010 Mathematics Subject Classification: 22E50
Key Words and Phrases: Local Langlands correspondence

⋅ 20th-F-10:45 − 11:20 Kac--Moody Eisenstein series (Kyu-Hwan Lee)

이규환((University of Connecticut))
Kyu-Hwan Lee, University of Connecticut

The theory of Eisenstein series on finite-dimensional Lie groups has played an important role in Langlands program. In this talk, we will define Eisenstein series on the Lie groups of infinite dimensional Lie algebras (Kac--Moody algebras) and consider some basic properties of the series. This is a joint work with Carbone, Garland, Liu and Miller.

2010 Mathematics Subject Classification: 20G44, 11F70
Key Words and Phrases: Eisenstein series, Kac--Moody groups

⋅ 20th-F-11:30 − 13:25   Chair: Young-Hoon Kiem (Seoul National University)

⋅ 20th-F-11:30 − 12:05 On the two-dimensional Jacobian conjecture (Jacob Glidewell, William E. Hurst, Kyungyong Lee, Li Li, George D. Nasr)

Jacob Glidewell((University of Alabama)), William E. Hurst((University of Alabama)), 이경용*((University of Alabama)), Li Li ((Oakland University)), George D. Nasr((University of Oregon))
Jacob Glidewell, University of Alabama, William E. Hurst, University of Alabama, Kyungyong Lee*, University of Alabama, Li Li, Oakland University, George D. Nasr, University of Oregon

We introduce one of notorious open problems from Smale's list: the Jacobian conjecture. The Jacobian conjecture states that if the Jacobian of a polynomial map is a nonzero constant, then the map is bijective. The condition of the Jacobian being equal to a constant can be translated to a system of (too many) polynomial equations. For the two-dimensional case, an elementary but promising approach, which is motivated by rank 2 cluster algebras, is to find ALL solutions systematically.

2010 Mathematics Subject Classification: 14R15, 14R10, 13F20
Key Words and Phrases: Jacobian conjecture, polynomial automorphism

⋅ 20th-F-12:05 − 12:40 Derived category of moduli space of vector bundles on a curve (Kyoung-Seog Lee, Han-Bom Moon)

이경석((University of Miami)), 문한봄*((Fordham University))
Kyoung-Seog Lee, University of Miami, Han-Bom Moon*, Fordham University

The derived category of a smooth projective variety is an invariant expected to encode birational geometric information of the variety. One way to study its structure is to divide it into smaller building blocks, and the semiorthogonal decomposition provides a systematic approach. In this talk, I will explain the current status of the study of the derived category of moduli spaces of vector bundles on a curve in the framework of semiorthogonal decomposition. In particular, I will describe how one can embed the derived category of the symmetric product of the base curve into the derived category of the moduli space and an implication in the Fano visitor problem.

2010 Mathematics Subject Classification: 14D20, 14F08
Key Words and Phrases: Moduli space, vector bundle, derived category, semiorthogonal decomposition, fano visitor

⋅ 20th-F-12:50 − 13:25 Non-vanishing of symmetric cube automorphic L-functions (Junehyuk Jung)

정준혁((Brown University))
Junehyuk Jung, Brown University

Jeff Hoffstein, Min Lee, and I recently proved that there are infinitely many Maass forms over an imaginary quadratic field whose corresponding symmetric cube $L$-functions do not vanish at the critical point $s=\frac{1}{2}$. In this talk, I will explain how one can prove this using a theory of cubic theta function and a rather straightforward spectral theory techniques.

2010 Mathematics Subject Classification: 11M99
Key Words and Phrases: Automorphic L functions

⋅ 20th-F-14:35 − 15:10   Chair: Jae-Hoon Kwon (Seoul National University)

⋅ 20th-F-14:35 − 15:10 The rational torsion subgroups of modular Jacobian varieties (Hwajong Yoo)

유화종((서울대))
Hwajong Yoo, Seoul National University

The rational torsion subgroup of an abelian variety over $\mathbb{Q}$, the field of rational numbers, is an important arithmetic invariant of a given variety. A well-known theorem of Barry Mazur completes the classification of possible rational torsion subgroups of elliptic curves over $\mathbb{Q}$. If the dimensions of abelian varieties are bigger than $1$, not much is known. One favorable case is also studied by Mazur, the Jacobian variety of the modular curve $X_0(N)$ for a prime $N$. In that case, the rational torsion subgroup is equal to the cuspidal subgroup, which is known as Ogg's conjecture. There are several attempts to generalize this conjecture. In this talk, we discuss some recent progress towards such a question.

2010 Mathematics Subject Classification: 11G18, 14G05, 14G35
Key Words and Phrases: Rational torsion subgroup, rational cuspidal subgroup, generalized Ogg's conjecture

⋅ 20th-F-15:30 − 16:40   Chair: Jae-Hoon Kwon (Seoul National University)

⋅ 20th-F-15:30 − 16:05 Modularity of defect one Serre weights (Daniel Le, Bao V. Le Hung, Stefano Morra, Chol Park, Zicheng Qian)

Daniel Le((Purdue Univeristy)), Bao V. Le Hung((Northwestern University)), Stefano Morra ((University of Paris 8)), 박철*((UNIST)), Zicheng Qian((Morningside Center of Mathematics))
Daniel Le, Purdue University, Bao V. Le Hung, Northwestern University, Stefano Morra, University of Paris 8, Chol Park*, UNIST, Zicheng Qian, Morningside Center of Mathematics

The weight part of Serre's conjecture is an important first step towards mod-p Langlands program. In this talk, we show the modularity of defect one Serre weights by determining the structure of potentially crystalline deformation rings of colength one. This is a joint work with Le, Le Hung, Morra, and Qian.

2010 Mathematics Subject Classification: 11F80, 11F03, 11F70
Key Words and Phrases: Serre weights, shadow weights, colength one deformation rings, shapes of Kisin modules

⋅ 20th-F-16:05 − 16:40 Extended crystals and quantum affine algebras (Euiyong Park)

박의용((서울시립대))
Euiyong Park, University of Seoul

In this talk, we talk about extended crystals for quantum groups. The notion of extended crystals is introduced for explaining the categorical crystal structure on the module category of a quantum affine algebra. We discuss how the set of the isomorphism classes of simple modules over a quantum affine algebra has an extended crystal structure. We then explain the Braid group action on an extended crystal. This talk is partially based on a joint work with M. Kashiwara (arXiv: 2111.07255 and 2207.11644).

2010 Mathematics Subject Classification: 05E10, 05E18, 17B37
Key Words and Phrases: Extended crystals, quantum affine algebras

⋅ 20th-F-17:00 − 18:10   Chair: Jae-Hoon Kwon (Seoul National University)

⋅ 20th-F-17:00 − 17:35 Algebraic Montgomery-Yang problem and cascade conjecture (DongSeon Hwang)

황동선((기초과학연구원 복소기하학연구단))
DongSeon Hwang, IBS-CCG

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on $\mathbb{S}^5$ admits at most $3$ non-free orbits. Koll\'ar as well as Fintushel and Stern formulated its algebraic version by considering its orbit space. The algebraic version is verified in many cases by the joint work of JongHae Keum with the speaker. In this talk, motivated by an observation that every known example of the remaining case has a special birational behavior called a cascade, we establish algebraic Montgomery-Yang problem assuming the cascade conjecture, which claims that every rational Q-homology projective planes with quotient singularities having ample canonical divisor admits a cascade. We also discuss some of its variants.

2010 Mathematics Subject Classification: 14J17, 14J26, 14J45
Key Words and Phrases: Montgomery-Yang problem, rational homology projective plane, cascade

⋅ 20th-F-17:35 − 18:10 Generalized logarithmic sheaf and Torelli problem (Sukmoon Huh, Simone Marchesi, Joan Pons-Llopis, Jean Vall\`es)

허석문*((성균관대)), Simone Marchesi((University of Barcelona)), Joan Pons-Llopis((Poli\-tecnico di Torino)), Jean Vall\`es((Universit \'ee de Pau et des Pays de l'Adour))
Sukmoon Huh*, Sungkyunkwan University, Simone Marchesi, University of Barcelona, Joan Pons-Llopis, Politecnico di Torino, Jean Vall\`es, Universit \'ee de Pau et des Pays de l'Adour

A logarithmic sheaf is the sheaf of differential one-forms with logarithmic poles along a divisor on a smooth projective variety and it was originally introduced by P. Deligne to define the mixed Hodge structure on the complement of the divisor. There have been two main streams of study on this object, one is to study the Torelli-type problem and the other is to study its freeness. In this talk we suggest a new setting for the Torelli-type problem and give answers to a few cases. This is a joint work with S. Marchesi, J. Pons-Llopis and J. Vall\`es.

2010 Mathematics Subject Classification: 14F06, 14J60, 14C34
Key Words and Phrases: Logarithmic sheaf, Torelli problem
Analysis and Probability

⋅ 20th-F-10:10 − 11:20   Chair: Seick Kim (Yonsei University)

⋅ 20th-F-10:10 − 10:45 Optimal estimates for the conductivity problems (Hongjie Dong, Yanyan Li, Zhuolun Yang)

20th-F-10:10 − 10:45
Hongjie Dong*, Brown University, Yanyan Li, Rutgers University, Zhuolun Yang, Brown University

In the first part of the talk, I will present some recent results (jointly with Yanyan Li and Zhuolun Yang) about the insulated conductivity problem with closely spaced inclusions in a bounded domain in $R^n$. The gradient of solutions may blow up as the distance between inclusions approaches to 0. We obtained an optimal gradient estimate of solutions in terms of the distance. In the second part, I will discuss a work in progress (jointly with Zhuolun Yang) regarding optimal estimates for higher derivatives of solutions of the conductivity problem with closely located circular inclusions in 2D, when the relative conductivities of inclusions have different signs. This improves a recent result of Ji and Kang.

2010 Mathematics Subject Classification: 35J15, 35Q74, 74E30, 74G70, 78A48
Key Words and Phrases: Optimal gradient estimates, high contrast coefficients, insulated conductivity problem, degenerate elliptic equation, maximum principle

⋅ 20th-F-10:45 − 11:20 Hele-Shaw with drift: Free boundary regularity (Inwon Kim, Yuming Paul Zhang)

김인원*((UCLA)), Yuming Paul Zhang((Auburn University))
Inwon Kim*, UCLA, Yuming Paul Zhang, Auburn University

We consider a Hele-Shaw type flow, which present a pressure-driven evolution of patches by Darcy's law. The problem arises classically from fluid dynamics, and more recently in population dynamics as well as in tumor growth models, introducing various external terms such as sources and drifts into the flow. Our focus will be in the regularity of the patch boundary that evolves in time. Roughly speaking, our result states that the patch boundary is smooth, if it is sufficiently close to a smooth profile in a unit space-time neighborhood. In the talk we will introduce the problem, discuss the regularizing mechanism in the problem, and discuss our result as well as other classical results in the literature.

2010 Mathematics Subject Classification: 35b65
Key Words and Phrases: Hele-Shaw flow, free boundary regularity

⋅ 20th-F-11:30 − 12:40   Chair: Seick Kim (Yonsei University)

⋅ 20th-F-11:30 − 12:05 Spectral heat content on a class of fractal sets for subordinate killed Brownian motions (Hyunchul Park, Yimin Xiao)

박현철*((SUNY New Paltz)), Yimin Xiao((Michigan State University))
Hyunchul Park*, SUNY New Paltz, Yimin Xiao, Michigan State University

In this talk, we study the spectral heat content (SHC) for subordinate (time-changed) killed Brownian motions on an open set with fractal boundaries. Recently, there have been growing interests on SHC for jump processes, but most works are for domains with sufficiently smooth boundaries. In this talk, we will focus on studying a small time asymptotic behavior of SHC on domains with fractal boundaries when the underlying processes are subordinate killed Brownian motions via $\frac{\alpha}{2}$-stable subordinator. There are three different decay regimes depending on the stability index $\alpha$ and the interior Minkowski dimension of the boundary of the underlying set. The main ingredients are the renewal theorem and a recent result on SHC on smooth domains. This is a joint work with Yimin Xiao.

2010 Mathematics Subject Classification: 35K05, 28A80, 60G52
Key Words and Phrases: Spectral heat content, subordinate killed Brownian motions, renewal theorem

⋅ 20th-F-12:05 − 12:40 A direction in optimal transport and Brownian motion (Young-Heon Kim)

김영헌((The University of British Columbia))
Young-Heon Kim, The University of British Columbia

In this talk we consider a few recent results arising from applying optimal transport theory to the study of optimal stopping of Brownian motion. This includes an application to the supercooled Stefan problem about the free boundary of the formation of ice.

2010 Mathematics Subject Classification: 49Q22, 60G40, 35R35
Key Words and Phrases: Optimal transport, Brownian motion, free boundaries, Stefan problem, optimal stopping

⋅ 20th-F-14:00 − 15:10   Chair: Seick Kim (Yonsei University)

⋅ 20th-F-14:00 − 14:35 The norm attaining operator theory on Banach space (Sun Kwang Kim)

김선광((충북대))
Sun Kwang Kim, Chungbuk National University

The aim of this talk is to introduce the theory of norm attaining operators on Banach spaces. There is a long history on this topic with beautiful results, we will see selected ones. Among them, we see some descriptions of geometries of Banach space like reflexivity and uniform convexity in terms of norm attaining operators. Especially, the reflexivity is equivalent to that all the functionals attains their norm which is a result of R. James (1964), and the Radon-Nikod\'ym property can be characterized by the denseness of norm attaining operators which is a result of J. Bourgain (1977). We mainly follow the developments of this theory, and we present new topics at the end.

2010 Mathematics Subject Classification: 46B20
Key Words and Phrases: Banach space, norm attaining operator

⋅ 20th-F-14:35 − 15:10 Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations (Seongyeon Kim, Yoonjung Lee, Ihyeok Seo)

김성연((고등과학원)), 이윤정((부산대)), 서이혁*((성균관대))
Seongyeon Kim, KIAS, Yoonjung Lee, Pusan National University, Ihyeok Seo*, Sungkyunkwan University

In this talk I will introduce weighted Strichartz estimates for the Schrodinger flow and discuss the sharpness thereof. Then I will explain how useful these estimates can be in the study of inhomogeneous Hartree equations in the critical case. The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical Hartree model. In particular, the critical case was unsolved until recently.

2010 Mathematics Subject Classification: 35B45, 35Q55
Key Words and Phrases: Weighted Strichartz estimates, nonlinear Schrodinger equations

⋅ 20th-F-15:30 − 16:40   Chair: Kyeong-Hun Kim (Korea University)

⋅ 20th-F-15:30 − 16:05 Dirichlet Heat kernel estimates for Markov processes with singular kernels (Kyung-Youn Kim, Lidan Wang)

김경윤*((National ChungHsing University)), Lidan Wang((Nankai University))
Kyung-Youn Kim*, National ChungHsing University, Lidan Wang, Nankai University

We discuss heat kernels for non-local operators which are transition densities of Markov processes corresponding to the operators as an infinitesimal generators in probability theory. We consider anisotropic Markov processes $Z:=(Z_1, \ldots, Z_d)$ where each coordinates $Z_i$ are $d$--independent $1$--dimensional processes whose characteristic function satisfies the weakly scaling condition. We obtain the heat kernel estimates on $\mathbb R^d$ and $C^{1,1}$--open set $D\subset \mathbb R^d$. This is the joint work with Lidan Wang.

2010 Mathematics Subject Classification: 31B25, 60J50
Key Words and Phrases: Markov jump process, Dirichelt heat kernel, first exit time, transition density, anisotropic process

⋅ 20th-F-16:05 − 16:40 Partition functions of determinantal and Pfaffian Coulomb gases (Sung-Soo Byun, Nam-Gyu Kang, Seong-Mi Seo)

변성수((고등과학원)), 강남규((고등과학원)), 서성미*((충남대))
Sung-Soo Byun, KIAS, Nam-Gyu Kang, KIAS, Seong-Mi Seo*, Chungnam National University

In this talk, I will introduce the two-dimensional determinantal and Pfaffian Coulomb gas models and discuss the large $N$ expansion of their free energies. Under the influence of external potentials that are radially symmetric, some correction terms in the expansion depend on whether the droplet is an annulus or a disc. I will explain how to obtain these correction terms in the expansion. This is joint work with Sung-Soo Byun and Nam-Gyu Kang.

2010 Mathematics Subject Classification: 60B20, 60G55, 60K35
Key Words and Phrases: Determinantal Coulomb gas, Pfaffian Coulomb gas, partition functions, random normal matrices, planar symplectic ensembles

⋅ 20th-F-17:00 − 18:10   Chair: Kyeong-Hun Kim (Korea University)

⋅ 20th-F-17:00 − 17:35 $L_p$-theories to second-order partial differential equations with unbounded and degenerate leading coefficients: an application of It\^o's stochastic calculus (Ildoo Kim, Kyeong-hun Kim)

김일두*((고려대)), 김경훈((고려대))
Ildoo Kim*, Korea University, Kyeong-hun Kim, Korea University

In this talk, we present a short history of $L_p$-theories to second-order partial differential equations and introduce recent developments handling degenerate and unbounded leading coefficients in weighted spaces, which could be obtained by applying probability theories.

2010 Mathematics Subject Classification: 60H15, 35R60, 35B65
Key Words and Phrases: Degenerate equations, unbounded coefficients, maximal $L_p$-regularity theory

⋅ 20th-F-17:35 − 18:10 Vorticity convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class (Joonhyun La, Chanwoo Kim)

라준현*((Imperial College London)), 김찬우((University of Wisconsin-Madison))
Joonhyun La*, Imperial College London, Chanwoo Kim, University of Wisconsin-Madison

It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations \begin{equation*} \mathrm{St} \partial_t F + v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) \end{equation*} toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded \begin{equation*} \begin{split} \partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\ \ \text{div }u =0. \end{split} \end{equation*} We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ($\omega \in L^{\mathfrak{p}}$ for any fixed $1 \leq {\mathfrak{p}}< \infty$). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in $L^{\mathfrak{p}}$ as ${\mathfrak{p}}\rightarrow \infty$ (localized Yudovich class).

2010 Mathematics Subject Classification: 35Q20, 35Q31
Key Words and Phrases: Hydrodynamic limit, Boltzmann equation, Euler equation
Geometry and Topology

⋅ 20th-F-10:10 − 11:20   Chair: Cheol-Hyun Cho (Seoul National University)

⋅ 20th-F-10:10 − 10:45 Homology cobordism and Heegaard Floer homology (Irving Dai, Kristen Hendricks, Jennifer Hom, Matthew Stoffregen, Linh Truong, Ian Zemke)

20th-F-10:10 − 10:45
Irving Dai, Stanford, Kristen Hendricks, Rutgers, Jennifer Hom*, Georgia Tech, Matthew Stoffregen, Michigan State University, Linh Truong, University of Michigan, Ian Zemke, Princeton

Under the operation of connected sum, the set of three-manifolds form a monoid. Modulo an equivalence relation called homology cobordism, this monoid (of homology spheres) becomes a group. What is the structure of this group? What families of three-manifolds generate, or don't generate, this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with various subsets of I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.

2010 Mathematics Subject Classification: 57Q60, 57R58
Key Words and Phrases: Homology cobordism, Heegaard Floer homology

⋅ 20th-F-10:45 − 11:20 The symplectic derivation Lie algebra and the mapping class group of a surface (Takuya Sakasai)

20th-F-10:45 − 11:20
Takuya Sakasai, The University of Tokyo

The symplectic derivation Lie algebra of the free Lie algebra plays an important role in the study of the mapping class group of a surface. We discuss applications of our computational results of this derivation algebra. In particular, we give an explicit description of the abelianization map of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

2010 Mathematics Subject Classification: 17B54, 57M27
Key Words and Phrases: Symplectic derivation, Johnson kernel, homology 3-sphere

⋅ 20th-F-11:30 − 12:40   Chair: Cheol-Hyun Cho (Seoul National University)

⋅ 20th-F-11:30 − 12:05 Revisit the theory of laminar groups (Hyungryul Baik, KyeongRo Kim)

백형렬*((카이스트)), 김경로((서울대학교))
Hyungryul Baik*, KAIST, KyeongRo Kim, Seoul National University

I will give a brief introduction to laminar groups which are groups of orientation-preserving homeomorphisms of the circle admitting invariant laminations. The term was coined by Calegari and the study of laminar groups was motivated by work of Thurston and Calegari-Dunfield. We present old and new results on laminar groups which tell us when a given laminar group is either Fuchsian or Kleinian.

2010 Mathematics Subject Classification: 57K32, 57M60, 20F65, 20F67
Key Words and Phrases: Lamination, 3-manfiold, taut foliation, veering triangulation, Fuchsian group, Kleinian group

⋅ 20th-F-12:05 − 12:40 On the second homology of planar graph braid groups (Byung Hee An, Ben Knudsen)

안병희*((경북대)), Ben Knudsen((Northeastern University))
Byung Hee An*, Kyungpook National University, Ben Knudsen, Northeastern University

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

2010 Mathematics Subject Classification: 55R80
Key Words and Phrases: Configuration spaces of graphs

⋅ 20th-F-14:00 − 15:10   Chair: Beomjun Choi (POSTECH)

⋅ 20th-F-14:00 − 14:35 Fukaya and Seidel's programs on mirror symmetry (Kwokwai Chan)

20th-F-14:00 − 14:35
Kwokwai Chan, The Chinese University of Hong Kong

Fukaya's program on SYZ mirror symmetry combined with Seidel's insight on homological mirror symmetry explains mirror symmetry as a topological duality at the large structure limits, followed by suitable deformations. I will explain this story and joint works with Conan Leung and Ziming Ma where we proved a modified version of one of Fukaya's main conjectures.

2010 Mathematics Subject Classification: 53D37
Key Words and Phrases: Mirror symmetry, deformation theory, Calabi-Yau manifold, scattering diagram

⋅ 20th-F-14:35 − 15:10 A nonexistence result for wing-like mean curvature flows in $\mathbb R^4$ (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits)

최경수*((고등과학원)), Robert Haslhofer((University of Toronto)), Or Hershkovits((Hebrew University of Jerusalem))
Kyeongsu Choi*, KIAS, Robert Haslhofer, University of Toronto, Or Hershkovits, Hebrew University of Jerusalem

Convex ancient mean curvature flows are singularity models of noncollasped mean curvature flow. In $\mathbb R^3$. the convex ancient flow converges to round cylinders as time goes back, and therefore the blow-down of each time slice of a complete convex ancient flow must be a half line. However, in $\mathbb R^4$, the convergence to cylinders can not exclude that possibility that some blow-downs are flat cones. In this talk, we discuss how to make a use of stability to exclude the flat cone blow-down scenarios.

2010 Mathematics Subject Classification: 53E10
Key Words and Phrases: Mean curvature flow, stability, Liouville theorem

⋅ 20th-F-15:30 − 16:40   Chair: Keomkyo Seo (Sookmyung Women's University)

⋅ 20th-F-15:30 − 16:05 Min-max constructions of free boundary minimal surfaces (Martin Man-Chun Li)

20th-F-15:30 − 16:05
Martin Man-Chun Li, The Chinese University of Hong Kong

Min-max theory is a powerful tool to construct unstable critical points to a general functional via mountain-pass arguments. It has been developed and applied to a wide range of areas in differential geometry and partial differential equations. In the past decade, we have witnessed tremendous success of the Almgren-Pitts min-max theory for minimal surfaces which has led to the resolution of several longstanding conjectures in geometry. This theory and its variants have recently been actively studied and further extended to many interesting new directions. In this talk, I will give a survey of some recent development on the min-max constructions of free boundary minimal surfaces. We will describe several different approaches and give a comparison between each of their own strengths and weaknesses. Along the way, we will mention some open problems in this research area. These works are substantially supported by grants from the Hong Kong Research Grant Council, and the NSFC Excellent Young Scientist Fund.

2010 Mathematics Subject Classification: 53C42, 49Q20, 58E12
Key Words and Phrases: Min-max theory, free boundary minimal surfaces, geometric measure theory, phase transition

⋅ 20th-F-16:05 − 16:40 A canonical diffeomorphism between closed manifolds with Ricci curvature bounded below (Shouhei Honda)

20th-F-16:05 − 16:40
Shouhei Honda, Tohoku University

In this talk, we prove that for any closed $n$-dimensional Riemannian manifold $N$ and any real number $K$, there exists $\epsilon>0$ such that if a closed $n$-dimensional Riemannian manifold $M$ with Ricci curvature bounded below by $K$ is $\epsilon$-Gromov-Hausdorff close to $N$, then there exists a (canonical) diffeomorphism $f$ from $M$ to $N$ with the sharp Lipschitz-H\"older estimate on $f$. This is a joint work with Yuanlin Peng (Tohoku University).

2010 Mathematics Subject Classification: 53C23
Key Words and Phrases: Ricci curvature

⋅ 20th-F-17:00 − 18:10   Chair: Juncheol Pyo (Pusan National University)

⋅ 20th-F-17:00 − 17:35 Heintze-Karcher’s inequality and Alexandrov’s theorem for capillary hypersurfaces (Xiaohan Jia, Guofang Wang, Chao Xia, Xuwen Zhang)

20th-F-17:00 − 17:35
Xiaohan Jia, Xiamen University, Guofang Wang, Albert-Ludwigs-University Freiburg, Chao Xia*, Xiamen University, Xuwen Zhang, Xiamen University

Heintze-Karcher’s inequality is an interesting geometric inequality which can be used to prove Alexandrov’s theorem on embedded closed CMC hypersurfaces. In this talk, we prove a Heintze-Karcher-type inequality for hypersurfaces with boundary in the half-space and a wedge. As application, we show Alexandrov-type theorem for embedded CMC capillary hypersurfaces. If time permits, we discuss the generalization to anisotropic capillary setting.

2010 Mathematics Subject Classification: 53C24, 35J25, 53C21
Key Words and Phrases: Heintze-Karcher's inequality, capillary hypersurface, CMC hypersurface, Alexandrov's theorem

⋅ 20th-F-17:35 − 18:10 Minimal annuli with free boundary in the unit ball (Pablo Mira)

20th-F-17:35 − 18:10
Pablo Mira, Polytechnic University of Cartagena

In this talk we will construct a family of compact free boundary minimal annuli immersed in the unit ball of Euclidean 3-space, the first such examples other than the critical catenoid. Their existence answers in the negative a problem of the theory that dates back to Nitsche in 1985. We will explain the geometry of these examples and discuss several open problems. Joint work with Isabel Fernandez and Laurent Hauswirth.

2010 Mathematics Subject Classification: 53A10
Key Words and Phrases: Minimal surfaces, free boundary, critical catenoid
Applied Mathematics

⋅ 20th-F-10:10 − 11:20   Chair: Jinsu Kim (POSTECH)

⋅ 20th-F-10:10 − 10:45 Comparison theorems for stochastic chemical reaction networks (Felipe Campos, Simone Bruno, Yi Fu, Domitilla Del Vecchio, Ruth Williams)

20th-F-10:10 − 10:45
Felipe Campos, University of California, San Diego, Simone Bruno, Massachusetts Institute of Technology, Yi Fu, University of California, San Diego, Domitilla Del Vecchio, Massachusetts Institute of Technology, Ruth Williams*, University of California, San Diego

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. We illustrate our results with applications to models of chromatin regulation and enzymatic kinetics.

2010 Mathematics Subject Classification: 60E15, 60J27, 92C40, 92C42
Key Words and Phrases: Stochastic comparison, biochemical reaction networks, monotonicity, uniformization

⋅ 20th-F-10:45 − 11:20 The relationship between deterministic and stochastic quasi-steady-state approximations (Jae Kyoung Kim)

김재경((카이스트 수리과학과 $/$ IBS 의생명수학그룹))
Jae Kyoung Kim, KAIST / IBS Biomedical Mathematics

The quasi steady-state approximation (QSSA) is frequently used to reduce deterministic models of biochemical networks. The resulting equations provide a simplified description of the network in terms of non-elementary reaction functions (e.g. Hill functions). Such deterministic reductions are frequently a basis for heuristic stochastic models in which non-elementary reaction functions are used as propensities of Gillespie algorithm. Despite the popularity of this heuristic stochastic simulations, it remains unclear when such stochastic reductions are valid. In this talk, I will present conditions under which the stochastic models with the non-elementary propensity functions accurately approximate the full stochastic models. If the validity condition is satisfied, we can perform accurate and computationally inexpensive stochastic simulation without converting the non-elementary functions to the elementary functions (e.g. mass action kinetics).

2010 Mathematics Subject Classification: 92-10
Key Words and Phrases: QSSA, stochastic, Gillespie algorithm, non-elementary function

⋅ 20th-F-11:30 − 12:40   Chair: Jae-Hun Jung (POSTECH)

⋅ 20th-F-11:30 − 12:05 The Gromov-Hausdorff distance between spheres (Facundo Mémoli, Sunhyuk Lim, Zane Smith)

20th-F-11:30 − 12:05
Facundo Mémoli*, The Ohio State University, Sunhyuk Lim, Max Planck Institute. Leipzig, Zane Smith, University of Minnesota

Distances such as the Gromov-Hausdorff distance and its Optimal Transport variants are nowadays routinely invoked in applications related to data classification. Interestingly, the precise value of these distances on pairs of canonical shapes is known only in very limited cases. In this talk, I will describe lower bounds for the Gromov-Hausdorff distance between spheres (endowed with their geodesic distances) which we prove to be tight in some cases via the construction of optimal correspondences. These lower bounds arise from a certain version of the Borsuk-Ulam theorem for discontinuous functions.

2010 Mathematics Subject Classification: 51F99, 53C23
Key Words and Phrases: Gromov-Hausdorff distance, Persistent homology

⋅ 20th-F-12:05 − 12:40 Geometric generation of Aesthetic shapes by integrable systems (Kenji Kajiwara)

20th-F-12:05 − 12:40
Kenji Kajiwara, Kyushu University

We first consider the log-aesthetic curves (LAC), which is a family of planar curves developed and used in the area of industrial design as the shape elements with built-in aesthetic nature. The LAC has been originally proposed by extracting the common properties from thousands of plane curves that car designers regard as aesthetic, which includes various spirals as special cases, such as the circle involute, the clothoid, the Nielsen's spiral, and the logarithmic spiral. We present a new mathematical framework of the LAC on the theory of integrable systems and the similarity geometry, a Klein geometry associated with the similarity transformation group. Using this framework, we show that the similarity curvature is governed by the stationary Burgers equation and that the LAC is characterized by the variational principle. Namely, the LAC can be regarded as the similarity geometry analogue of the celebrated Euler's elasticae in the Euclidean geometry. Based on this result, we present some generalizations of LAC: (i) integrable discretization, (ii) space curves, and (iii) surfaces. We expect that those curves and surfaces may be useful for generating “aesthetic shapes”.

2010 Mathematics Subject Classification: 37K10, 39A36, 53A04, 53A05, 53A35, 53A70
Key Words and Phrases: Elastica, log-aesthetic curve, similarity geometry, Burgers equation, mKdV equation, sine-Gordon equation, variational formulation

⋅ 20th-F-14:00 − 15:10   Chair: Woocheol Choi (Sungkyunkwan University)

⋅ 20th-F-14:00 − 14:35 Continuous-time analysis of AGM via conservation laws in dilated coordinate systems (Jaewook J. Suh, Gyumin Roh, Ernest K. Ryu)

서재욱((서울대)), 노규민((서울대)), 류경석*((서울대))
Jaewook J. Suh, Seoul National University, Gyumin Roh, Seoul National University, Ernest K. Ryu*, Seoul National University

We analyze continuous-time models of accelerated gradient methods through deriving conservation laws in dilated coordinate systems. Namely, instead of analyzing the dynamics of $X(t)$, we analyze the dynamics of $W(t)=t^\alpha(X(t)-X_c)$ for some $\alpha$ and $X_c$ and derive a conserved quantity, analogous to physical energy, in this dilated coordinate system. Through this methodology, we recover many known continuous-time analyses in a streamlined manner and obtain novel continuous-time analyses for OGM-G, an acceleration mechanism for efficiently reducing gradient magnitude that is distinct from that of Nesterov. Finally, we show that a semi-second-order symplectic Euler discretization in the dilated coordinate system leads to an $\mathcal{O}(1/k^2)$ rate on the standard setup of smooth convex minimization, without any further assumptions such as infinite differentiability.

2010 Mathematics Subject Classification: 90C25
Key Words and Phrases: Optimization, machine learning

⋅ 20th-F-14:35 − 15:10 Semi-anchored multi-step gradient method for nonconvex-nonconcave minimax optimization (Sucheol Lee, Donghwan Kim)

이수철((카이스트)), 김동환*((카이스트))
Sucheol Lee, KAIST, Donghwan Kim*, KAIST

Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this work, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem The resulting method, named SA-MGDA, is built upon the Bregman proximal point method.

2010 Mathematics Subject Classification: 90C47, 90C26, 65K05, 47J25, 68T05
Key Words and Phrases: Minimax problem, nonconvex-nonconcave problem, multi-step gradient descent ascent method, Bregman proximal point method

⋅ 20th-F-15:30 − 16:40   Chair: Sangwoon Yun (Sungkyunkwan University)

⋅ 20th-F-15:30 − 16:05 A two-phase proximal augmented Lagrangian method for high dimensional convex quadratic programming problems (Ling Liang, Xudong Li, Defeng Sun, Kim-Chuan Toh)

20th-F-15:30 − 16:05
Ling Liang, National University of Singapore, Xudong Li, Fudan University, Defeng Sun, Hong Kong Polytechnic University, Kim-Chuan Toh*, National University of Singapore

We aim to solve high dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality and inequality constraints. In order to solve the targeted problems to a desired accuracy efficiently, we develop a two-phase proximal augmented Lagrangian method, with Phase I to generate a reasonably good initial point to warm start Phase II to obtain an accurate solution efficiently. More specifically, in Phase I, based on the recently developed symmetric Gauss-Seidel (sGS) decomposition technique, we design a novel sGS based semi-proximal augmented Lagrangian method for the purpose of finding a solution of low to medium accuracy. Then, in Phase II, a proximal augmented Lagrangian algorithm is proposed to obtain a more accurate solution efficiently. Extensive numerical results evaluating the performance of our proposed algorithm against the highly optimized commercial solver Gurobi and the open source solver OSQP are presented to demonstrate the high efficiency and robustness of our proposed algorithm for solving various classes of large-scale convex QP problems.

2010 Mathematics Subject Classification: 90C06, 90C25, 90C90
Key Words and Phrases: Convex quadratic programming, augmented Lagrangian method

⋅ 20th-F-16:05 − 16:40 First order methods for the decentralized optimization problems (Woocheol Choi)

최우철((성균관대))
Woocheol Choi, Sungkyunkwan University

Decentralized optimization problems have received a lot of interest from various researchers since the problems arise in diverse areas containing machine learning, Signal processing, and distributed control. It is important to design a suitable decentralized algorithm for the problems considering various restrictions behind the problems. I will introduce gradient descent type algorithms for the decentralized problems and discuss their convergence properties.

2010 Mathematics Subject Classification: 90C25
Key Words and Phrases: Decentralized optimization, first-order optimization

⋅ 20th-F-17:00 − 18:10   Chair: Jae Kyoung Kim (KAIST)

⋅ 20th-F-17:00 − 17:35 Revealing regulatory rules of the DNA dynamics in a living cell using a continuous-time Markov chain (Jinsu Kim)

김진수((포항공대))
Jinsu Kim, POSTECH

Due to high spatial complexity, the dynamical behavior of DNA strands in a living cell has not been experimentally explored as much as the case of outside cells. To resolve the experimental limitations, we describe the random unwrapping and rewrapping behavior of DNA strands around nucleosome with a continuous-time Markov chain in order to catch the key regulatory rules governing it. We examine first passage times of the Markov chain as a function of protein binding sites in light of the well-known fact that protein binding at DNA strands can drastically influence the rate of rewrapping. Non-linearity of the first passage times allows us to find cooperativity of the DNA dynamics: the more DNA that has been unwrapped, the easier it is to continue wrapping.

2010 Mathematics Subject Classification: 92B05, 60J27
Key Words and Phrases: Markoch chain, first passage time, DNA, cooperativity

⋅ 20th-F-17:35 − 18:10 Robust perfect adaptation in systems and synthetic biology (Ankit Gupta, Mustafa Khammash)

20th-F-17:35 − 18:10
Ankit Gupta, ETH Zurich, Mustafa Khammash*, ETH Zurich

Adaptation is a recurring theme in biology. It allows a living system to survive and thrive in the face of unpredictable environments by maintaining key physiological variables at their desired levels through tight regulation. When one such variable is maintained at a certain value at the steady-state in spite of perturbations to a single input this property is called robust perfect adaptation (RPA). Adaptation is achieved in natural circuits by using integral control, a negative feedback strategy that performs mathematical integration to achieve structurally robust regulation. Here we solve the fundamental problem of maximal robust perfect adaptation (maxRPA), where for a designated output variable RPA is achieved in the presence of perturbations in almost all network parameters. We start by showing that the maxRPA property imposes specific structural constraints on the network. We then demonstrate that these constraints are characterized by simple linear-algebraic stoichiometric conditions. We use our results to obtain a novel Internal Model Principle (IMP) for biomolecular maxRPA networks, similar to the well-known IMP of control theory. Our results elucidate the universal requirements for maxRPA in biological network and present a novel foundation for studying adaptation in general biomolecular networks, with implications for both systems and synthetic biology.

2010 Mathematics Subject Classification: 92B05, 92-08
Key Words and Phrases: Adaptation, chemical reaction networks, control theory, mathematical biology
Discrete Mathematics and Mathematics of Computer Science

⋅ 20th-F-10:10 − 11:20   Chair: Jooyoung Lee (KAIST)

⋅ 20th-F-10:10 − 10:45 Secure sampling with sublinear communication (Seung Geol Choi)

최승걸((US Naval Academy))
Seung Geol Choi, US Naval Academy

Random sampling from specified distributions is an important tool with wide applications for analysis of large-scale data. In this paper we study how to randomly sample when the distribution is partitioned among two parties' private inputs. Of course, a trivial solution is to have one party send a (possibly encrypted) description of its weights to the other party who can then sample over the entire distribution (possibly using homomorphic encryption). However, this approach requires communication that is linear in the input size which is prohibitively expensive in many settings. In this paper, we investigate secure 2-party sampling with \emph{sublinear communication} for many standard distributions. We develop protocols for L1 and L2 sampling. Additionally, we investigate the feasibility of sublinear product sampling, showing impossibility for the general problem and showing a protocol for a restricted case of the problem. We additionally show how such product sampling can be used to instantiate a sublinear communication 2-party exponential mechanism for differentially-private data release.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Cryptography, secure sampling, sublinear computation

⋅ 20th-F-10:45 − 11:20 The effort and contribution of the crypto-community for NIST lightweight cryptography project (Donghoon Chang)

장동훈((IIIT-Delhi))
Donghoon Chang, IIIT-Delhi

In 2018, NIST (National Institute of Standards and Technology, USA) has initiated Lightweight Cryptography (LWC) project to solicit, evaluate, and standardize lightweight cryptographic algorithms focusing on Authenticated Encryption and Hash function that are suitable for use in constrained environments where the performance of current NIST cryptographic standards is not acceptable. In this talk, I would like to share the effort and contribution that the crypto-community around the world has made till now for the LWC project.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Lightweight cryptography, authenticated encryption, Hash function

⋅ 20th-F-11:30 − 12:40   Chair: Dongsu Kim (KAIST)

⋅ 20th-F-11:30 − 12:05 Combinatorial description for the Hall-Littlewood expansion of unicellular LLT and chromatic quasisymmetric polynomials (Meesue Yoo, Seung Jin Lee)

류미수*((충북대)), 이승진((서울대))
Meesue Yoo*, Chungbuk National University, Seung Jin Lee, Seoul National University

In this work, we obtain a Hall-Littlewood expansion of the chromatic quasisymmetric functions by using a Dyck path model and linked rook placements. By using the Carlsson-Mellit relation between the chromatic quasisymmetric functions and the unicellular LLT polynomials, this combinatorial description for the Hall-Littlewood coefficients of the chromatic quasisymmetric functions also gives the coefficients of the unicellular LLT polynomials expanded in terms of the modified transformed Hall-Littlewood polynomials. This is joint work with Seung Jin Lee.

2010 Mathematics Subject Classification: 05E05
Key Words and Phrases: Chromatic quasisymmetric polynomial, unicellular LLT polynomial, Hall-Littlewood polynomial, rook polynomial

⋅ 20th-F-12:05 − 12:40 Affine Gordon-Bender-Knuth identities for cylindric Schur functions (JiSun Huh, Jang Soo Kim, Christian Krattenthaler, Soichi Okada)

허지선((아주대)), 김장수*((성균관대)), Christian Krattenthaler((University of Vienna)), \linebreak Soichi Okada((University of Minnesota))
JiSun Huh, Ajou University, Jang Soo Kim*, Sungkyunkwan University, Christian Krattenthaler, University of Vienna, Soichi Okada, University of Minnesota

The Gordon-Bender-Knuth identities are determinant formulas for the sum of Schur functions of partitions with bounded height, which have interesting combinatorial consequences such as connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this talk we give an affine analog of the Gordon-Bender-Knuth identities, which are determinant formulas for the sum of cylindric Schur functions. We also consider combinatorial aspects of these identities. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and r-noncrossing and s-nonnesting matchings. This is joint work with JiSun Huh, Christian Krattenthaler, and Soichi Okada.

2010 Mathematics Subject Classification: 05E05, 05A15, 05A19
Key Words and Phrases: Cylindric tableau, Schur function, set partition

⋅ 20th-F-14:00 − 15:10   Chair: Seog-Jin Kim (Konkuk University)

⋅ 20th-F-14:00 − 14:35 Building the hierarchy of graph classes (Sang-il Oum)

엄상일((기초과학연구원 이산수학그룹~/~카이스트))
Sang-il Oum, IBS Discrete Mathematics Group / KAIST

We will give a survey on the classification of graph classes in terms of the transductions in monadic second-order logic. Blumensath and Courcelle (2010) characterized that every class of graphs is equivalent by transductions of the monadic second-order logic of the second kind to one of the following: class of all trees of height $n$ for an integer $n$, class of all trees, class of all paths, and class of all grids. They conjectured that there is a similar linear hierarchy of graph classes in terms of the monadic second-order logic of the first kind. We will discuss how a recent theorem of the speaker with O-joung Kwon, Rose McCarty, and Paul Wollan (2021) on the vertex-minor obstruction for shrub-depth and a theorem of the speaker with Bruno Courcelle (2007) on graphs of large rank-width and logical expression of vertex-minors solve some subproblems of their conjecture.

2010 Mathematics Subject Classification: 05C75
Key Words and Phrases: Graph class, monadic second-order logic, vertex-minor

⋅ 20th-F-14:35 − 15:10 Extremal results on 4-cycles (Jie Ma, Tianchi Yang)

20th-F-14:35 − 15:10
Jie Ma*, University of Science and Technology of China, Tianchi Yang, National University of Singapore

The study of the 4-cycle has an important enlightening effect on the development of Turan type problems, especially the degenerate cases. In this talk, we focus on two conjectures about 4-cycles: a conjecture of Erdos on the upper bound of the Turan number $ex(n,C_4)$, and a conjecture of Erdos-Simonovits on the supersaturation problem of 4-cycles. Joint with Tianchi Yang.

2010 Mathematics Subject Classification: 05C35, 05D99
Key Words and Phrases: Turan number, 4-cycle, supersaturation, Erdos-Simonovits conjecture

⋅ 20th-F-15:30 − 16:05   Chair: Seog-Jin Kim (Konkuk University)

⋅ 20th-F-15:30 − 16:05 Hypertrees in Steiner triple systems (Seonghyuk Im, Jaehoon Kim, Joonkyung Lee, Abhishek Methuku)

임성혁((카이스트)), 김재훈*((카이스트)), 이준경((한양대)), Abhishek Methuku((University of Birmingham))
Seonghyuk Im, KAIST, Jaehoon Kim*, KAIST, Joonkyung Lee, Hanyang University, Abhishek Methuku, University of Birmingham

A $3$-uniform hypertree is a linear $3$-graph where every pair of vertices has unique Berge-path between them. Elliott and R\"odl conjectured that an $n$-vertex Steiner triple system contains all possible $(1-o(1))n$-vertex hypertrees. We prove this conjecture.

2010 Mathematics Subject Classification: 05C65
Key Words and Phrases: Hypergraph, trees, Steiner triple system

⋅ 20th-F-16:05 − 16:40   Chair: Joohee Lee (Sungshin Women's University)

⋅ 20th-F-16:05 − 16:40 How to meet ternary LWE keys on Babai's nearest plane (Minki Hhan, Jiseung Kim, Changmin Lee, Yongha Son)

한민기((고등과학원)), 김지승*((전북대)), 이창민((고등과학원)), 손용하((삼성 SDS))
Minki Hhan, KIAS, Jiseung Kim*, Jeonbuk National University, Changmin Lee, KIAS, Yongha Son, Samsung SDS

A cryptographic primitive based on the Learning With Errors (LWE) problem with its variants is a promising candidate for the efficient quantum-resistant public key cryptosystem. The recent schemes use the LWE problem with a small-norm or sparse secret key for better efficiency. Such constraints, however, lead to more tailor-made attacks and thus are a trade-off between efficiency and security. Improving the algorithm for the LWE problem with the constraints thus has a significant consequence on the concrete security of schemes. In this talk, we will present a new hybrid attack on the LWE problem. This new attack combines the primal lattice attack and an improved MitM attack called Meet-LWE, answering an open problem posed by May [Crypto'21]. According to our estimation, the new hybrid attack performs better than the previous attacks for the LWE problems with a sparse ternary secret key, which plays the significant role for the efficiency of fully homomorphic encryption schemes. In terms of the technical part, we introduce a notion of matrix modulus reductions based on the careful analysis of Babai's nearest plane algorithm. The lattice reduction part of primal hybrid algorithms can be translated into the reduction from the LWE problem to the LWE-like matrix modulus equations. By generalizing Meet-LWE for the matrix modulus equation based on the Gram-Schmidt basis and the locality sensitive hashing, we obtain the new hybrid attack of lattice reduction and Meet-LWE.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: LWE, hybrid attacks, meet-in-the-middle algorithms

⋅ 20th-F-17:00 − 18:10   Chair: Joohee Lee (Sungshin Women's University)

⋅ 20th-F-17:00 − 17:35 Reducing the overhead of approximate homomorphic encryption (Seongkwang Kim)

김성광((삼성SDS))
Seongkwang Kim, Samsung SDS

Homomorphic encryption (HE) is a promising cryptographic primitive that enables computation over encrypted data, with a variety of applications including medical, genomic, and financial tasks. In Asiacrypt 2017, Cheon et al. proposed the CKKS scheme to efficiently support approximate computation over encrypted data of real numbers. HE schemes including CKKS, nevertheless, still suffer from slow encryption speed and large ciphertext expansion compared to symmetric cryptography. To address the issue of the ciphertext expansion and the client-side computational overhead, in 2011, Naehrig et al. proposed a hybrid framework, also called a transciphering framework. A transciphering framework converts a symmetric ciphertext into a homomorphic ciphertext on the server-side, reducing computational and communication overload on the client-side. In this talk, we introduce a transciphering framework for approximate homomorphic encryption and two families of HE-friendly ciphers suited to approximate computation.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Homomorphic encryption, transciphering framework, HE-friendly cipher

⋅ 20th-F-17:35 − 18:10 On the bit security of cryptographic primitives: Towards the definition from THE BOOK (Keewoo Lee)

이기우((서울대))
Keewoo Lee, Seoul National University

Bit security is a central concept in cryptography, which provides a simple but useful measure for the level of security. However, despite its importance, we still do not have a well-accepted formal definition of bit security. In this talk, we introduce the concept of bit security and a few hiccups of the conventional definition. Then, we review more recent definitions by Micciancio-Walter (Eurocrypt 2018) and Watanabe-Yasunaga (Asiacrypt 2021). Also, we discuss a new definition by the speaker, which (i) captures both search and decision primitives in a single framework like Micciancio-Walter and (ii) has a firm operational meaning like Watanabe-Yasunaga.

2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Bit security, security definitions, Hellinger distance

Trends in Number Theory

⋅ 19th-A-09:00 − 10:30 Chair: Hwajong Yoo (Seoul National University)

⋅ 19th-A-09:00 − 09:25 $p$-adic properties for Taylor coefficients of half-integral weight modular forms on $\Gamma_1(4)$ (Jigu Kim, Yoonjin Lee)

김지구*((이화여대)), 이윤진((이화여대))
Jigu Kim*, Ewha Womans University, Yoonjin Lee, Ewha Womans University

For an integer $m\ge 2$ and a prime $p\equiv 3 \pmod4$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ and CM points. In this talk, we show an affirmative answer to it for primes $p\ge5$. This is a joint work with Yoonjin Lee.

2010 Mathematics Subject Classification: 11F33, 11F37
Key Words and Phrases: Modular forms, Taylor coefficients, congruences

⋅ 19th-A-09:30 − 09:55 Cosmic Grothendieck-Galois theory (Jaehyeok Lee, Jae-Suk Park)

이재혁*((포항공대)), 박재석((포항공대))
Jaehyeok Lee*, POSTECH, Jae-Suk Park, POSTECH

Grothendieck unified the Galois theory of field extensions and the Galois theory of covering spaces by introducing the concept of Galois categories, which are categories equivalent to the category $G$-$\mathit{FSet}$ of finite left $G$-sets for some profinite group $G$. The first part of this talk is to introduce cosmic Galois categories, which are enriched categories with analogous properties. The base category for the enrichment is called a cosmos, which is a locally presentable closed symmetric monoidal category. The underlying mechanism of cosmic Galois theory is the yoga of enriched six-functor formalism in Wirthmuller context. Grothendieck suggested, Rivano developed and Deligne completed the concept of Tannakian categories $T$ over a field $k$, whose collection $\mathit{Fib}(T)$ of fiber functors forms an affine gerbe on the fpqc site of affine schemes over $k$, and that the given category $T$ is equivalent to the category of finite representations of the gerbe $\mathit{Fib}(T)$. The second part of this talk is to introduce cosmic Grothendieck categories, which are categories enriched in a cosmos with analogous properties. The underlying mechanism of cosmic Grothendieck theory is the yoga of enriched six-functor formalism in Grothendieck context.

2010 Mathematics Subject Classification: 11R32
Key Words and Phrases: Galois category, Tannakian category, cosmos, enriched category, six-functor formalism

⋅ 19th-A-10:00 − 10:30 Drinfeld level structures via prismatic Dieudonne theory (Gyujin Oh)

오규진((Columbia University))
Gyujin Oh, Columbia University

Despite the great advances in the subject of integral models for Shimura varieties, the construction of the integral model with a level structure deeper than the parahoric level has been limited. The major obstacle is that the definition of the Drinfeld level structure by Katz—Mazur is ill-behaved for p-divisible groups of dimension >1. We suggest a solution for certain deep p-levels using the prismatic Dieudonne theory recently developed by Anschutz—Le Bras.

2010 Mathematics Subject Classification: 11G18
Key Words and Phrases: Shimura varieties, prismatic Dieudonne theory

⋅ 19th-B-10:50 − 12:20 Chair: Myungjun Yu (Yonsei University)

⋅ 19th-B-10:50 − 11:15 Explicit constructions for Bhargava's ``Higher Composition Laws" (Seok Hyeong Lee)

이석형((서울대))
Seok Hyeong Lee, Seoul National University

Bhargava's ``Higher Composition Laws" give explicit one-to-one correspondence between rings of low ranks and certain integral forms. Generalizing Wood's extension of cubic and quartic ring parametrizations, we give a general algorithm of constructing rings out of well-posed integral forms which can be applied to all cases of Higher Composition Laws.

2010 Mathematics Subject Classification: 11R04, 13F20
Key Words and Phrases: Rings of low ranks, higher composition laws, global function rings

⋅ 19th-B-11:20 − 11:45 Average value of class numbers of cubic Kummer extensions over the rational function field (Jungyun Lee, Yoonjin Lee, Jinjoo Yoo)

이정연((강원대)), 이윤진((이화여대)), 유진주*((UNIST))
Jungyun Lee, Kangwon National University, Yoonjin Lee, Ewha Womans University, Jinjoo Yoo*, UNIST

We compute average value of class numbers of cubic function fields $K_m = k(\sqrt[3]{m})$, where $\mathbb{F}_q$ is a finite field with $q$ elements, $q \equiv 1 \pmod 3$, $k:=\mathbb{F}_q(T)$ is the rational function field, and $m \in \mathbb{F}_q[T]$ is a monic cube-free polynomial. For computation, we find the mean value of $|L(s,\chi)|^2$ evaluated at $s=1$ when $\chi$ goes through the primitive cubic Dirichlet characters of $\mathbb{F}_q[T]$. This is joint work with Jungyun Lee (Kangwon National University) and Yoonjin Lee (Ewha Womans University).

2010 Mathematics Subject Classification: 11M38, 11R29
Key Words and Phrases: L-function, mean value of class number, cubic function field

⋅ 19th-B-11:50 − 12:20 Regular triangular forms of rank exceeding 3 (Mingyu Kim)

김민규((성균관대))
Mingyu Kim, Sungkyunkwan University

For a polynomial $T(x)=x(x+1)/2$ and positive integers $a_1,a_2,\dots,a_k$, an integer-valued quadratic polynomial of the form $a_1T(x_1)+a_2T(x_2)+\cdots+a_kT(x_k)$ is called a triangular form of rank $k$. A triangular form is called regular if it represents all positive integers that are locally represented. In this talk, we find all regular triangular forms of rank exceeding 3.

2010 Mathematics Subject Classification: 11E12, 11E20
Key Words and Phrases: Regular triangular forms
Automorphic Forms and q-Series

⋅ 19th-C-15:00 − 16:50 Chair: Bo-Hae Im (KAIST)

⋅ 19th-C-15:00 − 15:25 Congruence properties for the number of partitions (Dohoon Choi, Youngmin Lee)

최도훈((고려대)), 이영민*((고등과학원))
Dohoon Choi, Korea University, Youngmin Lee*, KIAS

For a positive integer $n$, let $p(n)$ be the number of partitions of $n$. In 1960, Newman conjectured that for any non-negative integers $M$ and $r$ with $0\leq r\leq M-1$, there are infinitely many non-negative integers $n$ such that \[ p(n)\equiv r\pmod{M}. \] In this talk, we explain our results related to Newman's conjecture. Moreover, we introduce congruence properties for the number of $t$-core partitions, generalized Frobenius partitions, and so on.

2010 Mathematics Subject Classification: 11F33, 11P83
Key Words and Phrases: Partition function, Newman's conjecture

⋅ 19th-C-15:25 − 15:50 Riemann hypothesis for period polynomials attached to the derivatives of $L$-functions of cusp forms for $\Gamma_0(N)$ (Bo-Hae Im, Hojin Kim)

임보해((카이스트)), 김호진*((카이스트))
Bo-Hae Im, KAIST, Hojin Kim*, KAIST

In this talk we consider the period polynomials attached to the first derivatives of $L$-functions of newforms. We prove the Riemann hypothesis for all but finitely many of them, that is, all zeros of them lie on the circle $|z|=1/\sqrt{N}$ when the newform is of level $N$. This result extends the results by Jin, Ma, Ono and Soundararajan (2016) and by Diamantis and Rolen (2018, 2021). This is joint work with Bo-Hae Im.

2010 Mathematics Subject Classification: 11F11, 11F67
Key Words and Phrases: Period polynomials, modular forms

⋅ 19th-C-16:00 − 16:25 Generating functions for various types of core partitions (Hyunsoo Cho)

조현수((이화여대))
Hyunsoo Cho, Ewha Womans University

In this talk, we explain the generating functions for various types of core partitions which contains core partitions, self-conjugate core partitions, bar-core partitions, and so on. We also consider some related partition identities.

2010 Mathematics Subject Classification: 05A17, 11P81
Key Words and Phrases: Core partition, generating function, partition identity

⋅ 19th-C-16:25 − 16:50 The genus of quotients of several types of numerical semigroups (Hayan Nam, Kyeong Jun Lee)

남하얀*((덕성여대)), 이경준((연세대))
Hayan Nam*, Duksung Women's University, Kyeong Jun Lee, Yonsei University

Finding the Frobenius number and the genus of any numerical semigroup $S$ is a well-known open problem. Similarly, it has been studied to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this talk, by computing the Hilbert series of each type of numerical semigroup, we talk about an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.

2010 Mathematics Subject Classification: 11A99, 05A15, 20M14
Key Words and Phrases: Numerical semigroup, genus, hilbert series, quotient of numerical semigroup

⋅ 19th-D-17:00 − 18:15 Chair: Chang Heon Kim (Sungkyunkwan University)

⋅ 19th-D-17:00 − 17:25 Parity bias phenomena in ordinary integer partitions (Byungchan Kim, Eunmi Kim, Jeremy Lovejoy)

김병찬*((서울과기대)), 김은미((이화여대)), Jeremy Lovejoy((Universit\'e Paris Cit\'e))
Byungchan Kim*, SeoulTech, Eunmi Kim, Ewha Womans University, Jeremy Lovejoy, Universit\'e Paris Cit\'e

In a recent joint work with E. Kim and J. Lovejoy, we initiated researches on parity biases in integer partitions. Here the parity bias in ordinary integer partition refers to the inequality \[ p_o (n) > p_e (n) \] for all positive integers $n \neq 2$, where $p_o(n)$ (resp. $p_e (n)$) is the number of partitions of $n$ with more odd (resp. even) parts than even (resp. odd) parts. In this talk, we outline the results from a joint work with E. Kim and J. Lovejoy and more recent two refinements of the parity bias from another joint work with E. Kim.

2010 Mathematics Subject Classification: 11P81
Key Words and Phrases: Partitions, parity bias, q-series

⋅ 19th-D-17:25 − 17:50 On the basis for the space of composite integer level weakly holomorphic modular forms (Chang Heon Kim, Kyung Seung Lee)

김창헌((성균관대)), 이경승*((성균관대))
Chang Heon Kim, Sungkyunkwan University, Kyung Seung Lee*, Sungkyunkwan University

In this talk, we discuss the recipe for finding a basis of the space $M_{k}^{!}(N)$ of weakly holomorphic modular forms for composite integer level cases. In the meantime, the fact that there may be several inequivalent cusps in higher level cases is one of the obstacles in the study of finding the canonical basis of the space $M_{k}^{!}(N)$. Accordingly, in the previous studies, the spaces of forms having poles only at $\infty$ or $0$, such as $M_{k}^{\#}(N)$ or $M_{k}^{\flat}(N)$, or the cases where there is only one inequivalent cusp were mainly considered. We describe recent advances in the study of the basis problem.

2010 Mathematics Subject Classification: 11F03, 11F11
Key Words and Phrases: Weakly holomorphic modular form, Atkin-Lehner involution

⋅ 19th-D-17:50 − 18:15 Nonvanishing of L-functions associated to some Hecke characters of cyclotomic fields (Keunyoung Jeong, Yeong-Wook Kwon, Junyeong Park)

정근영((전남대)), 권영욱*((UNIST)), 박준영((UNIST))
Keunyoung Jeong, Chonnam National University, Yeong-Wook Kwon*, UNIST, Junyeong Park, UNIST

In this talk, we explain a nonvanishing result for central values of L-functions associated with some Hecke characters of cyclotomic fields. To obtain our result, we use a formula, due to T. Yang, expressing L-values in terms of local integrals, and then compute the local integral at each place. The most complicated is the computation at 2. As an application, we show that for each isogeny factor of Jacobians of some Fermat curves there are infinitely many twists whose analytic rank is zero. This is joint work with Keunyoung Jeong and Junyeong Park.

2010 Mathematics Subject Classification: 11R42, 11G10
Key Words and Phrases: Cyclotomic field, Hecke L-function, Jacobian of Fermat curve
Dynamics of Singular Structures in Fluids

⋅ 19th-A-09:00 − 10:30 Chair: Injee Jeong (Seoul National University)

⋅ 19th-A-09:00 − 09:30 Asymptotic profiles of the Navier-Stokes flow under fluid-solid interaction (Bum Ja Jin)

진범자((목포대))
Bum Ja Jin, Mokpo National University

We consider the motion of a rigid ball in the viscous incompressible fluid. The $L^q-L^r$ decay rate estimate for the fluid-structure semigroup operator has been already known, and it is almost the same as the one for the Stokes flow in the interior domain. Our aim is to derive the asymptotic profiles of the fluid and to investigate a wake region at large time.

2010 Mathematics Subject Classification: 33M33
Key Words and Phrases: Stokes, Navier-Stokes, asymptotic profile, decay rate, exterior

⋅ 19th-A-09:30 − 10:00 Some remarks on the magnetohydrodynamics (Jihoon Lee)

이지훈((중앙대))
Jihoon Lee, Chung-Ang University

In this talk, we introduce some mathematical modeling of the magnetohydrodynamics-the dynamics of the electrically conducting fluids. We provide a brief review some results on the well-posedness of the solutions to the magnetohydrodynamics. Also we provide a brief review of the well-posedness of the solutions to the magnetohydrodynamics equations with the rotation.

2010 Mathematics Subject Classification: 35Q35
Key Words and Phrases: Magnetohydrodynamics

⋅ 19th-A-10:00 − 10:30 Local regularity near boundary for the Stokes system in a half space (Kyung Keun Kang, Tongkeun Chang)

강경근*((연세대)), 장통근((연세대))
Kyung Keun Kang*, Yonsei University, Tongkeun Chang, Yonsei University

We construct a weak solution of the Stokes system in a half space, whose normal derivatives are bounded near boundary. The constructed solution satisfies no-slip condition everywhere on the boundary and its singularities are caused by a localized singular force. The solution is an extension of Sverak-Seregin's example that is a type of shear flow.

2010 Mathematics Subject Classification: 35Q35, 35B65
Key Words and Phrases: Stokes system, unbounded normal derivatives, half space

⋅ 19th-B-10:50 − 12:20 Chair: Kyudong Choi (UNIST)

⋅ 19th-B-10:50 − 11:20 Propagation of chaos in a two cancer cell phenotypes dynamics (Myeongju Chae, Jaewook Ahn, Young-Pil Choi, Jihoon Lee)

채명주*((한경대)), 안재욱((한경대)), 최영필((한경대)), 이지훈((한경대))
Myeongju Chae*, Hankyong National University, Jaewook Ahn, Hankyong National University, Young-Pil Choi, Hankyong National University, Jihoon Lee, Hankyong National University

In this talk we introduce a two population cancer model focused on a nonlocal cell to cell adhesion on multidimensional bounded domain. Based on the well-posedness results of the system we prove a simplified version of the model can be seen as the mean field limit of a stochastic particle system.

2010 Mathematics Subject Classification: 60K35, 35K57, 60J60, 82C22, 60F10
Key Words and Phrases: Non-local models, propagation of chaos, stochastic interacting particle systems, relative entropy method

⋅ 19th-B-11:20 − 11:50 Dynamics of geostrophic Bessel vortices (Sun-Chul Kim, Habin Yim, Sung-Ik Sohn)

김선철*((중앙대)), 임하빈((중앙대)), 손성익((강릉원주대))
Sun-Chul Kim*, Chung-Ang University, Habin Yim, Chung-Ang University, Sung-Ik Sohn, Gangneung-Wonju National University

The dynamics of geostrophic vortices described by modified Bessel functions is studied. We focus on the three-vortex case, where the possibility of self-similar motion and general dynamics for arbitrary strengths is studied. It is found that self-similar motions are limited to rigid rotations and self-similar triple collapse is essentially impossible. For a general description, trilinear coordinates are adopted and extended to the model of geostrophic vortices. The similarities and differences with the point-vortex model are emphasized. The physical regions in the phase plane cannot be directly identified, unlike the point-vortex case where the physical regions are identified as conic sections. The boundary of the physical region approaches the vertices of the triangle in trilinear coordinates in geostrophic vortices, while making a tangent with the sides of the triangle in point vortices. For positive values of the three vortex strengths, the boundary of the physical region is a closed concave curve, whereas it is a circle or ellipse in the case of the point vortex. Importantly, we find a new type of the phase trajectory: an open curve not intersecting with the physical region boundary. We also demonstrate the physical motions and trajectories of vortices for several representative cases of vortex strengths from numerical computations.

2010 Mathematics Subject Classification: 76B47
Key Words and Phrases: Geostrophic vortex, self-similar motion, trilinear coordinates, vortex collapse

⋅ 19th-B-11:50 − 12:20 Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches (Kyudong Choi, In-Jee Jeong, Deokwoo Lim)

최규동((UNIST)), 정인지((서울대)), 임덕우*((UNIST))
Kyudong Choi, UNIST, In-Jee Jeong, Seoul National University, Deokwoo Lim*, UNIST

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T} $. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $ \lbrace x_{1}<\alpha\rbrace,\; \alpha>0 $. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.

2010 Mathematics Subject Classification: 76B47, 35Q35
Key Words and Phrases: 2D Euler, vorticity distribution, stability, center of mass, rearrangement, perimeter, large time behavior
Nonlinear Dynamics in Classical and Quantum Models Arising from Mathematical Physics

⋅ 19th-A-09:00 − 10:30 Chair: Jinwook Jung (Chonbuk National University)

⋅ 19th-A-09:00 − 09:20 The 3D inviscid Boussinesq equations in the three-scale limits for rotating stratified fluids: Convergence and Devil's staircase paradox (Min Jun Jo, Junha Kim, Jihoon Lee)

조민준*((The University of British Columbia)), 김준하((고등과학원)), 이지훈((중앙대))
Min Jun Jo*, The University of British Columbia, Junha Kim, KIAS, Jihoon Lee, Chung-Ang University

Asymptotic regimes for the 3D inviscid Boussinesq equations were suggested in \textbf{BMNZ}, \emph{Theoret. Comput. Fluid Dynamics}, 9: 223--251 (1997) to analyze the effects of rotation and stratification on large-scale geophysical flows. In this article, we rigorously justify the convergence of the 3D inviscid Boussinesq equations to the QG (quasi-geostrophic) equations on any fixed time interval $[0,T]$ in the three-scale limits of rotation, stratification, and the ratio of the two effects. The ratio is essentially the ``Burger" number in geophysics, which we call here the pseudo-Burger number $\mu$. We show that such convergence happens for any positive $\mu\neq 1$. In contrast, when $\mu=1$, we prove non-convergence in the same space; the solutions to the Boussinesq equations stay away from the corresponding solutions to the QG equations, uniformly in rotation and stratification with the fixed ratio $\mu=1$. This implies that any convergence rates cannot be made uniform in $\mu$ near $\mu=1$, which can be regarded as the first partial rigorous resolution of Devil's staircase paradox that was originally cast in \textbf{BMNZ}. In view of the projections onto the eigenspaces for the linear propagator, we also specify a particular class of initial data which ensures the convergence regardless of $\mu$ even in a stronger sense.

2010 Mathematics Subject Classification: 35Q86, 76B70, 76U60
Key Words and Phrases: Rotating stratified fluids, quasi-geostrophic equations, Boussinesq equations

⋅ 19th-A-09:20 − 09:40 The Boltzmann equation in general solid torus domains (Chanwoo Kim, Gyounghun Ko, Donghyun Lee)

김찬우((University of Wisconsin)), 고경훈*((포항공대)), 이동현((포항공대))
Chanwoo Kim, University of Wisconsin, Gyounghun Ko*, POSTECH, Donghyun Lee, POSTECH

In this talk, we consider the Boltzmann equation in general solid torus domains with specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases such as an inflection grazing. Thus, it is important to analyze trajectories which cause grazings. We will provide new analysis to handle these trajectories in general solid torus domains.

2010 Mathematics Subject Classification: 35Q20, 82B40
Key Words and Phrases: Kinetic theory, Boltzmann equation, non-convex domain

⋅ 19th-A-09:50 − 10:10 Relativistic BGK model for gas mixtures (Myeong-Su Lee, Seok-Bae Yun, Byung-Hoon Hwang)

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

In this talk, we present a BGK-type model of the relativistic Boltzmann equation for gas mixtures, based on Marle's formulation. For this, we introduce a single global BGK-type operator for each species and determine the auxiliary parameters by imposing that our model satisfies the balance equations. And then we see that our model satisfies some basic properties. And we further see that our model can recover the classical BGK model in the Newtonian limit.In this talk, we present a BGK-type model of the relativistic Boltzmann equation for gas mixtures, based on Marle's formulation. For this, we introduce a single global BGK-type operator for each species and determine the auxiliary parameters by imposing that our model satisfies the balance equations. And then we see that our model satisfies some basic properties. And we further see that our model can recover the classical BGK model in the Newtonian limit.

2010 Mathematics Subject Classification: 35Q20, 35Q75, 82C40, 82D05
Key Words and Phrases: Boltzmann equation, BGK model, kinetic theory, gas mixtures, relativity

⋅ 19th-A-10:10 − 10:30 Passage from quantum to classical synchronization model (Gyuyoung Hwang)

황규영((서울대))
Gyuyoung Hwang, Seoul National University

We study the passage from quantum to classical dynamics in the presence of synchronizing force. Starting from the Schr\"odinger-Lohe model, we arrive at the Vlasov-Lohe model through the Wigner-Lohe model by taking the Wigner transform. This procedure, which is called the (semi)classical limit, has been intensively studied by many researchers and usually it is done by taking Wigner transform to one density function. However, due to presence of the coupled nonlinearity, we should deviate from the standard approach. We choose the Wigner matrix method to deal with the semiclassical analysis of the coupled nonlinear Schr\"odinger equations. This is a joint work with S.-Y., Ha and D. Kim.

2010 Mathematics Subject Classification: 81Q20
Key Words and Phrases: Semiclassical analysis, Wigner transform, synchronization, classical limit

⋅ 19th-B-10:50 − 12:20 Chair: Gi Chan Bae (Seoul National University)

⋅ 19th-B-10:50 − 11:10 On dispersive quantization and fractalization for the Kawahara equation (Seongyeon Kim)

김성연((고등과학원))
Seongyeon Kim, KIAS

In this talk, we present the dichotomy phenomena of solutions to the Kawahara equation with bounded variation initial data. These phenomena, called Talbot effect, are that at rational times the solution is quantized, while at irrational times it is a continuous nowhere differentiable function with fractal profile. Such in unknown for the Kawahara equation yet, which is a fifth-order KdV type equation. For the purpose, we obtain smoothing estimates for the nonlinear Duhamel solution, which, combined the known results on the linear solution, mathematically describes the Talbot effect.

2010 Mathematics Subject Classification: 35B45, 35Q35
Key Words and Phrases: Talbot effect, Kawahara equation

⋅ 19th-B-11:10 − 11:30 Asymptotic stability of the inviscid incompressible porous medium equation (Min Jun Jo, Junha Kim)

조민준((The University of British Columbia)), 김준하*((고등과학원))
Min Jun Jo, The University of British Columbia, Junha Kim*, KIAS

We consider the IPM equation in $\mathbb{R}^2$. It is well-known that the stratified state $(u_s, \rho_s, p_s)=(0, x_2, \frac12 x_2^2)$ is asymptotically stable. In this talk, we provide a neighborhood of $(u_s, \rho_s, p_s)$ consisting of stationary solutions to the IPM equation, where all elements are asymptotically stable. In particular, this is an improvement over the previous stability results.

2010 Mathematics Subject Classification: 35B40, 35A01
Key Words and Phrases: Temporal decay estimate, global well-posedness, IPM equation

⋅ 19th-B-11:40 − 12:00 Remarks on the Maxwell-Chern-Simons gauged model in $\mathbb{R}^{1+1}$ (Bora Moon)

문보라((한양대))
Bora Moon, Hanyang University

In this talk, we introduce the $(1+1)$-dimensional Maxwell-Chern-Simons $O(3)$ sigma model, which is obtained by the dimensional reduction of the $(2+1)$-dimensional Max\-well-Chern-Simons $O(3)$ sigma model. First, we study the local and global well-posedness of the model under the Lorenz gauge condition. Then we discuss the growth-in-time of the Sobolev norms for the solutions. This talk is based on joint work with Guanghui Jin.

2010 Mathematics Subject Classification: 35A01, 35B40, 35Q40
Key Words and Phrases: Maxwell-Chern-Simons model, well-posedness, growth-in-time of Sobolev norms

⋅ 19th-B-12:00 − 12:20 Scattering of cubic Dirac equations with a general class of Hartree-type nonlinearity for the critical Sobolev data (Seokchang Hong)

홍석창((중앙대))
Seokchang Hong, Chung-Ang University

Recently low-regularity behaviour of solutions to cubic Dirac equations with the Hartree-type nonlinearity has been extensively studied in somewhat a specific assumption on the structure of the nonlinearity. The key approach of the previous results is to exploit the null structure in the nonlinearity and the decay of the Yukawa potential. In this talk, we aim to go beyond; we investigate strong scattering property of cubic Dirac equations with quite a general class of the Hartree-type nonlinearity, which covers the Coulomb potential as well as the Yukawa potential, and the bilinear form, in which one cannot use the specific null structure. As a direct application, we also obtain the scattering for the boson-star equations with the scaling-critical Sobolev data.

2010 Mathematics Subject Classification: 35Q40
Key Words and Phrases: Dirac equation, Hartree-type nonlinearity, global well-posedness, scattering, the scale-invariant Sobolev space, angular regularity
Operator Theory on Reproducing Kernel Hilbert Spaces

⋅ 19th-C-15:00 − 16:10 Chair: In Hyoun Kim (Incheon National University)

⋅ 19th-C-15:00 − 15:20 Equivalence problems of Hankel operators (Sumin Kim)

김수민((성균관대))
Sumin Kim, Sungkyunkwan University

For an operator-valued symbol $\Phi\in L^2_s(\mathbb T, \mathcal{B}(E)) $, the Hankel operator $H_\Phi$ and the Toeplitz operator $T_{\Phi}$ on the vector-valued Hardy space $H^2(\mathbb T, E)$ are densely defined operators defined by $$ H_{\Phi} p:=J (I-P)(\Phi p) \quad \hbox{and} \quad T_{\Phi}p:= P(\Phi p) \quad(p \in \mathcal P_E), $$ where $P$ denotes the orthogonal projection that maps $L^2(\mathbb T, E)$ onto $H^2(\mathbb T, E)$ and $J$ denotes the unitary operator from $L^2(\mathbb T, E)$ to $L^2(\mathbb T, E)$ given by $(Jg)(z):=\overline{z} g(\overline{z})$ for $g \in L^2(\mathbb T, E)$. In this talk, we consider equivalence problems of Hankel and Toeplitz operators with operator-valued symbols. This talk is based on a joint work with In Sung Hwang and Woo Young Lee.

2010 Mathematics Subject Classification: 47B35, 42B30, 46E40
Key Words and Phrases: Hankel operators, Toeplitz operators, Hardy spaces

⋅ 19th-C-15:20 − 15:40 A distinction of $k$-hyponormal and weakly $k$-hyponormal weighted shifts (Mi Ryeong Lee)

이미령((대구가톨릭대))
Mi Ryeong Lee, Daegu Catholic University

Let $\alpha (x):\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},\ldots$ be a sequence with a real variable $x>0$ and let $W_{\alpha(x)}$ be the associated weighted shift with weight sequence $\alpha (x)$. In 2006, Exner-Jung-Park provided an algorithm to distinguish weak $k$-hyponormality and $k$-hyponormality of weighted shift $W_{\alpha(x)}$, and obtained $s_{n}>0$ for some low numbers $n=4,\ldots,10$, such that $W_{\alpha (s_{n})}$ is weakly $n$-hyponormal but not $n$-hyponormal. In this talk, we introduce a formula of $s_{n}$ (for all positive integer $n$) such that $W_{\alpha (s_{n})}$ is weakly $n$-hyponormal but not $n$-hyponormal, which improves Exner-Jung-Park's result above. This talk is based on a joint work with Chunji Li and Yiping Xiao.

2010 Mathematics Subject Classification: 47B37, 47B20
Key Words and Phrases: Subnormal, polynomially hyponormal, $k$-hyponormal, weakly $k$-hyponormal, weighted shift

⋅ 19th-C-15:50 − 16:10 Subnormality of Brownian-type operators with quasinormal entry (Il Bong Jung)

정일봉((경북대))
Il Bong Jung, Kyungpook National University

In 1995 Agler and Stankus introduced the notion of Brownian isometries of covariance $\sigma >0$ which is emerged from the study of the time shift operator of the modified Brownian motion process from one side. In this talk, we introduce operators that are represented by upper triangular $2\times 2$ block matrices having the $(2,2)$-entry $X$ as a place holder for operators and satisfy some algebraic constraints. We call such operators ``Brownian-type operators". The Brownian isometry of covariance $\sigma$ is contained in the class of Brownian-type operators. If $X$ is qusinormal, the associated Brownian-type operator is called ``Brownian-type operator with qasinormal entry" [or, briefly we say ``operator of class $ \mathcal{Q}$"]. We characterize the subnormality of operators of class $\mathcal{Q}$ by using the Taylor spectrum technique. In addition, we discuss some related topics. (This is a joint work with S. Chavan, Z. Jablonski, and J. Stochel.)

2010 Mathematics Subject Classification: 47B20, 47B37
Key Words and Phrases: Upper triangular $2\times 2$ block matrix, Taylor's spectrum, moment problem, Cauchy dual subnormality problem

⋅ 19th-D-16:30 − 17:10 Chair: In Sung Hwang (Sungkyunkwan University)

⋅ 19th-D-16:30 − 16:50 Algebraic properties of Toeplitz operators on the Newton space (Ji Eun Lee, Eungil Ko, Jongrak Lee)

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

The {\it Newton space} is defined by $$N^2({\mathbb P})=\big\{ f(z)=\sum_{n=0}^{\infty}a_n N_n(z)~:~\| f\|^2=\sum_{n=0}^{\infty}|a_n|^2<\infty \big\},$$ where the {\it $n$-th Newton polynomial} $N_n(z)$ is defined by $$N_n(z):=\frac{(-z)_n}{n!}=\begin{cases} 1~~~~\mbox{if}~n=0,\cr (-1)^n\frac{z(z-1)(z-2)\cdots(z-(n-1))}{n!}~\mbox{if}~n\geq 1. \end{cases}$$ In this paper, we study the Newton space $N^2({\mathbb P})$ which has the Newton polynomials as an orthonormal basis. We first investigate some relations between the orthonormal basis $\{z^n\}$ of the Hardy space $H^2(\mathbb D)$ and the orthonormal basis { $\{N_n\}$} of the Newton space $N^2({\mathbb P})$. Moreover, we study the algebraic properties of Toeplitz operators on Newton space $N^2({\mathbb P})$.

2010 Mathematics Subject Classification: 47B35, 47B15
Key Words and Phrases: Newton space, Newton polynomials, Toeplitz operator

⋅ 19th-D-16:50 − 17:10 Identifying Hardy classes of operator functions (Woo Young Lee)

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

In this talk, we consider the question: What is a circle companion of the Hardy space of the unit disk with values in the space of all bounded linear operators between separable Hilbert spaces? The question on the circle companion is to usually ask whether for each function $h$ on the unit disk, there exists a ``boundary function" $bh$ on the unit circle such that the mapping $h\mapsto bh$ is an isometric isomorphism between Hardy spaces of the unit disk and the unit circle with values in some Banach spaces. This work was collaborated with Raul Curto, In Sung Hwang and Sumin Kim.

2010 Mathematics Subject Classification: 42B30, 30H10, 46E40, 46E30
Key Words and Phrases: Operator-valued Hardy spaces, the analytic Radon-Nikod\'ym property, SOT measurable functions, strong Poisson integrals, strong boundary functions, circle companions
Recent Results in Geometric Structures on Manifolds

⋅ 19th-A-09:00 − 10:20 Chair: Yoonweon Lee (Inha University)

⋅ 19th-A-09:00 − 09:25 Curvature identities on Einstein manifolds (Yunhee Euh, Jihun Kim, Jeong Hyeong Park)

어윤희*((성균관대)), 김지훈((성균관대)), 박정형((성균관대))
Yunhee Euh*, Sungkyunkwan University, Jihun Kim, Sungkyunkwan University, Jeong Hyeong Park, Sungkyunkwan University

We recall a curvature identity for any 6-dimensional Riemannian manifold derived from the Chern-Gauss-Bonnet Theorem and Patterson's curvature identity on Riemannian manifolds based on the skew-symmetric properties of the generalized Kronecker delta. We provide some explicit formulas of Patterson's curvature identity on 5-dimensional and 6-dimensional Einstein manifolds. We explain that the curvature identities deduced from Patterson's results are the same as the curvature identities derived from the Chern-Gauss-Bonnet Theorem on Einstein manifolds. We also provide examples that support the theorems.

2010 Mathematics Subject Classification: 53B20
Key Words and Phrases: Einstein manifolds, curvature identities

⋅ 19th-A-09:25 − 09:50 Free boundary constant mean curvature surfaces in a convex domain (Keomkyo Seo)

서검교((숙명여대))
Keomkyo Seo, Sookmyung Women's University

We discuss a rigidity theorem for free boundary constant mean curvature surfaces in a convex domain in a 3-dimensional Riemannian manifold with sectional curvature bounded above by a constant, which states that such surface is homeomorphic to either a disk or an annulus under an appropriate pinching condition on the length of the traceless second fundamental form on the surface. This is joint work with Sung-Hong Min.

2010 Mathematics Subject Classification: 53C20, 53C42, 53A10
Key Words and Phrases: Free boundary, constant mean curvature

⋅ 19th-A-09:55 − 10:20 Some characterizations of harmonic manifolds (Jeong Hyeong Park)

박정형((성균관대))
Jeong Hyeong Park, Sungkyunkwan University

A Riemannian manifold $(M, g)$ is harmonic if a non-constant radial harmonic function exists in a punctured neighborhood at every point, or, equivalently if the volume density function centered at every point depends only on the distance from the center. There are many other characterizations of harmonic spaces. In this talk, we shall characterize harmonic manifolds by in terms of their volume density functions and in terms of the radial eigen-spaces of the Laplacian on functions and 1-forms.

2010 Mathematics Subject Classification: 53C21
Key Words and Phrases: Harmonic manifold, density function, radial eigen-spaces of the Laplacian

⋅ 19th-B-10:30 − 12:20 Chair: Jeong Hyeong Park (Sungkyunkwan University)

⋅ 19th-B-10:30 − 10:55 Gradient Ricci solitons with harmonic Weyl curvature in low dimension (Jongsu Kim)

김종수((서강대))
Jongsu Kim, Sogang University

In this lecture I will talk about gradient Ricci solitons and almost gradient Ricci solitons in three and four dimension. A gradient Ricci soliton is a Riemannian manifold $(M, g)$ and a smooth function $f$ satisfying $\nabla d f = -Rc + \lambda g $, where $Rc$ denotes the Ricci tensor of $g$ and $\lambda$ is a constant. If $(M, g, f)$ satisfies the equation with a function $\lambda$, then it is an almost gradient Ricci soliton. Gradient Ricci solitons are essential in Hamilton's Ricci flow theory as singularity models of the flow. I will present some quick review on recent developments on gradient Ricci solitons and then classifications of three and four dimensional (almost) gradient Ricci solitons with harmonic Weyl curvature. For instance, four dimensional gradient Ricci solitons with harmonic Weyl curvature are locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \times N_{\lambda}$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature ${\lambda} \neq 0$, a certain singular metric and a locally conformally flat metric. Finally I present some prospect for higher dimensional gradient Ricci solitons.

2010 Mathematics Subject Classification: 53C21, 53C25
Key Words and Phrases: Gradient Ricci soliton, harmonic Weyl curvature

⋅ 19th-B-10:55 − 11:20 Eigenvalue estimates for 3-Sasaki structures (Paul-Andi Nagy)

19th-B-10:55 − 11:20
Paul-Andi Nagy, Center for Complex Geometry, Institute for Basic Science(IBS),Daejeon

We prove optimal lower bounds for the first non-zero eigenvalue of the scalar sub-Laplaci\-an for 3-Sasaki metrics. In particular we give a description of the limiting space in terms of the automorphism algebra and using the 3-Sasaki moment map. This can be considered as an analogue of the Lichnerowicz-Matsushima estimate for K\"ahler-Einstein metrics and provides an essential improvement of existing estimates. Moreover we prove for the first time a lower bound for the spectrum of the Riemannian Laplacian on functions for all metrics in the canonical variation, only in terms of scalar curvature and dimension. In our proofs we use the interplay between Riemannian 3-Sasaki geometry and complex geometry on the associated hyperk\"ahler cone. Here we are also able to strengthen a result on the growth rate of harmonic functions, due to Conlon resp.~Hein \& Sun.

2010 Mathematics Subject Classification: 53C25, 53C26, 58C40, 35H10
Key Words and Phrases: Sub-Laplacian, 3-Sasaki structure, eigenvalue estimate, gap theorem

⋅ 19th-B-11:30 − 11:55 Geometric structures in contact manifolds (Jong Taek Cho)

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

In this talk, I would like to review the development of geometry on contact manifolds mainly in the Riemannian view point and to introduce some recent results.

2010 Mathematics Subject Classification: 53C25
Key Words and Phrases: Contact manifold, Riemannian geometry

⋅ 19th-B-11:55 − 12:20 The zeta-determinant of the generalized Dirichlet-to-Neumann operator associated to the Alvarez boundary condition (Yoonweon Lee)

이윤원((인하대))
Yoonweon Lee, Inha University

The gluing formula for the zeta-determinants of the Dolbeault Laplacians on a compact Riemann surface was proved by R. Wentworth by using the Alvarez boundary condition. The Alvarez boundary condition is defined by using the complex structure on a compact Riemann surface and in a case of a trivial line bundle it is reduced to the half Dirichlet and half Neumann boundary condition. He introduced a generalized Dirichlet-to-Neumann operator associated to the Alvarez boundary condition, which is a classical elliptic pseudodifferential operator of order zero, and used it to prove the gluing formula. In this talk, I’m going to describe the generalized Dirichlet-to-Neumann operator associated to the Alvares boundary condition on a trivial line bundle in terms of the classical Dirichlet-to-Neumann operators associated to the Dirichlet boundary condition and show that the gluing formula for the Dolbeault Laplacian is the same as the gluing formula for the Dirichlet and Neumann conditions.

2010 Mathematics Subject Classification: 58J52
Key Words and Phrases: Gluing formula of zeta-determinant, Dolbeault Laplacian, Alvarez boundary condition, generalized Dirichlet-to-Neumann operator
Applied Algebra and Optimization

⋅ 19th-B-10:50 − 12:20 Chair: Yoon Mo Jung (Sungkyunkwan University)

⋅ 19th-B-10:50 − 11:20 Active neuron least squares: A training method for rectified neural networks (Yeonjong Shin, Mark Ainsworth)

신연종*((카이스트)), Mark Ainsworth((Brown University))
Yeonjong Shin*, KAIST, Mark Ainsworth, Brown University

In this talk, we will present the Active Neuron Least Squares (ANLS), an efficient training algorithm for neural networks (NNs). ANLS is designed from the insight gained from the analysis of gradient descent training of NNs, particularly, the analysis of Plateau Phenomenon. The core mechanism is the option to perform the explicit adjustment of the activation pattern at each step, which is designed to enable a quick exit from a plateau. The performance of ANLS will be demonstrated and compared with existing popular methods in various learning tasks ranging from function approximation to solving PDEs.

2010 Mathematics Subject Classification: 65K10, 90C30
Key Words and Phrases: Neural network, least squares, network training, scientific machine learning

⋅ 19th-B-11:20 − 11:50 Trend filtering by adaptive piecewise polynomials (Juyoung Jeong, Yoon Mo Jung, Soo Hyun Kim, Sangwoon Yun)

정주영((성균관대 응용대수 및 최적화연구센터)), 정윤모((성균관대)), 김수현*((부산대)), 윤상운((성균관대))
Juyoung Jeong, Applied Algebra and Optimization Research Center, Sungkyunkwan University, Yoon Mo Jung, Sungkyunkwan University, Soo Hyun Kim*, Pusan National University, Sangwoon Yun, Sungkyunkwan University

Trend filtering is a regression problem to estimate underlying trends in time series data. It is necessary to investigate data in various disciplines. We propose a trend filtering method by adaptive piecewise polynomials. More specifically, we adjust the location and the number of breakpoints or knots to obtain a better fitting to given data. The numerical results on synthetic and real data sets show that it captures distinct features such as abrupt changes or kinks and provides a simplified form and brief summary of given data.

2010 Mathematics Subject Classification: 62J02
Key Words and Phrases: Trend filtering, piecewise polynomial regression, nonlinear regression, breakpoint merging

⋅ 19th-B-11:50 − 12:20 A shadowing property for smooth ADMM (Yoon Mo Jung, Bomi Shin, Sangwoon Yun)

정윤모((성균관대)), 신보미*((성균관대)), 윤상운((성균관대))
Yoon Mo Jung, Sungkyunkwan University, Bomi Shin*, Sungkyunkwan University, Sangwoon Yun, Sungkyunkwan University

In this talk, we study a shadowing property for optimization algorithm. We show that the ADMM flow generated by a coercive $C^2$ strongly convex objective function and a positive definite invertible real matrix has the eventual shadowing property. Also, if the ADMM flow generated by a $C^2$ convex objective function and a positive definite invertible real matrix has the eventual shadowing property, then the objective function has a unique minimizer.

2010 Mathematics Subject Classification: 37N40, 37B65
Key Words and Phrases: ADMM flow, shadowing property, first order methods

⋅ 19th-C-15:00 − 16:30 Chair: Soonhak Kwon (Sungkyunkwan University)

⋅ 19th-C-15:00 − 15:30 Extended commonality of paths and cycles via Schur convexity (Jang Soo Kim, Joonkyung Lee)

김장수((성균관대)), 이준경*((한양대 $/$ 기초과학연구원))
Jang Soo Kim, Sungkyunkwan University, Joonkyung Lee*, Hanyang University / IBS

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\geq 2^{1-e(H)}$ holds for every graphon $W:[0,1]^2\rightarrow [0,1]$, where $t_H(\cdot)$ denotes the homomorphism density of the graph $H$. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989). We prove a graph homomorphism inequality, which connects to old and new problems extending commonality of paths and cycles. Namely, whenever a graph $H$ is a path or a cycle and $W:[0,1]^2\rightarrow\mathbb{R}$ is a bounded symmetric measurable function, $t_H(W)+t_H(1-W)\geq t_{K_2}(W)^{e(H)} +t_{K_2}(1-W)^{e(H)}$ holds. This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons $W$ and odd cycles $H$. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.

2010 Mathematics Subject Classification: 05C60, 05D10, 05A20
Key Words and Phrases: Graph homomorphism, Ramsey multiplicity, symmetric polynomials

⋅ 19th-C-15:30 − 16:00 Odd coloring and proper conflict-free coloring of a sparse graph (Eun-Kyung Cho, Ilkyoo Choi, Hyemin Kwon, Boram Park)

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

In this talk, we will investigate the concepts and properties of an odd coloring and a proper conflict-free coloring of a graph. We will focus specifically on sparse graphs, such as a graph with a bounded maximum average degree, and a planar graph with a girth condition. Also, we will look at some interesting open questions related to these concepts. This talk is based on joint work with Ilkyoo Choi, Hyemin Kwon, and Boram Park.

2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: Odd coloring, proper conflict-free coloring, sparse graph

⋅ 19th-C-16:00 − 16:30 Vector-valued period polynomials with Hecke structures (Jihyun Hwang, Chang Heon Kim)

황지현*((이화여대)), 김창헌((성균관대))
Jihyun Hwang*, Ewha Womans University, Chang Heon Kim, Sungkyunkwan University

Period polynomials are one of major subjects in number theory because they are closely connected to L-functions associated to holomorphic cusp forms. To deal with period polynomials for congruence subgroups not the full modular group, we regard period polynomials as vector-valued functions. In this talk, we will deal with vector-valued period polynomials obtained from weakly holomorphic cusp forms, and observe the space of period polynomials with Hecke structures.

2010 Mathematics Subject Classification: 11F67
Key Words and Phrases: Period polynomials, weakly holomorphic modular forms, Hecke eigenforms
Data-Driven and Model-Driven Methods for Mathematical Biology

⋅ 19th-A-09:00 − 10:25 Chair: Jinsu Kim (POSTECH)

⋅ 19th-A-09:00 − 09:25 Scalable protein-DNA binding changer test for insertion and deletion of bases in the genome (Sunyoung Shin, Qinyi Zhou, Chandler Zuo, Yuannyu Zhang, Jian Xu, Min Chen)

신선영*((포항공대)), Qinyi Zhou((University of Texas at Dallas)), Chandler Zuo((University of Texas at Dallas)), Yuannyu Zhang((University of Texas Southwestern Medical Center)), Jian Xu((University of Texas Southwestern Medical Center)), Min Chen((University of Texas at Dallas))
Sunyoung Shin*, POSTECH, Qinyi Zhou, University of Texas at Dallas, Chandler Zuo, University of Texas at Dallas, Yuannyu Zhang, University of Texas Southwestern Medical Center, Jian Xu, University of Texas Southwestern Medical Center, Min Chen, University of Texas at Dallas

Mutations in the noncoding DNA, which represents approximately 99\% of the human genome, have been crucial to understand disease mechanisms through dysregulation of disease-associated genes. One key element in gene regulation that noncoding mutations mediate is the binding of proteins to DNA sequences. Insertion and deletion of bases (InDels) are the second most common type of mutations, following single nucleotide polymorphisms, that may impact protein-DNA binding. However, no existing methods can estimate and test the effects of InDels on the process of protein-DNA binding. We develop a novel statistical test, named binding changer test (BC test), using a Markov model to evaluate the impact of InDels and identify InDels altering protein-DNA binding. The test predicts binding changer InDels of regulatory significance with an efficient importance sampling algorithm generating background sequences in favor of large binding affinity changes. Simulation studies demonstrate its excellent performance. The application to human leukemia data uncovers candidate pathologic InDels on modulating MYC binding in leukemic patients. We develop R package atIndel, which is available on GitHub.

2010 Mathematics Subject Classification: 62M02, 68W25, 92-10, 92D10
Key Words and Phrases: Importance sampling, insertions and deletions, noncoding mutations, position weight matrix, transcription factor binding

⋅ 19th-A-09:30 − 09:55 Bayesian model calibration and sensitivity analysis for oscillating biological experiments (Youngdeok Hwang, Hang Jun Kim, Won Chang, Christian Hong, Steve MacEachern)

황영덕((City University of New York)), 김항준((University of Cincinnati)), 장원*((University of Cincinnati)), Christian Hong((University of Cincinnati)), Steve MacEachern((Ohio State University))
Youngdeok Hwang, City University of New York, Hang Jun Kim, University of Cincinnati, Won Chang*, University of Cincinnati, Christian Hong, University of Cincinnati, Steve MacEachern, Ohio State University

Most organisms exhibit various endogenous oscillating behaviors which provide crucial information as to how the internal biochemical processes are connected and regulated. Understanding the molecular mechanisms behind these oscillators requires interdisciplinary efforts combining both biological and computer experiments, as the latter can complement the former by simulating perturbed conditions with higher resolution. Harmonizing the two types of experiment, however, poses significant statistical challenges due to identifiability issues, numerical instability, and ill behavior in high dimension. This article devises a new Bayesian calibration framework for oscillating biochemical models. The proposed Bayesian model is estimated using an advanced MCMC which can efficiently infer the parameter values that match the simulated and observed oscillatory processes. Also proposed is an approach to sensitivity analysis approach based on the intervention posterior. This approach measures the influence of individual parameters on the target process by utilizing the obtained MCMC samples as a computational tool. The proposed framework is illustrated with circadian oscillations observed in a filamentous fungus, Neurospora crassa.

2010 Mathematics Subject Classification: 62P10, 92Bxx
Key Words and Phrases: Circadian cycle, differential equation, generalized multiset sampler, harmonic basis representation, intervention posterior, systematic biology

⋅ 19th-A-10:00 − 10:25 Efficient method for approximating traveling wave speeds via neural network approach (Hyung Ju Hwang, Sunwoo Hwang, Sung Woong Cho, Hwijae Son)

황형주*((포항공대)), 황선우((포항공대)), 조성웅((포항공대)), 손휘재((한밭대))
Hyung Ju Hwang*, POSTECH, Sunwoo Hwang, POSTECH, Sung Woong Cho, POSTECH, Hwijae Son, Hanbat National University

We discuss a neural network approach to approximating traveling wave speeds and traveling wave solutions for various kinds of partial differential equations via artificial neural networks. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter.

2010 Mathematics Subject Classification: 65N
Key Words and Phrases: Traveling wave, AI

⋅ 19th-B-10:50 − 12:15 Chair: Jinsu Kim (POSTECH)

⋅ 19th-B-10:50 − 11:15 Dramatic rescue of cancer-related cognitive changes in combined anticancer therapies (Yangjin Kim, Junho Lee, Jinsu Kim)

김양진*((건국대)), 이준호((건국대)), 김진수((원자력병원))
Yangjin Kim*, Konkuk University, Junho Lee, Konkuk University, Jinsu Kim, Korea Institute Radiological and Medical Sciences

Anti-cancer drugs including trastuzumab (TZB) typically induce serious side effects including significant cognitive changes. These cancer-related cognitive changes (CRCC) are regulated by an adverse biological process involving cancer stem cells (CSCs) and IL-6. Recent studies have reported that atorvastatin (ATV) may change the dynamic of cognitive impairment in a combination (TZB+ATV) therapy. In this study, we investigate the mutual interactions between cancer stem cells and the tumor cells that facilitate cognitive impairment during long term TZB therapy by developing a mathematical model that involves IL-6 and the key apoptotic regulation. The model predicts that the combination therapy can not only reduce tumor size but also reduce chemobrain effect. Several therapeutic approaches are shown. We also investigate the optimal strategies by using optimal control theory.

2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: Cancer, anti-cancer drugs, mathematical model

⋅ 19th-B-11:20 − 11:45 Simplifying complex chemical reaction networks (Yuji Hirono, Takashi Okada, Hiroyasu Miyazaki, Yoshimasa Hidaka)

19th-B-11:20 − 11:45
Yuji Hirono*, Asia Pacific Center for Theoretical Physics, Takashi Okada, iTHEMS, RIKEN, Hiroyasu Miyazaki, iTHEMS, RIKEN, Yoshimasa Hidaka, KEK Theory Center

Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In this talk, we discuss a method for the reduction of chemical reaction networks, by which important substructures can be extracted. Mathematical concepts such as homology and cohomology groups are found to be useful for characterizing the shapes of reaction networks and for tracking the changes of them under reductions. For a given chemical reaction network, we identify topological conditions on its subnetwork, reduction of which preserves the original steady state exactly. This method allows us to reduce a reaction network while preserving its original steady-state properties, thereby complex reaction systems can be studied efficiently.

2010 Mathematics Subject Classification: 92C42
Key Words and Phrases: Chemical reaction networks

⋅ 19th-B-11:50 − 12:15 Systematic inference identifies a major source of heterogeneity in non-Markovian cell signaling dynamics (Hyukpyo Hong, Dae Wook Kim, Jae Kyoung Kim)

홍혁표*((기초과학연구원)), 김대욱((University of Michigan)), 김재경((기초과학연구원))
Hyukpyo Hong*, Institute for Basic Science, Dae Wook Kim, University of Michigan, Jae Kyoung Kim, Institute for Basic Science

Identifying the sources of cell-to-cell variability in signaling dynamics is essential to understanding drug response variability and developing more effective therapeutics. However, it is challenging because many signaling intermediate reactions are experimentally unobservable. This can be overcome by replacing them with a single random time delay, but the resulting process is non-Markovian, making it difficult to infer cell-to-cell heterogeneity in reaction rates and time delays. In this talk, we present an efficient and scalable moment-based Bayesian method that infers cell-to-cell heterogeneity in the non-Markovian signaling process. We apply this method to single-cell expression profiles from promoters responding to various antibiotics and discovered a major source of cell-to-cell variability in antibiotic stress-signal response: the number of rate-limiting steps in signaling cascades. This knowledge can help identify more effective therapies that destroy all pathogenic or cancer cells.

2010 Mathematics Subject Classification: 92C42, 37N25, 62F15, 60K30
Key Words and Phrases: Queueing theory, Bayesian inference, non-Markovian stochastic process, cell-to-cell heterogeneity
Mathematical Logic and Its Applications

⋅ 19th-B-10:50 − 12:20 Chair: Byunghan Kim (Yonsei University)

⋅ 19th-B-10:50 − 11:20 NATP in lovely pairs and H-structures on geometric theories (Jinhoo Ahn, Joonhee Kim, Hyoyoon Lee, Junguk Lee)

안진후((고등과학원)), 김준희((연세대)), 이효윤*((연세대)), 이정욱((고등과학원))
Jinhoo Ahn, KIAS, Joonhee Kim, Yonsei University, Hyoyoon Lee*, Yonsei University, Junguk Lee, KIAS

It is known that if a complete theory $T$ is geometric, then being NTP$_2$ and NTP$_1$ are preserved under taking some dense/co-dense expansions. In this talk, we are interested in NATP; we show that being NATP is preserved to the theories of lovely pairs of models and H-structures on geometric theories. Also, we present a new criterion of having ATP in terms of the number of realizations of antichains and paths, whose proof is an ATP version for the SOP$_2$ case.

2010 Mathematics Subject Classification: 03C45
Key Words and Phrases: Antichain tree property, preservation, dense/co-dense expansion, lovely pair, H-structure

⋅ 19th-B-11:20 − 11:50 Dense/Co-dense expansions on vector spaces (Joonhee Kim)

김준희((연세대))
Joonhee Kim, Yonsei University

In joint work with JinHoo Ahn, Hyoyoon Lee, and Junguk Lee, we investigate how well NATP is preserved by taking model-theoretic constructions. As one of the results, we show that taking certain dense/co-dense expansions on vector spaces preserves having NATP. In recent work, Berenstein, d'Elbee, and Vassilev proved that if a theory of vector space $V$ over a field F has NIP (NTP1, NTP2, NSOP1), then its unary predicate expansion, whose interpretation of the predicate is a dense/co-dense R-submodule, still has NIP (NTP1, NTP2, NSOP1 resp.), where R is a subring of F. We show that the same construction also preserves NATP. That is if a vector space is NATP, then its expansion is also NATP.

2010 Mathematics Subject Classification: 03C45, 03C60
Key Words and Phrases: Model theory, classification theory, vector space, antichain tree property

⋅ 19th-B-11:50 − 12:20 Ax, Kochen, and Ershov meet ATP (Junguk Lee)

이정욱((고등과학원))
Junguk Lee, KIAS

In model theory of valued fields, one of most influential ideas is the Ax-Kochen-Ershov (AKE) principle, roughly saying that elementary properties of an unramified henselian valued field is controlled by elementary properties of its residue field and its value group. On the other hand, Ahn and Kim introduced a new dividing line, called antichain tree property (ATP), in Classification Theory in model theory. In a joint work with Ahn and Kim, we gave the AKE-style criterion for an unramified henselian valued field of equicharacteristic zero to be ATP. Namely, an unramified henselian valued field of equicharacteristic zero has ATP if and only if its residue field has ATP. In this talk, I will show that this type criterion is extended into the mixed characteristic case. Specially, in the case of perfect residue fields, an unramified henselian valued field of mixed characteristic has ATP if and only if its residue field has ATP. For example, the quotient field of the Witt ring of a perfect Frobenius field of characteristic p has no ATP even though it has TP2 and SOP, and so it does not belong to previously known tame classes.

2010 Mathematics Subject Classification: 03C45, 03C60
Key Words and Phrases: Antichain tree property, the AKE principle, Frobenius field

⋅ 19th-C-15:00 − 16:00 Chair: Joonhee Kim (Yonsei University)

⋅ 19th-C-15:00 − 15:30 So much logic underlies nowadays computers (Martin A Ziegler)

19th-C-15:00 − 15:30
Martin A. Ziegler, KAIST

Logic is sometimes seen as esoteric: by those ignorant of its many applications in Computer Science. We recall [doi:10.2307/2687775], and add further examples on how Logic is essential for nowadays computing applications.

2010 Mathematics Subject Classification: 03, 01, 68
Key Words and Phrases: Logic, computing

⋅ 19th-C-15:30 − 16:00 A type-theoretical interpretation of intuitionistic fixed point logic (Ulrich Berger, Sewon Park, Holger Thies, Hideki Tsuiki)

Ulrich Berger((Swansea University)), 박세원*((Kyoto University)), Holger Thies((Kyoto University)), Hideki Tsuiki((Kyoto University))
Ulrich Berger, Swansea University, Sewon Park*, Kyoto University, Holger Thies, Kyoto University, Hideki Tsuiki, Kyoto University

We present a way to embed the induction and coinduction operators and the corresponding proof rules from intuitionistic fixed point logic into a dependent type theory. The idea is implemented in Coq proof assistant using MetaCoq, a meta-programming plugin for Coq, for its realizability interpretation. We demonstrate its practicality by defining and computing various infinite representations of real numbers, including partial Gray codes.

2010 Mathematics Subject Classification: 03F55, 03D78, 03B38, 03B70
Key Words and Phrases: Intuitionistic fixed point logic, dependent type theory, real number computation, coq