Program and Abstracts
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- Plenary Lecture
- ⋅ 28th-O-15:20 − 16:10 Chair: Yongnam Lee (KAIST)
- ⋅ 28th-O-15:20 − 16:10 Birational geometry of Fano varieties (Jihun Park)
- 박지훈((기초과학연구원 \& 포항공대))
Jihun Park, IBS \& POSTECH
Fano varieties are among the most significant families of varieties in the classification of algebraic varieties. They have been the subject of extensive study for the last century and continue to be a crucial area of research in algebraic geometry today.
This talk aims to provide a brief history of Fano varieties, shedding light on their evolution in the field of algebraic geometry. In addition, it will present some contemporary research issues on Fano varieties, highlighting the recent advancements in this field.
2010 Mathematics Subject Classification: 14J45
Key Words and Phrases: Fano variety
- Public Lecture
- ⋅ 27th-O-17:00 − 18:00 Chair: Wanmo Kang (KAIST)
- ⋅ 27th-O-17:00 − 18:00 생성 인공지능 (Generative AI) (Jinwoo Shin)
- 신진우((카이스트))
Jinwoo Shin, KAIST
최근 GPT, Stable Diffusion 등과 같이 다양한 인공지능 분야 (영상, 언어, 음성 등)에서 생성 인공지능 모델과 관련 기술이 크게 각광을 받고 있다. 본 강연에서는 이러한 생성 모델의 기본이 되는 수학적인 원리를 소개하고, 이들을 활용한 다양한 응용 사례를 소개한다. 또한 이러한 생성 인공지능 기술이 인간수준의 지능을 가진 인공지능을 달성하기 위해 가지는 역할에 대해서 이야기한다.
2010 Mathematics Subject Classification:
Key Words and Phrases: 기계학습, 딥러닝
- Special Invited Lectures
- ⋅ 28th-O-16:20 − 17:00 Chair: Yoonweon Lee (Inha University)
- ⋅ 28th-O-16:20 − 17:00 {\scriptsize \rm[2022년 대한수학회 학술상 수상강연]} Renormalized volume and Liouville action (Jinsung Park)
- 박진성((고등과학원))
Jinsung Park, KIAS
The renormalized volume is an invariant for convex cocompact hyperbolic 3-manifolds
and the Liouville action is an invariant for conformal boundary Riemann surface of the
convex cocompact hyperbolic 3-manifolds. In this talk, some basic results including their relationship are explained.
2010 Mathematics Subject Classification: 32Q45, 57Q10, 58J28
Key Words and Phrases: Renormalized volume, Liouville action, hyperbolic 3-manifold
- ⋅ 28th-O-17:00 − 17:40 Chair: Beomjun Choi (POSTECH)
- ⋅ 28th-O-17:00 − 17:40 Well-posedness of mean curvature flow (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits, Brian White)
- 최경수*((고등과학원)), Robert Haslhofer((University of Toronto)), Or Hershkovits((The Hebrew University of Jerusalem)), Brian White((Stanford University))
Kyeongsu Choi*, KIAS, Robert Haslhofer, University of Toronto, Or Hershkovits, The Hebrew University of Jerusalem, Brian White, Stanford University
The mean curvature flow is an evolution of surfaces satisfying a geometric heat equation. The flow develops singularities, and changes the topology of the surfaces through singularities. Thus, the flow should be well-defined even in space-time neighborhoods of singularities.
In this talk, we discuss about how to define the flow at singularities by using surgery and weak solutions.
2010 Mathematics Subject Classification: 53E10, 35K93
Key Words and Phrases: Mean curvature flow, parabolic partial differential equations
- Invited Lectures
- ⋅ 29th-O-10:40 − 11:20 Chair: Young-Hoon Kiem (KIAS)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Algebra]} Descent in derived algebraic geometry (Chang-Yeon Chough)
- 조창연((서강대))
Chang-Yeon Chough, Sogang University
Among many different ways to introduce derived algebraic geometry is an interplay between ordinary algebraic geometry and homotopy theory. The infinity-category theory, as a manifestation of homotopy theory, supplies better descent results even for ordinary algebro-geometric objets, not to mention objects of interest in the derived setting. I’ll explain what this means in the first half. The second half will be devoted to my recent work on some excision and descent results for commutative ring spectra, generalizing Milnor excision for perfect complexes of ordinary commutative rings and $v$-descent for perfect complexes of locally noetherian derived stacks by Halpern-Leistner and Preygel, respectively. No prior experience on derived algebraic geometry is required for the talk.
2010 Mathematics Subject Classification: 18F20, 19E08, 14A30, 18N60, 55P43
Key Words and Phrases: Descent, excision, perfect complexes, infinity-categories, commutative ring spectra
- ⋅ 29th-O-11:30 − 12:10 Chair: Young-Hoon Kiem (KIAS)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Algebra]} Super Hamiltonian system in an algebraic framework (Sylvain Carpentier, Uhi Rinn Suh)
- Sylvain Carpentier((서울대)), 서의린*((서울대))
Sylvain Carpentier, Seoul National University, Uhi Rinn Suh*, Seoul National University
In an algebraic view point, a Hamiltonian system is an evolution equation described by a Poisson algebra structure. Particularly, in infinite dimensional cases, Hamiltonian systems can be investigated via a special type of Poisson algebras called Poisson vertex algebras. Moreover, if the system additionally has supersymmetries, we need to consider Poisson vertex algebra with supersymmetries, namely SUSY Poisson vertex algebra. In this talk, I will explain basic notions and properties of super Hamiltonian systems using the language of Poisson algebras.
2010 Mathematics Subject Classification: 35Q53
Key Words and Phrases: Poisson algebra, vertex algebra, Hamiltonian system, supersymmetry
- ⋅ 29th-O-10:40 − 11:20 Chair: Doyoon Kim (Korea University)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Analysis]} On dispersion managed nonlinear Schr\"{o}dinger equations (Young-Ran Lee)
- 이영란((서강대))
Young-Ran Lee, Sogang University
In this talk, the dispersion managed nonlinear Schr\"odinger equations and their averaged equations are considered. They arise naturally in modeling fiber-optics communication systems with periodically varying dispersion profile. We discuss the well-posedness, homogenization, existence and properties of ground states, and orbital stability of the set of ground states.
2010 Mathematics Subject Classification: 35A01, 35B27, 35Q51, 35Q55, 35Q60
Key Words and Phrases: Nonlinear Schr\"odinger equation, dispersion management
- ⋅ 29th-O-11:30 − 12:10 Chair: Hun Hee Lee (Seoul National University)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Analysis, 2022년 대한수학회 상산젊은수학자상 수상강연]} Fourier analysis on groups (Sang-Gyun Youn)
- 윤상균((서울대))
Sang-Gyun Youn, Seoul National University
One of the central topics in abstract harmonic analysis is Fourier analysis. The Fourier transforms on groups can be explained using convolution operators, and their analytic properties are heavily influenced by the underlying spaces and their algebraic structures. For example, the Fourier coefficients of integrable functions on non-abelian compact groups are no longer numbers, but they are matrices. The main aim of this talk is to provide an introduction to abstract harmonic analysis on groups, including the Plancherel theorem, the Hausdorff-Young inequality, Hardy-Littlewood inequalities, and Sobolev embedding properties.
2010 Mathematics Subject Classification: 43A15
Key Words and Phrases: Abstract harmonic analysis, group, Fourier analysis, convolution operator
- ⋅ 29th-O-10:40 − 11:20 Chair: Juncheol Pyo (Pusan National University)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Geometry]} Surface theories in the subspaces of the four-dimensional Lorentz space (Seong-Deog Yang)
- 양성덕((고려대))
Seong-Deog Yang, Korea University
In this talk, we review the surface theories in the 6 subspaces of the four-dimensional Lorentz space $\mathbb{L}^4$, which are the Euclidean three-space $\mathbb{E}^3$, hyperbolic three-space $\mathbb{H}^3(-1)$, Lorentzian three-space $\mathbb{L}^3$, de Sitter three-space $\mathbb{S}^3_1(1)$, the isotropic three-space $\mathbb{I}^3$, and the light cone three-space $\mathbb{Q}^3+$. They belong to
$$
\{ Riemannian, \ Lorentzian, \ degenerate \} \times \{ \ hyperplane, \ hypersphere \}.
$$
In particular we examine the structure of the various Bjorling representation formula for CMC 0 or 1 surfaces in them.
2010 Mathematics Subject Classification: Primary 53A10; Secondary 53B30
Key Words and Phrases: Subspaces of Lorentzian four-space, constant mean curvature, Bjorling representation formula
- ⋅ 29th-O-11:30 − 12:10 Chair: Se-Goo Kim (Kyung Hee University)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Topology]} Cappell-Shaneson homotopy 4-spheres (Min Hoon Kim)
- 김민훈((경북대))
Min Hoon Kim, Kyungpook National University
In 1976, Cappell and Shaneson constructed an infinite family of smooth homotopy 4-spheres, called Cappell-Shaneson homotopy 4-spheres.
Cappell-Shaneson homotopy 4-spheres are the most notable potential counterexamples to the smooth 4-dimensional Poincare conjecture and they are
related to other important conjectures including the Gluck, Schoenflies, slice-ribbon conjectures. In this talk, I will give a survey on
Cappell-Shaneson homotopy 4-spheres.
2010 Mathematics Subject Classification: 57N13
Key Words and Phrases: Homotopy 4-spheres
- ⋅ 29th-O-11:30 − 12:10 Chair: Ildoo Kim (Korea University)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Probability and Statistics]} Advances in the score-based generative models: Theory and application (Sungbin Lim)
- 임성빈((고려대))
Sungbin Lim, Korea University
Diffusion models have recently acquired significant attention in the field of generative modeling of machine learning research due to their various theoretical advantages and remarkable applications in artificial intelligence, such as Stable Diffusion and DALL-E. In this presentation, we first introduce the theoretical background of the score-based diffusion models and present the latest results of their applications to machine learning. We also present advanced score-based generative models based on the time reversal theory of L\'evy processes and diffusion theory in Hilbert space.
2010 Mathematics Subject Classification: 60J70 , 68T07, 60H15
Key Words and Phrases: Generative model, stochastic differential equation, machine learning
- ⋅ 29th-O-10:40 − 11:20 Chair: Seungil Kim (Kyung Hee University)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Applied Mathematics(including AI, Data Science), 2022년 대한수학회 상산젊은수학자상 수상강연]} Active Neuron least squares$:$ A training method for multivariate 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, and operator learning.
2010 Mathematics Subject Classification: 65K10, 90C30, 37N30
Key Words and Phrases: Neural networks, plateau phenomenon, network training and optimization
- ⋅ 29th-O-11:30 − 12:10 Chair: Seungil Kim (Kyung Hee University)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Applied Mathematics(including AI, Data Science)]} Toward bridging a connection between machine learning and applied mathematics (Youngjoon Hong)
- 홍영준((성균관대))
Youngjoon Hong, Sungkyunkwan University
This lecture explores the topics and areas that have guided my research in computational mathematics and deep learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and deep neural networks have achieved state-of-the-art results in a range of fields. The rapid expansion and outstanding success of deep learning and scientific computing have led to their applications across multiple disciplines. In this lecture, I will focus on connecting machine learning with applied mathematics, specifically discussing topics such as adversarial examples, generative models, and scientific machine learning.
2010 Mathematics Subject Classification: 68T07, 65N99, 65Z05, 78M32
Key Words and Phrases: Numerical analysis, machine learning, scientific machine learning, deep generative model
- ⋅ 29th-O-10:40 − 11:20 Chair: Jeongwook Chang (Dankook University)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Mathematical Education]} Why abduction is important in mathematics education? (Jihyun Hwang)
- 황지현((한국교원대))
Jihyun Hwang, Korea National University of Education
Abduction, the process of making based on observations, is a critical component of mathematics education. It enables students to formulate conjectures or theories that can be tested and proven through deductive reasoning. This is essential for mathematical modeling and proving, as abduction is often used to explore and discover the underlying structure of a problem, while deduction is used to confirm the results. However, students often struggle with abductive reasoning, as it requires a high level of cognitive flexibility and creativity. Therefore, it is important for educators to emphasize the importance of abductive reasoning in mathematics education and provide students with opportunities to develop this skill. Students who can do abductive inferences are more likely to have higher mathematics achievement in tests, as well as develop problem-solving skills and a sense of curiosity and inquiry. In addition, epistemic actions, such as exploring possibilities, and making hypotheses, are considered to be the top of the hierarchy of epistemic actions. By incorporating these actions into mathematics education, educators can help students develop a deeper understanding of mathematical concepts and become more effective problem solvers.
2010 Mathematics Subject Classification: 97C10
Key Words and Phrases: Abduction, deductio, mathematical modeling, epistemic actions, achievement
- ⋅ 29th-O-10:40 − 11:20 Chair: Sangwook Kim (Chonnam National University)
- ⋅ 29th-O-10:40 − 11:20 {\scriptsize \rm[Discrete Mathematics]} Combinatorial problems in convexity spaces (Andreas Holmsen)
- Andreas Holmsen((카이스트))
Andreas Holmsen, KAIST
The $(p,q)$ theorem is a famous generalization of Helly’s theorem on the intersections of convex sets. In the last two decades there has been great interest in generalizing the $(p,q)$ theorem to more general set systems. In this talk we review the recent developments in this area.
2010 Mathematics Subject Classification: 52A01, 52A35
Key Words and Phrases: Helly's theorem, geometric transversal theory, abstract convexity
- ⋅ 29th-O-11:30 − 12:10 Chair: Jooyoung Lee (KAIST)
- ⋅ 29th-O-11:30 − 12:10 {\scriptsize \rm[Cryptography]} 양자내성암호연구단 소개 (Nari Lee)
- 이나리((국가보안기술연구소))
Nari Lee, National Security Research Institute
1994년에 P. Shor가 소개한 양자 알고리즘은 양자컴퓨팅을 사용하여 다항식 시간 내에 인수분해나 이산대수 문제를 해결할 수 있는 것으로 알려져 있어 양자컴퓨터가 실용화된다면 기존 공개키암호에 심각한 위협이 될 수 있다. 양자컴퓨팅을 이용한 공격에도 안전할 것으로 기대되는 암호를 ‘양자내성암호’라고 한다. 2017년 시작된 미국 표준화 기구 NIST의 표준화 공모사업을 필두로 많은 국가에서 양자내성암호의 개발 및 표준화에 힘쓰고 있으며, 국내에서도 이와 관련된 분야의 활성화가 필요한 실정이다. 본 발표에서는 국가보안기술연구소가 국가정보원의 후원으로 2021년에 발족한 ‘양자내성암호연구단’의 설립 목적을 소개하고 현재까지의 연구단 주요 활동 내용과 진행사항 및 추후 계획에 대해 소개하고자 한다.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: 양자내성암호연구단, 양자내성암호, 양자컴퓨터
- Algebra
- ⋅ 28th-A-09:20 − 10:30 Chair: Uhi Rinn Suh (Seoul National University)
- ⋅ 28th-A-09:20 − 09:40 The theta cycles for modular forms modulo prime powers (Jigu Kim, Yoonjin Lee)
- 김지구*((연세대)), 이윤진((이화여대))
Jigu Kim*, Yonsei University, Yoonjin Lee, Ewha Womans University
Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers $p^m$, where $p\ge5$ and $m\ge2$. In this talk, we introduce modular forms with coefficients in a general ring and their mod $p^m$ filtrations. Then we study mod $p^m$ theta cycles. This is joint work with Yoonjin Lee.
2010 Mathematics Subject Classification: 11F33, 11F11
Key Words and Phrases: Reduction of modular forms modulo prime powers, theta cycles
- ⋅ 28th-A-09:50 − 10:10 Distribution of even and odd integers in gaps of numerical semigroups (KyeongJun Lee, Hayan Nam, Hyunsoo Cho)
- 이경준*((연세대)), 남하얀((덕성여대)), 조현수((이화여대))
KyeongJun Lee*, Yonsei University, Hayan Nam, Duksung Women's University, Hyunsoo Cho, Ewha Womans University
A numerical semigroup is a subset of the set of nonnegative integers which contains $0$ and is closed under addition, and whose complement is finite. The complement of a numerical semigroup is called the gap of a numerical semigroup. In this talk, we focus on the distribution of even and odd integers in gaps of numerical semigroups.
2010 Mathematics Subject Classification: 20M14
Key Words and Phrases: Quotient of numerical semigroups
- ⋅ 28th-A-10:10 − 10:30 Higher residues with Feynman path integral (Hoil Kim, Taejung Kim)
- 김호일*((경북대)), 김태정((한국교원대))
Hoil Kim*, Kyungpook National University, Taejung Kim, Korea National University of Education
We formulate an explicit realization of the canonical pairing in the negative cyclic homology of local and global matrix factorizations and we establish a formula of the Hirzebruch-Riemann-Roch by introducing a twisted de Rham valued Todd class.
We also characterize its relation to Saito's higher residue pairings, reproving the conjecture of Shklyarov.
We also mention its relation to Feynman path integral in physics.
2010 Mathematics Subject Classification: 13D09, 13D45, 14C17, 14C40, 14Q65
Key Words and Phrases: Higher residue, Feynman path integral, Chern character, matrix factorization, HRR formula
- ⋅ 29th-D-09:20 − 10:30 Chair: Chang-Yeon Chough (Sogang University)
- ⋅ 29th-D-09:20 − 09:40 Koll\'{a}r conjecture on weighted homogeneous surface singularities (Jaekwan Jeon, Dongsoo Shin)
- 전재관*((충남대)), 신동수((충남대))
Jaekwan Jeon*, Chungnam National University, Dongsoo Shin, Chungnam National University
J. Koll\'{a}r conjectured that every irreducible component of the deformation space of a rational singularity is parametrized by a certain partial resolution of the singularity, which is known as $P$-resolution (generally, $P$-modification).
H. Park and D. Shin propose a strategy for proving the conjecture for sandwiched singularities. A sandwiched singularity is a normal singularity that birational to $\mathbb{C}^2$. The singularity is rational and contains quotient surface singularities, weighted homogeneous surface singularities, rational singularities with reduced fundamental cycles, etc.
I will introduce the strategy and show that the conjecture holds for weighted homogeneous surface singularities under some conditions.
2010 Mathematics Subject Classification: 14D15
Key Words and Phrases: Deformation, weighted homogeneous surface singularity, sandwiched singularity, $P$-resolution
- ⋅ 29th-D-09:50 − 10:10 A new formula of the determinant tensor and bounds of its tensor rank (Jeong-Hoon Ju, Taehyeong Kim, Yeongrak Kim)
- 주정훈*((부산대)), 김태형((부산대)), 김영락((부산대))
Jeong-Hoon Ju*, Pusan National University, Taehyeong Kim, Pusan National University, Yeongrak Kim, Pusan National University
The determinant tensor $\det_n$ for $n \times n$ matrices is a classical object in linear and multilinear algebra, and its tensor rank measures the complexity of the determinant as a function. However, its tensor rank is widely unknown, in particular, for $n \geq 4$. So it is meaningful to improve the upper bound of its tensor rank. A direct method to improve the upper bound is to find a more efficient formula of $\det_n$. In this talk, we present a new formula of $\det_4$ which is derived by Least Absolute Shrinkage and Selection Operator (LASSO) technique. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system. Finally, considering some symmetries in that formula, we present a generalized formula for $\det_{2k}$ so that $Crank(\det_n) \leq rank(\det_n) \leq \frac{n!}{2^{\lfloor(n-2)/2\rfloor}}$ when the base field is not of characteristic $2$.
2010 Mathematics Subject Classification: 14N07, 15A15, 62J07
Key Words and Phrases: Determinant, tensor rank, LASSO
- ⋅ 29th-D-10:10 − 10:30 A geometry of the moduli space of Higgs pairs over an irreducible nodal curve of arithmetic genus one (Sang-Bum Yoo)
- 유상범((공주교대))
Sang-Bum Yoo, Gongju National University of Education
Higgs bundles on an elliptic curve were intensively studied by E. Franco, O. Garcia-Prada and P. E. Newstead. In this talk, we introduce their description of the moduli space of Higgs bundles on an elliptic curve and all fibers of the Hitchin map defined on the moduli space. Next, we describe the moduli space of Higgs pairs over an irreducible nodal curve of arithmetic genus one and all fibers of the Hitchin map on this moduli space, for which we use the moduli space of generalized parabolic Hitchin pairs that U. N. Bhosle constructed.
2010 Mathematics Subject Classification: 14D22, 14H60, 14E05, 14E15
Key Words and Phrases: Higgs pair, Hitchin map, irreducible nodal curve
- Analysis
- ⋅ 28th-A-09:00 − 10:30 Chair: Dong Hyun Cho (Kyonggi University)
- ⋅ 28th-A-09:00 − 09:15 On a new class of Kirchhoff equations involving the $p$-Laplacian (Seol Vin Kim, Yun-Ho Kim)
- 김설빈*((서울대)), 김연호((상명대))
Seol Vin Kim*, Seoul National University, Yun-Ho Kim, Sangmyung University
In this talk, we are concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional $p(\cdot)$-Laplacian equations of Kirchhoff-Schr\"{o}dinger type with concave--convex nonlinearities when the convex term does not require the \linebreak Ambrosetti-Rabinowitz condition. The main aim of the present talk, on a new class of the Kirchhoff term, is to discuss the multiplicity result of non-trivial solutions by using the dual fountain theorem as the main tool.
2010 Mathematics Subject Classification: 35B33, 35D30, 35J20, 35J60, 35J66
Key Words and Phrases: Fractional $p$-Laplacian, Kirchhoff function, weak solution, dual fountain theorem
- ⋅ 28th-A-09:15 − 09:30 Wavelet series expansion in Hardy spaces with approximate duals (Youngmi Hur, Hyojae Lim)
- 허영미((연세대)), 임효재*((연세대))
Youngmi Hur, Yonsei University, Hyojae Lim*, Yonsei University
In this talk, we study a wavelet series expansion in Hardy space \(H^p\), \(0<p \leq 1\), using approximate duals, the generalized concept of duals.
First, I introduce the wavelet frame operator and provide sufficient conditions to be bounded and invertible on the Hardy space \(H^p\), \(0<p \leq 1\), using the appropriately generalized Calder\'{o}n-Zygmund operator for the full range of Hardy spaces. Based on these conditions, I offer an additional computable condition for the functions to be the approximate duals. Further, I obtain the wavelet series expansion in the Hardy space \(H^p\) with the approximate duals by applying the generalized result of Frazier and Jawerth and the (non-dyadic) Littlewood-Paley theory. Finally, I show that the Mexican hat function can be used as an example for the Hardy space \(H^p\) with \(1/2<p \leq 1\).
2010 Mathematics Subject Classification: 42C40, 42C15
Key Words and Phrases: Hardy spaces, approximate duals, wavelet frame operator, wavelet series
- ⋅ 28th-A-09:30 − 09:50 A necessary and sufficient condition for the existence of global solutions to reaction-diffusion systems (Jaeho Hwang, Soon-Yeong Chung)
- 황재호*((서강대)), 정순영((서강대))
Jaeho Hwang*, Sogang University, Soon-Yeong Chung, Sogang University
In this paper, we study the existence and nonexistence of the global solution to the following nonlinear reaction-diffusion system with time-dependent reactions under the Dirichlet boundary:
\[
\begin{cases}
u_{t}=\Delta u+ (1+t)^{-\delta} e^{\alpha t}v^{p},\,\,&\mbox{ in }\Omega\times (0,t^{*}),\\
v_{t}=\Delta v+ (1+t)^{-\delta} e^{\beta t}u^{q},\,\,&\mbox{ in }\Omega\times (0,t^{*}),
\end{cases}
\]
where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with a smooth boundary $\partial \Omega$, $\alpha,\beta\in\mathbb{R}$, $p,q>0$, and $\delta \in\mathbb{R}$. The purpose of this paper is to give the critical exponent
\[
(pq)^{*}:=1+\frac{\max\{\alpha+\beta p, \beta+\alpha q,0\}}{\lambda_{1}}
\]
and the subcritical exponent
\[
(p,q)^{*}:=\begin{cases}
\delta_{\infty}+q-1,&\mbox{ if }\alpha+\beta p> \beta+\alpha q,\\
\delta_{\infty}+p+q-1,&\mbox{ if }\alpha+\beta p= \beta+\alpha q,\\
\delta_{\infty}+p-1,&\mbox{ if }\alpha+\beta p< \beta+\alpha q,
\end{cases}
\]
where $\delta_{\infty}:=\begin{cases}
\frac{1}{\delta},&\delta>0,\\
\infty,&\delta\leq 0.
\end{cases}$ and $\lambda_{1}$ is the first Dirichlet eigenvalue for the Laplace operator. In fact, the critical exponents $(pq)^{*}$ and $(p,q)^{*}$ lead a necessary and sufficient condition for the existence and nonexistence of the global solutions.
2010 Mathematics Subject Classification: 35B44, 35K51, 35K57
Key Words and Phrases: Blow-up, global existence, parabolic system
- ⋅ 28th-A-09:50 − 10:10 On weighted estimates for Haar multipliers (Daewon Chung)
- 정대원((계명대))
Daewon Chung, Keimyung University
In this talk, necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multiplies $$T_{w,\sigma}^{t}f(x)=\sum_{I\in\mathcal{D}}\sigma_I\left(\frac{w(x)}{\langle w\rangle_I}\right)^t\langle f, h_I\rangle h_I(x)\quad(t\in\mathbb{R})$$ are uniformly (on the choice of signs $\sigma$) bounded from $L^2(u)$ into $L^2(v)$ are provided. These dyadic operators have symbols $s(x,I)=\sigma_I(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\mathbb{R}$ and the frequency variable $I\in\mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $\sigma_I=\pm 1$, and $\langle w\rangle_I$ stands for the integral average of $w$ on $I$. We will also discuss about the simplified cases of $u=v=1$, $u=v$ and $\sigma=1$.
2010 Mathematics Subject Classification: 42A45
Key Words and Phrases: t-Haar multipliers, martingale transform, two-weighted inequalities
- ⋅ 28th-A-10:10 − 10:30 An evaluation formula for generalized conditional Wiener integrals with its applications (Dong Hyun Cho)
- 조동현((경기대))
Dong Hyun Cho, Kyonggi University
Let $C^{\mathbb B}[a,b]$ denote an analogue of Wiener space over paths in abstract Wiener space $\mathbb B$, the space of $\mathbb B$-valued continuous functions on $[a,b]$. In this talk, we derive a simple formula evaluating generalized conditional Wiener integrals over $C^{\mathbb B}[a,b]$ with an infinite dimensional vector-valued conditioning function. As applications, we evaluate the generalized conditional Wiener integrals of various functions containing the time integral which is interested in quantum mechanics and Feynman integral theory.
2010 Mathematics Subject Classification: 28C20, 42B10, 46G12, 46T12, 60G15
Key Words and Phrases: Abstract Wiener space, analogue of Wiener space, conditional Wiener integral, time integral, Wiener space, Wiener space over paths in abstract Wiener space
- ⋅ 29th-D-09:00 − 10:30 Chair: Changwook Yoon (Chungnam National University)
- ⋅ 29th-D-09:00 − 09:15 Multiplicity result of solutions to the double phase anisotropic variational problems with a variable exponent (JunHyuk Ahn, Yun-Ho Kim)
- 안준혁*((상명대)), 김연호((상명대))
JunHyuk Ahn*, Sangmyung University, Yun-Ho Kim, Sangmyung University
This talk is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti-Rabinowitz condition. The aim of the present talk, on a new class of superlinear term which is different from the previous related works, is to discuss the multiplicity result of non-trivial solutions by applying the dual fountain theorem as the main tool. In particular, our main result is obtained without assuming the conditions on the nonlinear term at infinity.
2010 Mathematics Subject Classification: 35B38, 35J20, 35J62
Key Words and Phrases: Double phase equations, variable exponent Orlicz-Sobolev spaces, variational methods, multiple solutions
- ⋅ 29th-D-09:15 − 09:30 A gradient flow for the Porous medium equations with Dirichlet boundary conditions (Geuntaek Seo, Dongkwang Kim, Dowan Koo)
- 서근택*((연세대)), 김동광((연세대)), 구도완((연세대))
Geuntaek Seo*, Yonsei University, Dongkwang Kim, Yonsei University, Dowan Koo, Yonsei University
We consider the gradient flow structure of the porous medium equations with nonnegative constant boundary conditions. We prove that weak solutions to the equations can be obtained by the variational steepest descent scheme by considering an entropy functional with respect to Wb2 distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94 (2010), pp.~107--130]. In addition, we establish an energy dissipation inequality.
2010 Mathematics Subject Classification: 35A15, 49K20, 49Q22
Key Words and Phrases: Wasserstein space, entropy functional, gradient flows, porous medium equation
- ⋅ 29th-D-09:30 − 09:50 A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space (Jae-Hwan Choi, Ildoo Kim, Jin Bong Lee)
- 최재환*((카이스트)), 김일두((고려대)), 이진봉((서울대))
Jae-Hwan Choi*, KAIST, Ildoo Kim, Korea University, Jin Bong Lee, Seoul National University
This talk focuses on the study of initial value problems with (time-measurable) pseudo-differential operators in weighted $L_p$-spaces. We consider initial data given in generalized Besov spaces, and the regularity assumptions on the symbols of our pseudo-differential operators depend on the dimension of the space-variable and weights. These regularity assumptions are characterized based on some properties of weights, but no regularity condition is given with respect to the time variable. Our main result shows the uniqueness, existence, and maximal regularity estimates of a solution $u$ in weighted Sobolev spaces with variable smoothness. We also emphasize that a weight given in our estimates with respect to the time variable is beyond the scope of Muckenhoupt's class.
This presentation is based on joint work with Ildoo Kim and Jin Bong Lee.
2010 Mathematics Subject Classification: 35B30, 35S05, 35B65, 47G30
Key Words and Phrases: Initial value problem, (time-measurable) pseudo-differential operator, Muckenhoupt’s weight, variable smoothess
- ⋅ 29th-D-09:50 − 10:10 On Kirchhoff-Schrodinger equations involving the $p(\cdot)$-Laplace-type operator (Yun-Ho Kim)
- 김연호((상명대))
Yun-Ho Kim, Sangmyung University
This talk is devoted to deriving several multiplicity results of nontrivial weak solutions to Kirchhoff-Schr\"{o}dinger equations involving the $p(\cdot)$-Laplace-type operator. The aims of this talk are stated as follows. First, under some conditions on a nonlinear term, we show that our problem has a sequence of infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the problem on a new class of nonlinear term. The primary tools to obtain such multiplicity results are the fountain theorem and the dual fountain theorem, respectively.
2010 Mathematics Subject Classification: 35D30, 35J20, 35J60, 35J92, 47J30
Key Words and Phrases: $p(x)$-Laplace type, variable exponent Lebesgue-Sobolev spaces, weak solution, fountain theorem, dual fountain theorem
- ⋅ 29th-D-10:10 − 10:30 Existence and asymptotic properties of aerotaxis model with the Fokker–Planck type diffusion (Changwook Yoon, Jihoon Lee)
- 윤창욱*((충남대)), 이지훈((중앙대))
Changwook Yoon*, Chungnam National University, Jihoon Lee, Chung-Ang University
In this talk, we consider Fokker–Planck type diffusion aerotaxis equations which
resemble the usual aerotaxis equations. In view of well-posedness, its own special
diffusion structure enables us to take advantage. In two dimensions, we show
the existence of global classical solutions under some appropriate conditions. In
addition, the stabilization property of the solution is studied, as time approaches
infinity. In three dimensions, we prove the existence of global weak solutions.
2010 Mathematics Subject Classification: 35K55
Key Words and Phrases: Keller–Segel
- Geometry
- ⋅ 28th-C-14:20 − 15:00 Chair: Keomkyo Seo (Sookmyung Women's University)
- ⋅ 28th-C-14:20 − 14:40 Some characterizations of ellipsoids in ${\mathbb E}^{n+1}$ and an extension of Schneider's theorem (Dong-Soo Kim)
- 김동수((전남대))
Dong-Soo Kim, Chonnam National University
Suppose that $M$ is a strictly convex and closed hypersurface in
${\mathbb E}^{n+1}$ with the origin $o$ in its interior. We consider the homogeneous function $g$ of positive degree $d$ satisfying $M=g^{-1}(1)$.
Then, for a positive number $h$ the level hypersurface $g^{-1}(h)$ of $g$ is a homothetic hypersurface of $M$ with respect to the origin $o$.
In this paper, for tangent hyperplanes $\Phi_h$ to $g^{-1}(h)$ ($0<h<1$)
we study the $(n+1)$-dimensional volume of the region enclosed by $\Phi_h$ and the hypersurface $M$, etc..
As a result, with the aid of the theorem of Blaschke and Deicke for proper affine hypersphere centered at the origin, we establish some characterizations for ellipsoids in ${\mathbb E}^{n+1}$. As a corollary, we extend Schneider's characterization for ellipsoids in ${\mathbb E}^{3}$.
Finally, for further study we raise a question for elliptic paraboloids which was originally conjectured by Golomb.
2010 Mathematics Subject Classification: 52A20, 53A05, 53A07, 53C45
Key Words and Phrases: Ellipsoid, proper affine hypersphere, volume, cone, strictly convex, homothetic hypersurface, Gauss-Kronecker curvature
- ⋅ 28th-C-14:40 − 15:00 Killing normal Jacobi operator for Hopf real hypersurfaces in complex Grassmannians of rank 2 (Hyunjin Lee, Young Jin Suh, Changhwa Woo)
- 이현진*((조선대)), 서영진((경북대)), 우창화((부경대))
Hyunjin Lee*, Chosun University, Young Jin Suh, Kyungpook National University, Changhwa Woo, Pukyong National University
As the generalized notion of parallelism for any symmetric (1,1)-type tensor field $T$
of a real hypersurface in Kaehler manifolds, we introduce the notion of cyclic parallelism. This is equal to the notion of Killing tensor field. In this talk, by using this notion with respect to the normal Jacobi operator, we want to give some classification results of Hopf real hypesurfaces in the complex Grassmannians of rank 2.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: Complex Grassmannians of rank two, complex (hyperbolic) two-plane Grassmannians, Killing tensor field, cyclic parallelism, normal Jacobi operator
- ⋅ 29th-D-09:20 − 10:30 Chair: Dae Won Yoon (Gyeongsang National University)
- ⋅ 29th-D-09:20 − 09:40 Rigidity for weighted area-minimizing hypersurface via weighted scalar curvature (Sanghun Lee, Juncheol Pyo, Sangwoo Park)
- 이상훈*((부산대)), 표준철((부산대)), 박상우((부산대))
Sanghun Lee*, Pusan National University, Juncheol Pyo, Pusan National University, Sangwoo Park, Pusan National University
In this talk, we study rigidity for weighted area-minimizing hypersurface. First, we review previous rigidity results related to scalar curvature. In particular, we focus on the splitting theorem for area-minimizing hypersurface. Second, we consider Weighted manifold and introduce our results of weighted scalar curvature rigidity.
2010 Mathematics Subject Classification: 53C20, 53A10
Key Words and Phrases: Rigidity, weighted manifold, f-area minimizing, weighted scalar curvature
- ⋅ 29th-D-09:50 − 10:10 Integrals on the intersections of horospheres in asymptotically harmonic spaces (Sinhwi Kim, JeongHyeong Park)
- 김신휘*((성균관대)), 박정형((성균관대))
Sinhwi Kim*, Sungkyunkwan University, JeongHyeong Park, Sungkyunkwan University
It is shown that volume-preserving mappings between all pairs of two points in an asymptotically harmonic manifold can be constructed. If the asymptotically harmonic manifold satisfies the visibility condition, it is also shown that the mappings which preserve distances in some directions can be constructed. We use such mappings to prove that some integrals on the intersection of horospheres are independent of the differences between the values of the corresponding Busemann functions.
2010 Mathematics Subject Classification: 53C25, 53C30
Key Words and Phrases: Asymptotically harmonic manifold, Busemann function, horosphere, visibility manifold
- ⋅ 29th-D-10:10 − 10:30 Some classification theorems of $\lambda$-translating solitons (Eungmo Nam)
- 남응모((부산대))
Eungmo Nam, Pusan National University
$\lambda$-translating solitons are natural generalizations of translating solitons which are special solutions of the mean curvature flow.
Since translating solitons often occur as some type of singularities, it is important to study ($\lambda$-)translating solitons.
Moreover, in terms of variational viewpoint, $\lambda$-translating solitons are related to isoperimetric problems.
In this talk, I will give some classification results on these solitons.
More specifically, under the constancy of the mean curvature, a $\lambda$-translating soliton must be a hyperplane or split into constant mean curvature submanifold and Euclidean line. Furthermore, under the constancy of norm of the second fundamental form, a graphical $\lambda$-translating soliton must be a hyperplane.
These results hold both on the Euclidean space and Minkowski space.
2010 Mathematics Subject Classification: 53C42
Key Words and Phrases: $\lambda$-translating solitons, classification theorems
- Topology
- ⋅ 29th-D-08:40 − 10:30 Chair: Min Hoon Kim (Kyungpook National University)
- ⋅ 29th-D-08:40 − 09:00 A topological characterization of symplectic fillings of Seifert 3-manifolds (Hakho Choi, Jongil Park)
- 최학호*((서울대)), 박종일((서울대))
Hakho Choi*, Seoul National University, Jongil Park, Seoul National University
In this talk, we discuss minimal symplectic fillings of a Seifert 3-manifold $Y$ with a canonical contact structure. After a review of the classification scheme for minimal symplectic fillings of $Y$, I’ll explain surgery descriptions of the fillings together with relations between the fillings and Milnor fibers of the normal complex surface singularity corresponding to $Y$. This is a joint work with Jongil Park.
2010 Mathematics Subject Classification: 53D05, 57R17, 32S25
Key Words and Phrases: Seifert $3$-manifold, symplectic filling, rational blowdowns, Milnor fibers
- ⋅ 29th-D-09:00 − 09:20 Rational ruled surfaces as symplectic hyperplane sections (Myeonggi Kwon, Takahiro Oba)
- 권명기*((순천대)), Takahiro Oba((Osaka University))
Myeonggi Kwon*, Sunchon National University, Takahiro Oba, Osaka University
In this talk we are interested in the following question: For a given rational ruled surface, which symplectic manifolds may contain it as a symplectic hyperplane section? We discuss some non-embeddability results on the question and its consequences in fillability of contact manifolds.
2010 Mathematics Subject Classification: 53D05
Key Words and Phrases: Symplectic hyperplane sections, symplectic fillings, contact manifolds
- ⋅ 29th-D-09:20 − 09:40 Symplectic Torelli classes with positive entropy (Joontae Kim, Myeonggi Kwon)
- 김준태*((서강대)), 권명기((순천대))
Joontae Kim*, Sogang University, Myeonggi Kwon, Sunchon National University
A topological entropy of a diffeomorphism on a smooth manifold measures its complexity in terms of volume growth. By a Torelli class of a symplectic manifold, we mean a symplectic mapping class which is homologically trivial. In this talk, we show that there is a symplectic K3 surface which admits Torelli classes with positive topological entropy. This is joint work in progress with Myeonggi Kwon.
2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Entropy, Torelli class, K3 surface, Floer homology
- ⋅ 29th-D-09:50 − 10:10 Structural stability and shadowing for flows (Keonhee Lee)
- 이건희((충남대))
Keonhee Lee, Chungnam National University
In this talk, we discuss a relationship between the structural stability and another type of shadowing (called L-shadowing) for flows. In the first part, we characterize the L-shadowing flows on compact metric spaces by using the local stable and unstable sets, and show that any L-shadowing flow admits the spectral decomposition. In the second part, we prove that any structurally stable $C^1$ vector field on a compact smooth manifold has the L-shadowing property, and study the $C^1$ interior of all L-shadowing $C^1$ vector fields. Finally we present open problems in this direction.
\begin{thebibliography}{9}
\bibitem{a1} N. H. Du, K. Lee, and N. Nguyen, {\it Structural stability and L-shadowing for flows}, preprint.
\bibitem{a2} K. Lee and N. Nguyen, {\it Spectral decomposition and $\Omega$-stability of flows with expanding measures}, J. Differential Equations {\bf 269} (2020), 7574--7604.
\bibitem{a3} K. Lee and C. A. Morales, and M. J. Pacifico, {\it Singular strange attractors beyond the boundary of hyperbolic flows}, J. Differential Equations {\bf 345} (2023), 104--129.
\end{thebibliography}
2010 Mathematics Subject Classification: 37C50, 37D20
Key Words and Phrases: Flow, hyperbolicity, L-shadowing, structural stability
- ⋅ 29th-D-10:10 − 10:30 Spectral decomposition for homeomorphisms on non-metrizable totally disconnected spaces (Jumi Oh)
- 오주미((성균관대))
Jumi Oh, Sungkyunkwan University
In this talk, we introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.
2010 Mathematics Subject Classification: 37D05, 37D20
Key Words and Phrases: Expansive, shadowing, Smale's spectral decomposition theorem
- Probability and Statistics
- ⋅ 29th-D-09:00 − 10:30 Chair: Ildoo Kim (Korea University)
- ⋅ 29th-D-09:00 − 09:20 An $L_{q}(L_{p})$-theory for parabolic equations with anisotropic non-local operators (Jae-Hwan Choi, Jaehoon Kang, Daehan Park)
- 최재환((카이스트)), 강재훈((서울대)), 박대한*((고등과학원))
Jae-Hwan Choi, KAIST, Jaehoon Kang, Seoul National University, Daehan Park*, KIAS
In this presentation, we provide an $L_{q}(L_{p})$-regularity theory for parabolic equations with non-loccal operator
$$
\partial_{t}u(t,x) = \mathcal{L}u(t,x) + f(t,x) ,\quad t>0,\, x\in \mathbb{R}^{d}.
$$
Here, the Fourier multiplier of the non-local operator $\mathcal{L}$ is not radially symmetric. For example, we can take
$$
\mathcal{L}u(x) = \sum_{i=1}^{\ell} -(-\Delta_{x_{i}})^{\delta_{i}/2}u(x), \quad x\in \mathbb{R}^{d}
$$
where $d=\sum_{i}^{\ell}d_{i}$, and $-(-\Delta_{x_{i}})^{\delta_{i}/2}$ is the (fractional) Laplacian of order $\delta_{i}\in(0,2]$ in $\mathbb{R}^{d_{i}}$, and thus the Fourier multiplier of $\mathcal{L}$ is given as $\sum_{i}^{\ell}|\xi_{i}|^{\delta_{i}}$. That is, $\mathcal{L}$ is an operator with different differentiability in each direction. We study the operator $\mathcal{L}$ under the framework of subordinate Brownian motion and Calder\'on-Zygmund theory.
2010 Mathematics Subject Classification: 35K05, 35K08, 47G20, 60G51
Key Words and Phrases: Parabolic equations, anisotropic operators, subordinate Brownian motions, heat kernel estimation
- ⋅ 29th-D-09:20 − 09:40 Solvability and H\"older regularity for stochastic Burgers' equations with time fractional derivatives and multiplicative space-time white noise (Beom-Seok Han)
- 한범석((카이스트))
Beom-Seok Han, KAIST
We propose a solvability for stochastic Burgers' equations that involve time fractional derivatives and multiplicative white noise in time and space. Consider
$$
\partial_t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^i u u_{x^i} + \partial_t^\beta\int_0^t \sigma(u)dW_t,\,t>0;\,\,u(0,\cdot) = u_0,
$$
where $\alpha\in(0,1)$, $\beta < 3\alpha/4+1/2$, and $d< 4 - 2(2\beta-1)_+/\alpha$. The derivatives $\partial_t^\alpha$ and $\partial_t^\beta$ are the Caputo fractional derivatives of order $\alpha$ and $\beta$, the process $W_t$ is an $L_2(\mathbb R^d)$-valued cylindrical Wiener process, and the coefficients $a^{ij}, b^i, c$ and $\sigma(u)$ are random.
The H\"older regularity of the solution is demonstrated. For example, for any constant $T<\infty$, small $\epsilon>0$, and almost sure $\omega\in\Omega$, we have
$$ \sup_{x\in\mathbb R^d}|u(\omega,\cdot,x)|_{C^{\left[ \frac{\alpha}{2}\left( \left( 2-(2\beta-1)_+/\alpha-d/2 \right)\wedge1 \right)+\frac{(2\beta-1)_{-}}{2} \right]\wedge 1-\epsilon}([0,T])}<\infty
$$
and
$$ \sup_{t\leq T}|u(\omega,t,\cdot)|_{C^{\left( 2-(2\beta-1)_+/\alpha-d/2 \right)\wedge1 - \epsilon}(\mathbb R^d)} < \infty.
$$
The H\"older regularity of the solution in time changes behavior at $\beta = 1/2$. For example, if $\beta\geq1/2$, then the H\"older regularity of the solution in time is $\alpha/2$ times the one in space.
2010 Mathematics Subject Classification: 35R11, 26A33, 60H15, 35R60
Key Words and Phrases: Stochastic partial differential equation, time fractional derivatives, stochastic Burgers' equation, time fractional Burgers' equation, space-time white noise, H\"older regularity
- ⋅ 29th-D-09:50 − 10:10 Heat kernel estimates for Dirichlet forms vanishing at the boundary (Soobin Cho, Panki Kim, Renming Song, Zoran Vondracek)
- 조수빈*((서울대)), 김판기((서울대)), Renming Song((University of Illinois Urbana-Cham\-paign)), Zoran Vondracek((University of Zagreb))
Soobin Cho*, Seoul National University, Panki Kim, Seoul National University, Renming Song, University of Illinois Urbana-Champaign, Zoran Vondracek, University of Zagreb
In this talk, I will discuss heat kernel estimates for jump-type Dirichlet forms and corresponding Markov jump processes with jump kernels vanishing at the boundary. I will give an overview of the results on heat kernel estimates for jump processes in spaces with boundaries. Then I will present some new features of jump processes whose jump kernel vanishes at the boundary. Examples of such models include spectral fractional Laplacians. The talk is based on joint work with Panki Kim, Renming Song, and Zoran Vondracek.
2010 Mathematics Subject Classification: Primary 60J35, 60J45; Secondary 31C25, 35K08, 60J50, 60J76
Key Words and Phrases: Markov processes, Dirichlet forms, jump kernel, killing potential, heat kernel estimates
- ⋅ 29th-D-10:10 − 10:30 Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators (Haesung Lee)
- 이해성((한국과학영재학교))
Haesung Lee, Korea Science Academy of KAIST
We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by $L^1$-uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions $C=(c_{ij})_{1 \leq i,j \leq d}$ that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding It\^{o}-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions.
2010 Mathematics Subject Classification: 31C25, 60J46, 47D60
Key Words and Phrases: $L^1$-uniqueness, invariant measure, recurrence, transience, Dirichlet forms
- Applied Mathematics(including AI, Data Science)
- ⋅ 29th-D-09:00 − 10:00 Chair: Seunggyu Lee (Korea University)
- ⋅ 29th-D-09:00 − 09:20 Portfolio optimization in an illiquid market with transaction costs and search frictions (Tae Ung Gang, Jin Hyuk Choi)
- 강태웅*((카이스트)), 최진혁((울산과학기술원))
Tae Ung Gang*, KAIST, Jin Hyuk Choi, UNIST
We consider an optimal investment problem to maximize expected power-utility of the random terminal wealth in an market with two types of illiquidity: transaction costs and search frictions. In the market model, we suppose that an investor can trade only at arrival times of a Poisson process, and pays proportional transaction costs for purchasing or selling stocks. Furthermore, the random terminal time is exponentially distributed which is independent of the Poisson process and Brownian motion. We characterize a unique optimal trading strategy in terms of buy region, no-trade region, and sell region. Furthermore, we provide asymptotic expansions on small transaction costs and small search frictions for boundaries of the no-trade region and value function. The asymptotic expansions on small transaction costs and small search frictions show that the width of no-trade region widens and the diminishing effect of the value function increases as the transaction costs increase and the search frictions decrease. Moreover, the ﬁrst order of the width of no-trade region and value reduction are represented by transaction costs parameter times search frictions parameter and transaction costs parameter times square root of search frictions parameter, respectively.
2010 Mathematics Subject Classification: 93E20
Key Words and Phrases: Stochastic control, optimal investment, transaction costs, search frictions
- ⋅ 29th-D-09:20 − 09:40 G1 Hermite interpolation method for spatial PH curves with PH planar projections (Yoonae Song, Soo Hyun Kim, Hwan Pyo Moon)
- 송윤애((동국대)), 김수현*((부산대)), 문환표((동국대))
Yoonae Song, Dongguk University, Soo Hyun Kim*, Pusan National University, Hwan Pyo Moon, Dongguk University
The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the $G^1$ Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for $G^1$ interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary $G^1$ data in general, however, we show the feasibility of the system for the $G^1$ data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.
2010 Mathematics Subject Classification: 65D17, 68U07
Key Words and Phrases: PH curve, PHoPH curve, quaternion representation, $G^1$ Hermite interpolation, Monte-Carlo simulation
- ⋅ 29th-D-09:40 − 10:00 Mixed virtual volume method for elliptic problems and flux recovery (Gwanghyun Jo, Do Young Kwak)
- 조광현*((군산대)), 곽도영((카이스트))
Gwanghyun Jo*, Kunsan National University, Do Young Kwak, KAIST
We define mixed virtual volume methods by integrating the mixed systems with judiciously chosen test functions on each element [1]. Our scheme can be converted into symmetric and positive definite (SPD) system for the primary variable. Once the primary variable is obtained, the Darcy velocity can be recovered easily on each element. The optimal error estimates are proven which are supported by numerical experiments.
\begin{thebibliography}{9}
\bibitem{b1} G. Jo and Do Y. Kwak, {\it Mixed virtual volume methods for elliptic problems}, Comput. Math. Appl. {\bf 113} (2022), 345--352.
\end{thebibliography}
2010 Mathematics Subject Classification: 65M08 , 65M60
Key Words and Phrases: Virtual element method, nonconforming virtual element method, polygonal mesh, local velocity recovery
- Discrete Mathematics
- ⋅ 29th-D-08:40 − 10:30 Chair: Sang June Lee (Kyung Hee University)
- ⋅ 29th-D-08:40 − 09:00 Orthogonal matroids over tracts (Tong Jin, Donggyu Kim)
- Tong Jin((Georgia Institute of Technology)), 김동규*((카이스트 \& 기초과학연구원 이산수학그룹))
Tong Jin, Georgia Institute of Technology, Donggyu Kim*, KAIST \& IBS-DIMAG
We generalize Baker-Bowler's theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, and orthogonal vector sets, and establish basic properties on functoriality, duality, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we prove that an orthogonal matroid is regular if and only if it is representable over $\mathbb{F}_2$ and $\mathbb{F}_3$, which was originally shown by Geelen (1996), and that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over $\mathbb{F}_3$ and $\mathbb{F}_4$.
This is joint work with Tong Jin.
2010 Mathematics Subject Classification: 05B35
Key Words and Phrases: Matroid, delta-matroid, orthogonal Grassmannian, partial field, hyperfield
- ⋅ 29th-D-09:00 − 09:20 Recognizing internally $4$-connected pinch-graphic matroids (Bertrand Guenin, Cheolwon Heo)
- Bertrand Guenin((University of Waterloo)), 허철원*((고등과학원))
Bertrand Guenin, University of Waterloo, Cheolwon Heo*, KIAS
Even-cycle matroids are elementary lifts of graphic matroids.
An even-cycle matroid is pinch-graphic if it has a signed-graph representation with a blocking pair.
We present a polynomial algorithm to check if an internally 4-connected binary matroid is pinch-graphic.
This is joint work with Bertrand Guenin at University of Waterloo.
2010 Mathematics Subject Classification: 05B35, 05C22 , 05C85
Key Words and Phrases: Binary matroid, signed graph, even-cycle matroid, pinch-graphic matroid, recognition algorithm
- ⋅ 29th-D-09:20 − 09:40 Analysis on vulnerability of the Korean power grid by distinguishing generators and substations (Jae Hyun Park, Sang June Lee, Byungchan Kim)
- 박재현*((경희대)), 이상준((경희대)), 김병찬((서울과기대))
Jae Hyun Park*, Kyung Hee University, Sang June Lee, Kyung Hee University, Byungchan Kim, Seoul National University of Science and Technology
A power grid is a graph in which vertices are generators and substations, and edges are transmission lines. Power grids have been studied without distinction between generators and substations. However, distinguishing between generators and substations are necessary for a proper understanding of power grids. Thus, we define a concept as a gs-graph by distinguishing generators and substations. As a data, we use the Korean power grid (KPG).
Next, we define corresponding measures such as efficiency and characterized path lengths, etc, with respect to gs-graphs. Then we also define new reference graphs using ER random graph and BA scale-free graph in view of gs-graphs. It turns out that the new reference graphs are more similar to the KPG than previous reference graphs in some sense.
We perform static analysis and dynamic analysis by deleting vertices from the KPG and reference graphs to measure vulnerability. For each analysis, there are 3 scenarios such as random vertex removal, degree-based vertex removal, betweenness-based vertex removal. For each scenario, we use 5 measures to understand the KPG and reference graphs and to check what is the reference graph most close to the KPG.
2010 Mathematics Subject Classification: 05C82
Key Words and Phrases: Power grid, complex analysis, random graph
- ⋅ 29th-D-09:50 − 10:10 On $Q$-integral graphs with fixed $Q$-spectral radius (Semin Oh, Jeong Rye Park, Jongyook Park, Yoshio Sano)
- 오세민*((부산대)), 박정례((경북대)), 박종육((경북대)), Yoshio Sano((University of Tsukuba))
Semin Oh*, Pusan National University, Jeong Rye Park, Kyungpook National University, Jongyook Park, Kyungpook National University, Yoshio Sano, University of Tsukuba
The signless Laplacian $Q$ of graph $G$ is the sum of the degree matrix of $G$ and the adjacency matrix of $G$. In this talk, we consider the problem of determining the $Q$-integral graphs, i.e., the graphs with integral signless Laplacian spectrum. We call the largest eigenvalues of $Q$ the $Q$-spectral radius. We find all such graphs with $Q$-spectral radius 5 and obtain partial results for the next natural case, with $Q$-spectral radius 6.
2010 Mathematics Subject Classification: 05C50, 05C85
Key Words and Phrases: Graph spectrum, integral graph, spectral radius, signless Laplacian matrix
- ⋅ 29th-D-10:10 − 10:30 Laplacian spectrum of $k$-symmetric graphs (Sunyo Moon, Hyungkee Yoo)
- 문선요((고등과학원)), 유형기*((이화여대))
Sunyo Moon, KIAS, Hyungkee Yoo*, Ewha Womans University
For some positive integer $k$,
if the finite cyclic group $\mathbb{Z}_k$ can freely act on a graph $G$,
then we say that $G$ is $k$-symmetric.
In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices.
But if a graph has a pendant vertex, then it is at most 1-connected.
In this talk, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1.
We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are integers.
2010 Mathematics Subject Classification: 15A18, 05C50
Key Words and Phrases: Laplacian eigenvalue, $k$-symmetric graph, $k$-symmetric join
- Cryptography
- ⋅ 28th-B-10:30 − 12:00 Chair: Kyung-Ah Shim (NIMS)
- ⋅ 28th-B-10:30 − 10:50 A key recovery protocol for multiparty threshold ECDSA schemes (Myungsun Kim, Sangrae Cho, Seongbong Choi, Young-Seob Cho, Soohyung Kim, Hyung Tae Lee)
- 김명선((가천대)), 조상래((한국전자통신연구원)), 최성봉*((중앙대)), 조영섭((한국전자통신연구원)), 김수형((한국전자통신연구원)), 이형태((중앙대))
Myungsun Kim, Gachon University, Sangrae Cho, Electronics and Telecommunications Research Institute, Seongbong Choi*, Chung-Ang University, Young-Seob Cho, Electronics and Telecommunications Research Institute, Soohyung Kim, Electronics and Telecommunications Research Institute, Hyung Tae Lee, Chung-Ang University
Recently, threshold ECDSA schemes have received much attention from the security community, due to the need of efficient key management in the blockchain system. For the practical use of threshold cryptosystem, a key recovery protocol is essential for users who lost their own secret shares to recover them. It was studied for a long time in the proactive secret-sharing area, but the main aim of recent studies in that area is to achieve stronger security and so they are immoderate for the currently existing threshold ECDSA schemes.
In this paper, we provide a new key recovery protocol for threshold ECDSA schemes that is secure against static corruptions by malicious adversaries, as in the common adversary model of the state-of-the-art threshold ECDSA schemes. Our proposed protocol reduces both the computational and communication costs to $O(t^2)$ from $O(t^3)$ where $t$ is the threshold of the schemes, that is, the minimum number of users required for generating a valid signature. According to our experimental results, when $t=2$ with 128-bit security, while the previous result takes 10.46 ms in total for all computations (excluding the transmission time on the network), our protocol takes 4.21 ms, which improves by a factor of about 2.48 times. The advantage of our protocol over the previous result is bigger when $t$ is larger. For example, when $t=9$ with 128-bit security, while the previous result requires 333.42 ms in total for all computations, our protocol requires 56.61 ms, which outperforms the previous result by a factor of about 5.89 times.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Recovery protocol, proactive security, threshold ECDSA, secret-sharing
- ⋅ 28th-B-10:50 − 11:10 SilverMask: face template protection with fine-grained noise-correction (Minsu Kim, Seunghun Paik, Seongae Baek, Sangyoon Shin, Sunpill Kim, Jaehong Seo)
- 김민수*((한양대)), 백승훈((한양대)), 백성애((한양대)), 신상윤((한양대)), 김선필((한양대)), 서재홍((한양대))
Minsu Kim*, Hanyang University, Seunghun Paik, Hanyang University, Seongae Baek, Hanyang University, Sangyoon Shin, Hanyang University, Sunpill Kim, Hanyang University, Jaehong Seo, Hanyang University
Face recognition is widely used in practice. As the threat of face template leakage increases, the necessity for secure face template protection arises. If we use face recognition as an authentication system, one of the major benefits is that it does not require the user to memorize any secret information such as a password. In such an application without using a secret key, tools for protecting users' templates are quite restricted. Fuzzy commitment (FC) scheme is a promising tool for this situation. However, its performance is only confirmed with relatively easy datasets and is still far from a practical solution compared to existing non-protected face recognitions. In this paper, we propose a new FC-based protection called \emph{SilverMask} that can be combined with any face recognition model trained with cosine similarity-based loss. One of the two main techniques is a new real-valued error-correcting code (ECC) with well-spread codeword set, \textit{i.e.}, a codeword set with large minimum distance between codewords. Since the well-spreadness of the codeword set is directly associated with the performance degradation of FC, our ECC itself drastically improves the performance for several benchmarks. In addition, as a second contribution, we present a new training method for minimizing performance degradation by proposing a novel loss function, called ICC loss. This loss function compensates for performance degradation due to FC-based protection methods. To analyze the performance of SilverMask using ICC loss, we experiment with four representative datasets (LFW, AgeDB-30, CFP, and IJB-C) and achieve state-of-the-art results with a 115-bit security level. In particular, our proposal improves the TAR by 38.33\% on the LFW dataset compared to the previous state-of-the-art FC-based template protections. To facilitate future research, our code is available from github.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Fuzzy commitment
- ⋅ 28th-B-11:20 − 11:40 Toward practical lattice-based proof of knowledge from Hint-MLWE (Duhyeong Kim, Dongwon Lee, Jinyeong Seo, Yongsoo Song)
- 김두형((Intel Labs)), 이동원 ((서울대)), 서진영*((서울대)), 송용수((서울대))
Duhyeong Kim, Intel Labs, Dongwon Lee, Seoul National University, Jinyeong Seo*, Seoul National University, Yongsoo Song, Seoul National University
In the last decade, zero-knowledge proof of knowledge protocols have been extensively studied to achieve active security of various cryptographic protocols. However, the existing solutions simply seek zero-knowledge for both message and randomness, which is an overkill in many applications since protocols may remain secure even if some information about randomness is leaked to the adversary.
We develop this idea to improve the state-of-the-art proof of knowledge protocols for RLWE-based public-key encryption and BDLOP commitment schemes. In a nutshell, we present new proof of knowledge protocols without using noise flooding or rejection sampling which are provably secure under a computational hardness assumption, called Hint-MLWE. We also show an efficient reduction from Hint-MLWE to the standard MLWE assumption.
Our approach enjoys the best of two worlds because it has no computational overhead from repetition (abort) and achieves a polynomial overhead between the honest and proven languages. We prove this claim by demonstrating concrete parameters and compare with previous results. Finally, we explain how our idea can be further applied to other proof of knowledge providing advanced functionality.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Zero-knowledge, proof of plaintext knowledge, BDLOP, Hint-MLWE
- ⋅ 28th-B-11:40 − 12:00 A privacy-preserving framework for monitoring the distribution process of controlled substances (Kyuhwan Lee, Hyeonbum Lee, Taeho Jung, Jaehong Seo)
- 이규환*((한양대)), 이현범((한양대)), 정태호((University of Notre dame)), 서재홍((한양대))
Kyuhwan Lee*, Hanyang University, Hyeonbum Lee, Hanyang University, Taeho Jung, University of Notre dame, Jaehong Seo, Hanyang University
We present a framework using membership proof and hierarchical blockchain architecture for monitoring the distribution process of legally controlled substances, such as narcotics or psychotropic drugs, which must be handled by authorized subjects; whose quantity should be controlled strictly.
In this framework, basically, two types of membership proofs are generated. First type of membership proof prove that identifier (ID) of subjects who transfer or take over controlled substances is included in the permission list. The other type of membership proof prove that the subjects own substances assigned unique serial numbers. These membership proofs are hided by random values of subjects and pushed to the lower layer of hierarchical blockchain.
The multiple proofs recorded in the lower layer are aggregated into one proof by subjects who can access other subjects’ ID information. Then, aggregated proofs pushed to the upper layer of hierachical blockchain. To verify one aggregated proof allows for multiple proofs in the lower layer to be verified simultaneously, increasing throughput.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Zero-knowledge proof, membership proof, blockchain, hierarchical blockchain
- ⋅ 28th-C-13:10 − 15:00 Chair: Minkyu Kim (National Security Research Institute)
- ⋅ 28th-C-13:10 − 13:30 A unified framework of HE for multiple parties with non-interactive setup (Hyesun Kwak, Dongwon Lee, Yongsoo Song, Sameer Wagh)
- 곽혜선*((서울대)), 이동원((서울대)), 송용수((서울대)), Sameer Wagh((Devron Corporation))
Hyesun Kwak*, Seoul National University, Dongwon Lee, Seoul National University, Yongsoo Song, Seoul National University, Sameer Wagh, Devron Corporation
Homomorphic Encryption (HE), first constructed in 2009, is a class of encryption schemes that enables computation over encrypted data. Variants of HE in the context of multiple parties have led to the development of two different lines of HE schemes -- Multi-Party Homomorphic Encryption (MPHE) and Multi-Key Homomorphic Encryption (MKHE). These primitives cater to different applications and each approach has its own pros and cons. At a high level, MPHE schemes tend to be much more efficient but require the set of computing parties to be fixed throughout the entire operation, frequently a limiting assumption. On the other hand, MKHE schemes tend to have poor scaling (quadratic) with the number of parties but allow us to add new parties to the joint computation anytime since they support computation between ciphertexts under different keys.
In this work, we formalize a new variant of HE called Multi-Group Homomorphic Encryption (MGHE). Stated informally, an MGHE scheme provides seamless integration between MPHE and MKHE, and combines the best of both these primitives. In an MGHE scheme, a group of parties generates a public key jointly which results in compact ciphertexts and efficient homomorphic operations, similar to MPHE. However, unlike MPHE, it also supports computations on encrypted data under different keys, a property enjoyed by MKHE schemes.
We provide a concrete construction of such an MGHE scheme from the BFV scheme.
The public key generation procedure of our scheme is fully non-interactive so that the set of computing parties does not have to be determined and no information about other parties is needed in advance of individual key generation.
At the heart of our construction is a novel refactoring of the relinearization key to avoid interaction as typically needed. We also implement our scheme and demonstrate that this generalization does not incur any additional overhead and in fact, can be more performant than existing MPHE and MKHE schemes.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Homomorphic encryption, ring learning with errors
- ⋅ 28th-C-13:30 − 13:50 IDFace: Efficient and secure identification for face images (Sunpill Kim, Seunghun Paik, Hwang Chan Woo, Dongsu Kim, Junbum Shin, Jae Hong Seo)
- 김선필((한양대)), 백승훈((한양대)), 황찬우*((한양대)), 김동수((한양대)), 신준범((CRYPTOLAB)), 서재홍((한양대))
Sunpill Kim, Hanyang University, Seunghun Paik, Hanyang University, Hwang Chan Woo*, Hanyang University, Dongsu Kim, Hanyang University, Junbum Shin, CRYPTOLAB, Jae Hong Seo, Hanyang University
Over the past decade, deep learning technology has improved not only biometric recognition systems but also the attack methods against them such as recovering original biometric data from corresponding feature templates. Although encrypting biometric templates is a natural approach for template protection, it usually prevents effective use of templates such as for searching. This restriction has been a main bottleneck for designing practical and secure identification protocols with large databases.
In this study, we propose a new efficient identification protocol with template protection, called IDFace. The IDFace is designed on the basis of two novel techniques for efficient searching on a (encrypted) biometric database with a cosine similarity metric. The first technique is a template representation transformation that sharply reduces a unit cost for matching test even before encrypting templates. The second is a space-efficient encoding that reduces wasted space from the encryption algorithm, and thus saves the number of operations on encrypted templates.
We demonstrate the effectiveness of our techniques through experiments. IDFace can identify a biometric template from among a database of 1 million encrypted templates in less than a second, which is at most 92x faster than the previous best result using fully homomorphic encryption.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Biometric template protection, face recognition, homomorphic encryption
- ⋅ 28th-C-13:50 − 14:10 Accelerating homomoprhic encryption via level-aware key-switching (Intak Hwang, Jinyeong Seo, Yongsoo Song)
- 황인탁*((서울대)), 서진영((서울대)), 송용수((서울대))
Intak Hwang*, Seoul National University, Jinyeong Seo, Seoul National University, Yongsoo Song, Seoul National University
A key-switching operation is a commonly used technique for constructing non-linear homomorphic operations in lattice-based homomorphic encryption (HE) schemes.
Although it provides useful functionality, it remains as a main performance bottleneck.
Its computational complexity mainly depends on the level of input ciphertexts and which decomposition function used during the key-switching operation.
However, the previous method by Han and Ki (CT-RSA 2020) only uses a single decomposition function for all levels, so it does not provide the optimal performance at some levels.
In this paper, we present a new key-switching algorithm for leveled HE schemes.
Compared to the previous method, our method enables one to use multiple decomposition functions. As a result, one can choose the best-performing decomposition function at each level to obtain optimal complexity. We achieve this by enlarging the class of possible decomposition functions and observing their algebraic properties.
Our method improves the performance of key-switching operation both theoretically and concretely. In asymptotic analysis, our method reduces computational overhead for NTT operations from quadratic scale to linear scale.
In concrete performance, our method achieves at most 2.4x speed-up compared to the previous method. Also, our method has an advantage in evaluating large-depth arithmetic circuits. For example, our method attains 1.5x speed-up in case of evaluating the CKKS bootstrapping circuit.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Homomorphic encryption, gadget decomposition, key switching
- ⋅ 28th-C-14:20 − 14:40 CPA-to-CCA generic transformation for lattice based KEM without random oracle (Jaeseon Kim, Hyang-Sook Lee, Jeongeun Park)
- 김재선*((이화여대)), 이향숙((이화여대)), 박정은((imec-COSIC KU Leuven))
Jaeseon Kim*, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University, Jeongeun Park, imec-COSIC KU Leuven
In the random oracle model, cryptosystems can easily be designed and proved their security. Random oracle is actually implemented using a secure hash function because it is hard to build that. But it is mistrust because secure hash function does not imply random oracle. Therefore, achieving IND-CCA secure KEM in the standard model is an active research topic even though they are less efficient so various research on this topic is being studied at the same time. We optimized one of them to be applied to lattice-based cryptosystems, and as a result, we propose the first generic transformation to convert a lattice-based IND-CPA secure KEM to IND-CCA. Our generic transformation reduced the complexity of encapsulation, the most expensive part of the previous paper down to constant from poly(n), where n is the security parameter via using uniform hash functions. Furthermore, we provide an instantiation based on the lattice problem.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Chosen ciphertext security, lattice-based cryptography, instantiation
- ⋅ 28th-C-14:40 − 15:00 Improved universal thresholdizer from threshold fully homomorphic encryption (Jung Hee Cheon, Wonhee Cho, Jiseung Kim)
- 천정희((서울대)), 조원희*((서울대)), 김지승((전북대))
Jung Hee Cheon, Seoul National University, Wonhee Cho*, Seoul National University, Jiseung Kim, Jeonbuk National University
The Universal Thresholdizer (CRYPTO'18) is a cryptographic scheme that facilitates the transformation of any cryptosystem into a threshold cryptosystem, making it a versatile tool for threshold cryptography. For instance, this primitive enables the black-box construction of a one-round threshold signature scheme based on the Learning with Error problem, as well as a one-round threshold chosen ciphertext attack-secure public key encryption, by being combined with non-threshold schemes.
The compiler is constructed in a modular fashion and includes a compact threshold fully homomorphic encryption, a non-interactive zero-knowledge proof with preprocessing, and a non-interactive commitment.
An instantiation of the Universal Thresholdizer can be achieved through the construction of a compact threshold fully homomorphic encryption. Currently, there are two threshold fully homomorphic encryptions based on linear secret sharing, with one using Shamir's secret sharing and the other using the $\{0,1\}$-linear secret sharing scheme ($\{0,1\}$-LSSS). The former fails to achieve compactness as the size of its ciphertext is $O(N\log N)$, where $N$ is the number of participants in the distributed system. Meanwhile, the latter provides compactness, with a ciphertext size of $O(\log N)$, but requires $O(N^{4.3})$ share keys on each party, leading to high communication costs.
In this paper, we propose a communication-efficient Universal Thresholdizer by revisiting the threshold fully homomorphic encryption. Our scheme reduces the number of share keys required on each party to $O(N^{2+o(1)})$ while preserving the ciphertext size of $O(\log N)$. To achieve this, we introduce a new linear secret sharing scheme called TreeSSS, which requires a smaller number of shared keys and satisfies compactness. As a result, the Threshold Fully Homomorphic Encryption underlying our linear secret sharing scheme has fewer shared keys during the setup algorithm and reduced communication costs during the partial decryption algorithm. Moreover, the construction of a Universal Thresholdizer can be achieved through the use of TreeSSS, as it reduces the number of shared keys compared to previous constructions. Additionally, TreeSSS may be of independent interest, as it improves the efficiency in terms of communication costs when used to replace $\{0,1\}$-LSSS.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: Threshold cryptography, secret sharing, fully homomorphic encryption, universal thresholdizer
- ⋅ 29th-D-09:00 − 10:30 Chair: Kyung Chul Jeong (National Security Research Institute)
- ⋅ 29th-D-09:00 − 09:20 Divide and prange: Revisiting the Prange's information set decoding for LPN over large fields (Jiseung Kim, Changmin Lee)
- 김지승*((전북대)), 이창민((고등과학원))
Jiseung Kim*, Jeonbuk National University, Changmin Lee, KIAS
The learning parity with noise (LPN) problem has been extensively utilized in classical cryptography to construct cryptographic primitives. A natural extension of the LPN problem, namely LPN over large fields, involves a large field modulus instead of a binary modulus. This variant has proven useful in constructing cryptographic primitives, such as indistinguishability obfuscations and pseudorandom correlation generators.
Despite its wide-ranging applications, the concrete security of LPN over large fields remains unclear. In practice, most primitives have employed specific parameters to ensure security against the BKW attack and its variants, which are the current best attacks for solving the standard LPN problem in asymptotic and concrete settings. The BKW attack and its variants primarily aim to reduce the dimension of a secret vector by collecting collision vectors in a binary space. However, the reduction techniques employed in these attacks are inefficient for solving LPN over large fields, as obtaining collision vectors over a large prime field $\mathbb{F}_q$ requires exponential time in $\log q$. In contrast to prevailing knowledge, the Gaussian elimination attack is currently the best attack for solving LPN over large fields.
In this study, we undertake a comprehensive analysis of the Prange's information set decoding algorithm, which is considered the simplest algorithm for solving the learning parity with noise (LPN) problem. Our focus is on providing an improved performance for the parameter regime commonly used in pseudorandom correlation generators. We present a detailed examination of the algorithm and provide a thorough analysis of its complexity. Our results show that the proposed algorithm outperforms existing methods in this parameter regime, making it a viable alternative for cryptographic applications that rely on pseudorandom correlation generators.
2010 Mathematics Subject Classification: 94A60, 11T71
Key Words and Phrases: Learning parity with noise, concrete security
- ⋅ 29th-D-09:20 − 09:40 Multi-user security of EWCDM and DWCDM (Yeongmin Lee, Jooyoung Lee, Woohyuk Chung, Wonseok Choi)
- 이영민*((카이스트)), 이주영((카이스트)), 정우혁((카이스트)), 최원석((고등과학원))
Yeongmin Lee*, KAIST, Jooyoung Lee, KAIST, Woohyuk Chung, KAIST, Wonseok Choi, KIAS
In Crypto~2016, Cogliati and Seurin proposed to enhance the security of the Wegman-Carter MAC scheme by adding an extra block cipher call to it, and the resulting MAC scheme, dubbed Encrypted Wegman-Carter with Davies-Meyer (EWCDM), has been proved to be secure beyond the birthday bound. In Crypto~2018, Datta~\textit{et al.} proposed its single-key variant, dubbed Decrypted Wegman-Carter with Davies-Meyer (DWCDM), and proved that it enjoys the same level of security as EWCDM.
In FSE~2021, Datta~\textit{et al.} proved that EWCDM and DWCDM are both secure up to $O(2^{\frac{3n}{4}})$ MAC queries and $O(2^{n})$ verification queries in the nonce respecting setting. However, their security claims have been made only in the single-user setting. By a hybrid argument, their multi-user security would degrade as the number of users grows.
In this paper, we improve their provable security in the multi-user setting. We consider two different types of metrics on the number of queries; one is the total number of queries, and the other is the number of maximum queries that an adversary can make to each user. In terms of the total number of queries, we proved that EWCDM and DWCDM are secure up to $O(2^{\frac{3n}{4}})$ MAC queries, \textit{independently of the number of users}. In terms of the maximum number of queries per user, their mu-security degrades more slowly compared to the bounds from standard hybrid arguments. For example, EWCDM is secure up to the birthday bound until the number of users reaches $2^n$. Our security proof is based on a novel combination of the chi-squared method and the coefficient-H technique.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Message authentication code, Wegman-Carter MAC, multi-user security, chi-squared method, mirror theory
- ⋅ 29th-D-09:50 − 10:10 Vectorized linear approximations for attacks on reduced variant of SNOW 3G (Jung Hee Cheon, Wonhee Cho, Minsik Kang)
- 천정희((서울대)), 조원희((서울대)), 강민식*((서울대))
Jung Hee Cheon, Seoul National University, Wonhee Cho, Seoul National University, Minsik Kang*, Seoul National University
SNOW 3G is a stream cipher designed in 2006 by ETSI/SAGE, serving in both 3GPP and 4G LTE as one of the standard algorithms for data confidentiality and integrity protection. In our work, we perform a linear cryptanalysis of $\text{SNOW 3G}^{\oplus}$, the reduced variant of SNOW 3G, in which the 32-bit adders are replaced with exclusive-OR. We first derive 24-bit and one-byte linear approximations of the finite state machine in SNOW 3G with a bias around $2^{-17.61}$ and $2^{-22.83}$, respectively. We then exploit the approximations to launch a correlation attack against $\text{SNOW 3G}^{\oplus}$ resulting in an expected complexity of $2^148$. We also explore how to launch a distinguishing attack within an expected complexity of $2^{128}$ using 24-bit linear approximations.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: Stream cipher, SNOW 3G, linear cryptanalysis
- ⋅ 29th-D-10:10 − 10:30 XOCB: Beyond-birthday-bound secure authenticated encryption mode with rate-one computation (ByeongHak Lee, Seongha Hwang, Zhenzhen Bao, Akiko Inoue, Kazuhiko Minematsu, Jooyoung Lee)
- 이병학*((삼성SDS)), 황성하((삼성SDS)), Zhenzhen Bao((Zhongguancun Laboratory)), Akiko Inoue((NEC)), Kazuhiko Minematsu((NEC)), 이주영((삼성SDS))
ByeongHak Lee*, Samsung SDS, Seongha Hwang, Samsung SDS, Zhenzhen Bao, Zhongguancun Laboratory, Akiko Inoue, NEC, Kazuhiko Minematsu, NEC, Jooyoung Lee, Samsung SDS
We present a new block cipher mode of operation for authenticated encryption (AE), dubbed XOCB, that has the following features: (1) beyond-birthday-bound (BBB) security based on the standard pseudorandom assumption of the internal block cipher if the maximum block length is sufficiently smaller than the birthday bound, (2) rate-1 computation, and (3) supporting any block cipher with any key length. Namely, XOCB has effectively the same efficiency as the seminal OCB while having stronger quantitative security without any change in the security model or the required primitive in OCB. Although numerous studies have been conducted in the past, our XOCB is the first mode of operation to achieve these multiple goals simultaneously.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: Authenticated encryption, block cipher, OCB, beyond-birthday-bound security
- Probability and Mathematical Physics
- ⋅ 28th-A-09:00 − 10:30 Chair: Nam-Gyu Kang (KIAS)
- ⋅ 28th-A-09:00 − 09:20 Intermittency and stochastic heat equations (Kunwoo Kim)
- 김건우((포항공대))
Kunwoo Kim, POSTECH
Intermittency is defined as localized bursts of high-frequency activity, and is a phenomenon commonly observed in nature. In the context of stochastic heat equations, intermittency refers to the property that the solution tends to develop tall peaks that are distributed over small regions (intermittent islands), which are separated by large areas where the solution is small (valleys). In this talk, we will discuss how to formalize intermittency mathematically, specifically by showing that intermittent islands form multi-fractals, and by providing an upper bound on the size of the valleys.
2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: Intermittency, stochastic heat equation, valleys, multi-fractal
- ⋅ 28th-A-09:20 − 09:40 Energy landscape of Ising model without external field (Seonwoo Kim, Insuk Seo)
- 김선우((서울대)), 서인석*((서울대))
Seonwoo Kim, Seoul National University, Insuk Seo*, Seoul National University
In this talk, we present our recent result on the analysis of energy landscape of the Ising model on lattice. In particular, for the Ising model with external fields, it has been known from 90s that there exists a sharp saddle structure between two metastable state. In particular, this structure can be characterized in terms of the critical droplet. On the other hand, the model without external fields known to be difficult since such a characterization is unavailable and moreover it is expected that the saddle structure is extremely complicated. We will explain our recent results on the characterization of this complicated saddle structure of the Ising model without external field.
2010 Mathematics Subject Classification: 60J28
Key Words and Phrases: Ising model, energy landscape, Gibbs measure
- ⋅ 28th-A-09:50 − 10:10 Universality of Poisson-Dirichlet law for log-correlated fields (Kyeongsik Nam, Shirshendu Ganguly)
- 남경식*((카이스트)), Shirshendu Ganguly((University of California, Berkeley))
Kyeongsik Nam*, KAIST, Shirshendu Ganguly, University of California, Berkeley
It is widely conjectured that the Poisson-Dirichlet behavior appears in the low-temperature disordered system, in a universal sense. However, this principle has been verified only for the particular models which are exactly solvable, such as the 2D Gaussian free field. In this talk, I will talk about the universal Poisson-Dirichlet behavior which holds for the general log-correlated Gaussian fields. Based on the joint work with Shirshendu Ganguly.
2010 Mathematics Subject Classification: 60G15, 82B41, 60G70, 60G55
Key Words and Phrases: Log-correlated fields, Gaussian multiplicative chaos, Poisson-Dirichlet law, freezing
- ⋅ 28th-A-10:10 − 10:30 Planar equilibrium measure problem in the quadratic fields with a point charge (Sung-Soo Byun)
- 변성수((고등과학원))
Sung-Soo Byun, KIAS
In this talk, I will introduce an equilibrium problem with logarithmic interaction under the influence of quadratic potentials with a point charge. I will present explicit shapes of the associated droplets, which particularly shows that their topology reveals a phase transition. I will then outline the idea of the proof based on the so-called conformal mapping method.
2010 Mathematics Subject Classification: 82B26
Key Words and Phrases: Planar equilibrium measure problem, Hele-Shaw flow, two-dimensional Coulomb gases, conformal mapping method
- ⋅ 28th-B-10:50 − 12:20 Chair: Insuk Seo (Seoul National University)
- ⋅ 28th-B-10:50 − 11:10 Gibbsianness of determinantal point processes and its applications (Hyun Jae Yoo)
- 유현재((한경대))
Hyun Jae Yoo, Hankyong National University
We discuss the Gibbsianness of determinantal point processes (DPPs). For some restricted class of DPPs, namely when the largest spectrum of the kernel operators are strictly less than 1, we can show that the (discrete) DPPs are Gibbs measures for certain potentials. Then we construct the Glauber and Kawasaki dynamics for the systems leaving the DPPs invariant.
2010 Mathematics Subject Classification: 82B20
Key Words and Phrases: Determinantal point processes, Gibbs measure, stochastic dynamcs
- ⋅ 28th-B-11:10 − 11:30 Phase transition of the free energy in spherical spin glass (Ji Oon Lee)
- 이지운((카이스트))
Ji Oon Lee, KAIST
Spherical spin glass model is a variant of the Sherrington-Kirkpatrick model, where the spin variable is uniformly distributed on a hypersphere. The properties of the free energy in this model, including its limit and fluctuation, can be proved by applying recent results of random matrix theory, which establishes phase transition in various models. In this talk, I will introduce a general strategy for the analysis of the free energy in spherical spin glass model and explain several distinct phase transition phenomena.
2010 Mathematics Subject Classification: 82B44, 60B20
Key Words and Phrases: Spin glass, phase transition, free energy, random matrix
- ⋅ 28th-B-11:40 − 12:00 Markov chains with slow mixing in biochemical systems (Jinsu Kim)
- 김진수((포항공대))
Jinsu Kim, POSTECH
Markov chain mixing times indicate the time needed for the probability distribution of the Markov chain to converge to its stationary distribution. The mixing time of Markov chains is not only of theoretical significance but also has practical value. It serves as a useful criterion for estimating the long-term behavior of a stochastic system using stochastic simulation algorithms. For a Markov chain on a countable state space, one main interest is the growth of the mixing times as a function of the initial condition of the Markov chain. In this presentation, we provide non-exponentially ergodic Markov chains arising in biochemical reaction systems whose mixing is slower than the rate of order $|x|^m$ for some integer $m \ge 2$. Moreover, for the sharpness of the mixing time estimation, we establish the upper bound of the first hitting times that are closely linked to the mixing times.
2010 Mathematics Subject Classification: 60J27, 92B05
Key Words and Phrases: Markov chains, mixing times, non-exponential ergodicitiy
- ⋅ 28th-B-12:00 − 12:20 Metastability of overdamped Langevin dynamics (Jungkyoung Lee, Insuk Seo, Claudio Landim)
- 이중경*((고등과학원)), 서인석((서울대)), Claudio Landim((IPAM))
Jungkyoung Lee*, KIAS, Insuk Seo, Seoul National University, Claudio Landim, IPAM
Overdamped Langevin dynamics exhibit metastability when there are multiple critical points in the potential function.
So far, Markov chain descriptions of metastable transitions between global minima on a single time scale have been studied.
In this talk, we will explain the metastability of the dynamics on multiple time scales, and briefly introduce a proof based on the resolvent equation approach.
2010 Mathematics Subject Classification: 60F10, 60J60
Key Words and Phrases: Metastability, Langevin dynamics, mathematical physics
- Recent Results on Commutative Algebra and Related Fields
- ⋅ 28th-B-10:50 − 11:30 Chair: Nam Kyun Kim (Hanbat National University)
- ⋅ 28th-B-10:50 − 11:10 On $\phi$-pure theories (Younes El Haddaoui, Hwankoo Kim, Najib Mahdou)
- Younes El Haddaoui((University S. M. Ben Abdellah Fez)), 김환구*((호서대)), Najib Mahdou((University S. M. Ben Abdellah Fez))
Younes El Haddaoui, University S. M. Ben Abdellah Fez, Hwankoo Kim*, Hoseo University, Najib Mahdou, University S. M. Ben Abdellah Fez
The notion of purity plays a fundamental role in the theory of abelian groups. Zhao introduced the concepts of nonnil relatively divisible submodules and nonnil pure submodules. In this talk, we present some additional properties of nonnil-pure submodules by defining the nonnil-pure exact sequences, which generalize the classical definition of purity due to P. M. Cohn. Next, inspired by the work of Fieldhouse, we will generalize the $\phi$-von Neumann regular rings introduced and studied by Tang, Wang, and Zhao to the modules.
2010 Mathematics Subject Classification: 13C12
Key Words and Phrases: Nonnil-pure exact sequence, nonnil-pure submodule, $\phi$-flat module, nonnil-injective module, $\phi$-u projective module, nonnil-FP-injective module, $\phi$-I-purity, $\phi$-II-purity and $\phi$-III
- ⋅ 28th-B-11:10 − 11:30 On generalized graded rings (Dongkyu Kim, Jung Wook Lim)
- 김동규*((경북대)), 임정욱((경북대))
Dongkyu Kim*, Kyungpook National University, Jung Wook Lim, Kyungpook National University
In this talk, we introduce the concept of generalized graded rings and some properties on it.
The class of generalized graded rings contains those of graded rings and generalized power series rings.
More precisely, we define three kinds of homogeneous type ideals and examine differences among these concepts.
2010 Mathematics Subject Classification: 13A02, 13A15, 13B30, 13F25
Key Words and Phrases: Generalized graded ring, homogeneous ideal, quasi-homogeneous ideal
- ⋅ 28th-B-11:40 − 12:20 Chair: Gangyong Lee (Chungnam National University)
- ⋅ 28th-B-11:40 − 12:00 Descriptions of radicals in Ore extensions (Nam Kyun Kim)
- 김남균((한밭대))
Nam Kyun Kim, Hanbat National University
In this paper, we first introduce the $\Omega$-prime ideal and the $\Omega$-prime radical of a ring $R$, to obtain connections between the prime radical of the Ore extension $R[x;\sigma,\delta]$ and the $\Omega$-prime radical of the base ring $R$.
Based on these results, we next give definitions of the $\Omega$-LS-prime ideal, the $\Omega$-strongly prime ideal and the $\Omega$-uniformly strongly prime ideal of a ring $R$ to provide formulas for the LS-prime radical, the strongly prime radical and the
uniformly strongly prime radical of the Ore extension.
2010 Mathematics Subject Classification: 16N40, 16S36
Key Words and Phrases: Ore extension, prime radical, Levitzki radical, strongly prime radical, uniformly strongly prime radical
- ⋅ 28th-B-12:00 − 12:20 Some properties of abelian C-regular rings (Chang Ik Lee, Yang Lee, Sangwon Park)
- 이창익*((부산대)), Yang Lee((Yanbian University)), 박상원((동아대))
Chang Ik Lee*, Pusan National University, Yang Lee, Yanbian University, Sangwon Park, Dong-A University
In this talk we study the structure of right regular commutators, and call a ring $R$ {\it strongly C-regular} if $ab-ba\in (ab-ba)^2R$ for any $a, b\in R$. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring $R$, (i) if $R/W(R)$ is commutative then $R$ is commutative; and (ii) every prime factor ring of $R$ is either a commutative domain or a noncommutative division ring, where $W(R)$ is the Wedderburn radical of $R$.
2010 Mathematics Subject Classification: 16E50, 16U80, 16N40, 16N20, 16N60
Key Words and Phrases: Commutator, strongly C-regular ring, right regular, commutator ideal
- ⋅ 28th-C-13:30 − 14:10 Chair: Hwankoo Kim (Hoseo University)
- ⋅ 28th-C-13:30 − 13:50 On piecewise prime modules (Gangyong Lee, S. Tariq Rizvi)
- 이강용*((충남대)), S. Tariq Rizvi((The Ohio State University))
Gangyong Lee*, Chungnam National University, S. Tariq Rizvi, The Ohio State University
An important topic of study in Ring Theory is that of prime rings and prime ideals because these
notions help provide the description of structures of rings
As the class of piecewise prime rings is one of the special class of quasi-Baer rings,
the piecewise prime rings have a general triangular matrix representation with prime rings on the
diagonal. A quasi-Baer ring is said to be
\emph{piecewise prime} (\emph{PWP}) if the ring has a complete set of
triangulating idempotents. The notion of PWP rings was introduced by
Birkenmeier-Heatherly-Kim-Park in 2000.
In this talk, we introduce and characterize a piecewise prime module (simply, a PWP module) via its endomorphism
ring. We discuss that every direct summand of a PWP module is a PWP module. It is proven that any direct sum of copies of a PWP module is always a PWP module. Last, we show that every corner ring $eRe$ of a PWP ring $R$ is also a PWP ring.
This talk is based on a joint work with S. Tariq Rizvi.
\begin{thebibliography}{9}
\bibitem{c1} G. F. Birkenmeier, H. E. Heatherly, J. Y. Kim, and J. K. Park, {\it Triangular matrix representations}, J. Algebra {\bf 230} (2000), no. 2, 558--595.
\bibitem{c2} G. Lee and S. T. Rizvi, {\it Direct sums of quasi-Baer modules}, J. Algebra {\bf 456} (2016), 76--92.
\bibitem{c3} G. Lee and S. T. Rizvi, {\it The structure of piecewise prime modules}, Commun. Algebra {\bf 48} (2020), no. 7, 2750-–2765.
\end{thebibliography}
2010 Mathematics Subject Classification: 16D70, 16S50, 16P60, 16D80
Key Words and Phrases: Prime rings, piecewise prime rings, piecewise prime module
- ⋅ 28th-C-13:50 − 14:10 Elasticity of factorizations in numerical semigroup rings (Hyun Seung Choi)
- 최현승((경북대))
Hyun Seung Choi, Kyungpook National University
An integral domain $R$ is said to be \textit{atomic} if each nonzero nonunit of $R$ is a product of finitely many irreducible elements. In that case, the \textit{elasticity} of $R$ is defined as
$$\scalebox{0.95}{$\displaystyle\rho(R)=\sup\{\tfrac{m}{n}: x_1\cdots x_n=y_1\cdots y_m, \textnormal { each } x_1,\dots, x_n, y_1,\dots, y_m\textnormal{ are irreducible elements of }R \}$}.$$
The computation of elasticity of $R$ when $R$ is a numerical semigroup ring over a field was the main topic of a series of papers by David F. Anderson et al. in 1990s. In those papers, an upper bound was given in terms of the Davenport constant of an abelian group associated to the base field and the numerical semigroup (which turned out to be the class group of $R$), and obtained the exact values of such constants for specific numerical semigroups. We will present the exact value of the Davenport constant of the class group of an arbitrary numerical semigroup ring over a field, thereby taking up where the aforementioned authors left off.
2010 Mathematics Subject Classification: 13A05, 13F15
Key Words and Phrases: Numerical semigroup rings, factorization, elasticity
- ⋅ 28th-C-14:15 − 15:00 Chair: Gyu Whan Chang (Incheon National University)
- ⋅ 28th-C-14:15 − 14:30 On zero-sum free sequences contained in random subsets of finite cyclic groups (Sang June Lee, Jun Seok Oh)
- 이상준((경희대)), 오준석*((제주대))
Sang June Lee, Kyung Hee University, Jun Seok Oh*, Jeju National University
Let $C_n$ be a cyclic group of order $n$. A sequence $S$ of length $\ell$ over $C_n$ is a sequence $S = g_1 \boldsymbol{\cdot} g_2 \boldsymbol{\cdot} \ldots \boldsymbol{\cdot} g_{\ell}$ of $\ell$ elements in $C_n$, where repetition of elements is allowed and their order is disregarded. We say that $S$ is a zero-sum sequence if $\Sigma_{i=1}^{\ell} a_i = 0$ and that $S$ is a zero-sum free sequence if $S$ contains no zero-sum subsequence. In 2000, Gao obtained a construction of all zero-sum free sequences of length $n-1-k$ over $C_n$ for $0\leq k \leq \left\lfloor \frac{n}{3} \right\rfloor$.
In this talk, we consider a generalization for a random subset of $C_n$. Let $R=R(C_n,p)$ be a random subset of $C_n$ obtained by choosing each element in $C_n$ independently with probability $p$. Let $N^R_{n-1-k}$ be the number of zero-sum free sequences of length $n-1-k$ in $R$. Also, let $N^R_{n-1-k,d}$ be the number of zero-sum free sequences of length $n-1-k$ having $d$ distinct elements in $R$. We obtain the expectations of $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ for $0\leq k\leq \left\lfloor \frac{n}{3} \right\rfloor$ and show that $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ are asymptotically almost surely (a.a.s.) concentrated around their expectations when $k$ is fixed. Moreover, we provide two ways to compute the expectations using partition numbers and a recurrence formula.
2010 Mathematics Subject Classification: 11B50, 11B30, 05D40
Key Words and Phrases: Zero-sum free sequence, Kim--Vu polynomial concentration
- ⋅ 28th-C-14:30 − 14:45 Rings and radicals related to $n$-primary (Chen Hongying, Yang Lee)
- Chen Hongying*((Pusan National University)), Yang Lee((Yanbian University))
Chen Hongying*, Pusan National University, Yang Lee, Yanbian University
This article concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal $I$ of a ring $R$ is called $n$-primary (resp., $T$-primary) provided that $AB \subseteq I$ for ideals $A,B$ of $R$ implies that $(A+I)/I$ or $(B+I)/I$ is nil of index $n$ (resp., $(A+I)/I$ or $(B+I)/I$ is nil) in $R/I$, where $n \geq 1$. It is proved that for a proper ideal $I$ of a principal ideal domain $R$, $I$ is T-primary if and only if $I$ is of the form $p^kR$ for some prime element $p$ and $k \geq 1$ if and only if $I$ is $2$-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a $T$-primary ideal $I$ of a ring $R$, $R/I$ is prime when the Wedderburn of $R/I$ is zero. In addition we provide a method of constructing strictly descending chain of $n$-primary radicals from any domain, where the $n$-primary radical of a ring $R$ means the intersection of all the $n$-primary ideals of $R$.
2010 Mathematics Subject Classification: 16D25, 16S36, 16U80
Key Words and Phrases: $n$-primary ring, $T$-primary ring, $n$-primary radical, $T$-primary radical, prime ring, polynomial ring, matrix ring
- ⋅ 28th-C-14:45 − 15:00 On $S$-$n$-absorbing ideals (Hyungtae Baek, Hyun Seung Choi)
- 백형태*((경북대)), 최현승((경북대))
Hyungtae Baek*, Kyungpook National University, Hyun Seung Choi, Kyungpook National University
In 2011, Anderson and Badawi generalized the concept of prime ideals and in 2020, Hamed and Malek generalized the concept of prime ideals using multiplicative sets.
In this talk, for a commutative ring with identity $R$ and a multiplicative subset $S$ of $R$,
we define an {\it $S$-$n$-absorbing ideals} generalizing these and examine following problems:
\begin{enumerate}
\item[(1)]
If $I$ is an $S$-$n$-absorbing ideal of $R$,
then is $IR_S$ an $n$-absorbing ideal of $R_S$?
What about the converse?
\item[(2)]
When is each ideal $I$ of $R$ disjoint from $S$ an $S$-$n_I$-absorbing ideal for some $n_I \in \mathbb{N}$?
\end{enumerate}
2010 Mathematics Subject Classification: 13A15
Key Words and Phrases: $S$-$n$-absorbing ideal
- Trends in Algebraic Geometry
- ⋅ 28th-C-13:30 − 14:50 Chair: DongSeon Hwang (IBS-CCG)
- ⋅ 28th-C-13:30 − 13:50 Characterization of algebraic varieties via absolute complexity (Dae-Won Lee)
- 이대원((이화여대))
Dae-Won Lee, Ewha Womans University
The absolute complexity of a log pair $(X,\Delta)$ is defined as $\dim X+\rho(X)-d$, where $d$ is the sum of the coefficients of $\Delta$. In this talk, we give characterizations of projective spaces and Fano type varieties via absolute complexity. Moreover, we will provide a criteria for the non-existence of lc $-(K_X+\Delta)$-minimal models by using absolute complexity.
2010 Mathematics Subject Classification: 14J45, 14E30
Key Words and Phrases: Absolute complexity, minimal model program
- ⋅ 28th-C-14:00 − 14:20 Cylinders in del Pezzo surfaces of low degrees (Jaehyun Kim)
- 김재현((이화여대))
Jaehyun Kim, Ewha Womans University
Cylinder is an $\mathbb{A}^1$-ruled Zariski open subset in a normal projective variety over some affine variety. If the boundary of cylinder is defined by an effective member in numerical class of given divisor, then the cylinder is called polar for the divisor. The existence of such a structure has deep connections to certain group actions on the corresponding affine cone. From this point of view, the ample polar cylindricity of del Pezzo surfaces of degrees at least 3 has been extensively studied in many areas. In this talk, we focus on the ample polar cylinders in smooth del Pezzo surfaces of lower degrees and explain a new method to characterize them.
2010 Mathematics Subject Classification: 14D06
Key Words and Phrases: Ample polar cylinder, Fujita invariant, Fujita rank
- ⋅ 28th-C-14:30 − 14:50 Invariant subvarieties with small dynamical degree (Yohsuke Matsuzawa, Sheng Meng, Takahiro Shibata, De-Qi Zhang, Zhong Guolei)
- Yohsuke Matsuzawa((Osaka Metropolitan University)), Sheng Meng((East China Normal University)), Takahiro Shibata((National Fisheries University)), De-Qi Zhang((National University of Singapore)), Zhong Guolei*((IBS))
Yohsuke Matsuzawa, Osaka Metropolitan University, Sheng Meng, East China Normal University, Takahiro Shibata, National Fisheries University, De-Qi Zhang, National University of Singapore, Zhong Guolei*, IBS
Given a dominant self-morphism $f$ of an algebraic variety, we consider the set of $f$-periodic subvarieties of small dynamical degree and the subset of maximal elements which are $f$-invariant. In this talk, I will give some asymptotic upper bounds of such subset in terms of the first dynamical degree and the dimension. This is a joint work with Yohsuke Matsuzawa, Sheng Meng, Takahiro Shibata and De-Qi Zhang.
2010 Mathematics Subject Classification: 14J50, 08A35, 32H50, 37B40, 11G10, 14M25
Key Words and Phrases: Small dynamical degree, periodic subvariety, algebraic group, (semi-)abelian variety, toric variety, polarized endomorphism
- ⋅ 29th-D-09:00 − 10:20 Chair: Eui Sung Park (Korea University)
- ⋅ 29th-D-09:00 − 09:20 Birational geometry of generalized Hessenberg varieties and the generalized \linebreak Shareshian-Wachs conjecture (Young-Hoon Kiem, Donggun Lee)
- 김영훈((고등과학원)), 이동건*((기초과학연구원))
Young-Hoon Kiem, KIAS, Donggun Lee*, IBS
Hessenberg varieties are subvarieties of flag varieties with interesting properties in both algebro-geometric and combinatorial perspectives. The Shareshian-Wachs conjecture connects their cohomology with the chromatic quasi-symmetric functions of the associated graphs, which are refinements of the chromatic polynomials. In this talk, we introduce generalized Hessenberg varieties and study their birational geometry via blowups. As a result, natural maps from Hessenberg varieties to projective spaces or the permutohedral varieties are decomposed into explicit blowups and projective bundle maps. As a byproduct, we also provide an elementary proof of the Shareshian-Wachs conjecture and its natural generalization. This is joint work with Prof. Young-Hoon Kiem.
2010 Mathematics Subject Classification: 14M15, 05E10
Key Words and Phrases: Hessenberg varieties, blowups, chromatic symmetric functions, Shareshian-Wachs conjecture
- ⋅ 29th-D-09:30 − 09:50 A decomposition for bounding on Castelnuovo-Mumford regularity of monomial ideals (Jaewoo Jung, Greg Blekherman)
- 정재우*((기초과학연구원 복소기하학연구단)), Greg Blekherman((School of Mathematics at Georgia Institute of Technology))
Jaewoo Jung*, IBS-CCG, Greg Blekherman, School of Mathematics at Georgia Institute of Technology
Recall that the Betti numbers of a variety (or its defining ideal) are the ranks of free modules in the minimal free resolution of the variety. The (graded) Betti numbers of square-free monomial ideals can be investigated combinatorially through the Stanley-Reisner correspondence. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the monomial ideals in terms of properties on the corresponding simplicial complexes.
In this talk, we focus on the square-free quadratic monomial ideals corresponding to simple graphs. Our graph decomposition theorem provides a bound on the regularity of a monomial ideal. By combining the theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of quadratic monomial ideals.
2010 Mathematics Subject Classification: 14P99, 05C90, 13D02
Key Words and Phrases: Stanley-Reisner correspondences, monomial ideals, Castelnuovo-Mumford regularity
- ⋅ 29th-D-10:00 − 10:20 Projective manifolds with big tangent bundles (Jeong-Seop Kim)
- 김정섭((고등과학원))
Jeong-Seop Kim, KIAS
A certain positivity of a tangent bundle restricts the geometry of the underlying projective manifold. Due to Mori, the projective spaces are the only projective manifolds with ample tangent bundles. Also, as Campana and Peternell conjectured, rational homogeneous spaces are the only known examples of Fano manifolds with nef tangent bundles. In this talk, I will introduce some recent progress on the questions of big and pseudo-effective tangent bundles, including the case of ruled varieties over smooth projective curves.
2010 Mathematics Subject Classification: 14H60
Key Words and Phrases: Tangent bundle, positivity, ruled variety
- Algebraic Number Theory and Related Topics
- ⋅ 28th-C-13:40 − 14:50 Chair: Bo-Hae Im (KAIST)
- ⋅ 28th-C-13:40 − 14:10 Refined formulas for Bloch-Kato Selmer groups of modular forms (Chan-Ho Kim)
- 김찬호((고등과학원))
Chan-Ho Kim, KIAS
We discuss certain explicit formulas for Bloch-Kato Selmer groups of (critical twists of) modular forms and the relation with the classical Tamagawa number conjecture.
2010 Mathematics Subject Classification: 11F67, 11G40, 11R23
Key Words and Phrases: Selmer groups, modular forms, Tamagawa number conjecture
- ⋅ 28th-C-14:20 − 14:50 On the Galois structure of units in totally real p-rational number fields (Zakariae Bouazzaoui, Donghyeok Lim)
- Zakariae Bouazzaoui((Moulay Ismail University)), 임동혁*((이화여대))
Zakariae Bouazzaoui, Moulay Ismail University, Donghyeok Lim*, Ewha Womans University
The theory of factor-equivalence of integral lattices gives a far-reaching relationship between the Galois module structure of units of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the group of units of $K$ by the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. By using the strong arithmetic properties of totally real $p$-rational number fields, we prove that non-abelian $p$-rational $p$-extensions of $\mathbb{Q}$ do not have Minkowski units, which extends the results of Burns to non-abelian number fields. We also study the relative Galois module structure of $E_{L}$ for varying Galois extensions $L/K$ of totally real $p$-rational number fields with a fixed Galois group $G$. In that case, we prove that there are finitely many $\mathbb{Z}_{p}[G]$-lattices $X_{i}$ such that for any $L$, $E_{L} \otimes_{\mathbb{Z}} \mathbb{Z}_{p}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n} \oplus X_{i}$ for some $X_{i}$ and $n \in \mathbb{N}$. This is a joint work with Zakariae Bouazzaoui.
2010 Mathematics Subject Classification: 11R80, 11R33
Key Words and Phrases: $p$-rational number field, Galois module structure of units, factor equivalence, regulator constant
- ⋅ 29th-D-09:20 − 10:30 Chair: Sungmun Cho (POSTECH)
- ⋅ 29th-D-09:20 − 09:50 Orbital integrals for classical Lie algebra and smoothening (Yuchan Lee, Sungmun Cho, Taeyeoup Kang)
- 이유찬*((포항공대)), 조성문((포항공대)), 강태엽((포항공대))
Yuchan Lee*, POSTECH, Sungmun Cho, POSTECH, Taeyeoup Kang, POSTECH
Orbital integral is a fundamental object of the trace formula in the context of Langlands program. A traditional method to study orbital integral is through Bruhat-Tits building or Shalika germs.
In this talk, we will propose another philosophy to study orbital integrals using smooth\-ening. Applied to $\mathfrak{gl}_{n}$, we will provide a closed formula of the orbital integral for $\mathfrak{gl}_{n}$ with $n=2,3$ and a new lower bound for a general $n$. We will also introduce our ongoing work to treat other classical Lie algebras(i.e. $\mathfrak{u}_{n}$, $\mathfrak{so}_{n}$ or $\mathfrak{sp}_{2n}$). This is a joint work with Sungmun Cho and Taeyeoup Kang.
2010 Mathematics Subject Classification: 11F72, 11S80, 14B05
Key Words and Phrases: Trace formula, orbital integral, smooth integral model
- ⋅ 29th-D-10:00 − 10:30 Primitively 2-universal $\mathbb Z$-lattices of rank 6 (Jongheun Yoon, Byeong-Kweon Oh)
- 윤종흔*((서울대)), 오병권((서울대))
Jongheun Yoon*, Seoul National University, Byeong-Kweon Oh, Seoul National University
A positive definite quadratic lattice over $\mathbb Z$ is called primitively n-universal if it primitively represents any positive definite n-ary quadratic lattice over $\mathbb Z$. In this paper, we prove that the minimal rank of a primitively 2-universal $\mathbb Z$-lattice is 6, and we provide a complete classification of a primitively 2-universal $\mathbb Z$-lattices of rank 6.
2010 Mathematics Subject Classification: 11E10, 11E20
Key Words and Phrases: Primitive n-universality
- Analytic Number Theory and Related Topics
- ⋅ 28th-A-09:10 − 10:30 Chair: Jun Ho Whang (Seoul National University)
- ⋅ 28th-A-09:10 − 09:40 Algebraic independence of the Eisenstein series and common zeros of quasimodular forms (Bo-Hae Im, Hojin Kim, Wonwoong Lee)
- 임보해((카이스트)), 김호진((카이스트)), 이원웅*((카이스트))
Bo-Hae Im, KAIST, Hojin Kim, KAIST, Wonwoong Lee*, KAIST
In 1996, Nesterenko proved the algebraic independence between the values of the Eisenstein series and the exponential function, which is a generalization of Chudnovski's result, or Mahler's result on the algebraic independence on the Eisenstein series as meromorphic functions in 1969. In this talk, We introduce how Nesterenko's argument is improved and generalized, and prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke triangle groups. We apply this result to the study on the zeros of quasimodular forms.
2010 Mathematics Subject Classification: 11J89, 11F11
Key Words and Phrases: Modular forms, quasimodular forms, algebraic independence
- ⋅ 28th-A-10:00 − 10:30 Zeta functions of integral quiver representations and their properties (Seungjai Lee, Christopher Voll)
- 이승재*((서울대)), Christopher Voll((Universit\"at Bielefeld))
Seungjai Lee*, Seoul National University, Christopher Voll, Universit\"t Bielefeld
Since Grunewald, Segal, and Smith introduced the zeta functions of groups and rings in 1988, zeta functions have become important tools in studying properties of various algebraic structures. In this talk, we introduce zeta functions of integral quiver representations and discuss recent developments we made on this new subject. In particular, we show how they generalize and unify previous work on zeta functions of groups and rings. This is joint work with Christopher Voll.
2010 Mathematics Subject Classification: 11M41, 16G20, 17B05
Key Words and Phrases: Zeta functions of algebraic structures, Quiver representations
- ⋅ 28th-B-10:50 − 12:10 Chair: Dohyeong Kim (Seoul National University)
- ⋅ 28th-B-10:50 − 11:20 Infinite families of class groups of quadratic fields with 3-rank at least one: quantitative bounds (Siyun Lee, Yoonjin Lee, Jinjoo Yoo)
- 이시윤((이화여대)), 이윤진((이화여대)), 유진주*((울산과학기술원))
Siyun Lee, Ewha Womans University, Yoonjin Lee, Ewha Womans Univeristy, Jinjoo Yoo*, UNIST
We find a complete criterion for the 3-rank difference between the parametric families of quadratic number fields $K$ and its associated quadratic field $\widetilde{K}$ to be one. Using Scholz criteria and parametric families $K$, we improve an effective lower bound on the number of imaginary quadratic fields whose absolute discriminants are less than or equal to $X$ and whose ideal class groups have 3-rank at least one. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two. This is joint work with Siyun Lee and Yoonjin Lee.
2010 Mathematics Subject Classification: 11R29, 11R11, 11R45
Key Words and Phrases: Quadratic number field, class group, Scholz theorem, 3-rank
- ⋅ 28th-B-11:40 − 12:10 Regularity properties of Brjuno functions associated with classical continued fractions (Seul Bee Lee, Stefano Marmi)
- 이슬비*((기초과학연구원 기하학 수리물리 연구단)), Stefano Marmi((Scuola Normale Superiore))
Seul Bee Lee*, IBS-CGP, Stefano Marmi, Scuola Normale Superiore
An irrational number is called a Brjuno number if the sum of the series of $log(q_{n+1})/q_n$ converges, where $q_n$ is the denominator of the n-th principal convergent of the regular continued fraction. The importance of Brjuno numbers comes from the study of analytic small divisor problems in dimension one. In 1988, J.-C. Yoccoz introduced the Brjuno function which characterizes the Brjuno numbers to estimate the size of Siegel disks. In this talk, we introduce Brjuno-type functions associated with by-excess (negative), odd and even continued fractions. Then we discuss the $L^p$ and the Hölder regularity properties of the difference between the classical Brjuno function and the Brjuno-type functions. This is joint work with Stefano Marmi.
2010 Mathematics Subject Classification: 37F50, 37E05, 11J70, 11K50
Key Words and Phrases: Brjuno function, Brjuno condition, H\"older continuity, continued fractions
- Several Complex Variables and Related Topics
- ⋅ 28th-C-13:30 − 14:10 Chair: Kang-Hyurk Lee (Gyeongsang National University)
- ⋅ 28th-C-13:30 − 13:50 Invariant weighted Bergman metrics on bounded domains (Sungmin Yoo)
- 유성민((인천대))
Sungmin Yoo, Incheon National University
In this talk, we study the weighted Bergman kernels and metrics on bounded domains. In particular, we focus on the biholomorphically invariant weighted Bergman metrics. As special examples, we study two weighted Bergman metrics, which were introduced by Tian and Tsuji respectively. We also show the $C^0$ estimates of the kernels and metrics of them using the $L^2$ d-bar estimates.
2010 Mathematics Subject Classification: 32A36, 32F45
Key Words and Phrases: Weighted Bergman kernels, Bergman metrics, Invariant metrics
- ⋅ 28th-C-13:50 − 14:10 Exponentially weighted Bergman kernels: a survey (Soohyun Park)
- 박수현((부산대))
Soohyun Park, Pusan National University
This presentation concerns Bergman kernel estimates for holomorphic function spaces with exponential weights $e^{-\phi}$. In 1991, Christ derived kernel estimates by means of the Laplacian of the function $\phi$. We will provide a brief overview of analogous results on certain one and higher dimensional complex spaces. Then, we will discuss methods for obtaining Christ-type estimates of the kernel of the weighted Bergman space on the unit ball.
2010 Mathematics Subject Classification: 30H20, 32A25
Key Words and Phrases: Exponential weight, Bergman space, Bergman kernel
- ⋅ 28th-C-14:20 − 15:00 Chair: Young-Jun Choi (Pusan National University)
- ⋅ 28th-C-14:20 − 14:40 $L^2$ holomorphic jet extension over complex hyperbolic spaces forms with finite volume (Seungjae Lee, Aeryeong Seo)
- 이승재*((기초과학연구원)), 서애령((경북대))
Seungjae Lee*, IBS, Aeryeong Seo, Kyungpook National University
In this talk, we explore a relationship between sections of $L^2$ symmetric powers on smooth quotients of the $n$-dimensional complex unit ball under torsion-free and cofinite lattices and weighted $L^2$ holomorphic functions on certain ball fiber bundles over these quotients. Specifically, we develop an $L^2$ version of the Hodge decomposition for the $\bar \partial$-operator by applying the Donnelly-Fefferman method in conjunction with the Bochner-Kodaira-Nakano formula. This allows us to construct a weighted holomorphic function on the ball bundles from any given $L^2$ symmetric differential of degree $m \geq n+1$. This work is joint work with A. Seo of Kyungpook National University.
2010 Mathematics Subject Classification: 32A05, 32W05, 32Q05
Key Words and Phrases: Cofinite volume quotient of complex unit ball, $L^2$-Hodge decomposition for $\bar \partial$-operator, holomorphic ball bundle
- ⋅ 28th-C-14:40 − 15:00 On Nevanlinna-Pick interpolation in the unit disc (Jisoo Byun, Hyunil Choi)
- 변지수((경남대)), 최현일*((부산대))
Jisoo Byun, Kyungnam University, Hyunil Choi*, Pusan National University
Nevanlinna and Pick solved the interpolation problem independently, proving that an interpolating function exists if and only if the matrix defined in terms of given data is positive semi-definite. After Sarason's work, the Nevanlinna-Pick interpolation problem is generalized following a functional analytic way. In this talk, we present a geometric reinterpretation of Nevanlinna-Pick interpolation problem. This is joint work with J. Byun at Kyungnam University.
2010 Mathematics Subject Classification: 32E30
Key Words and Phrases: Nevanlinna-Pick interpolation, automorphism, unit disc
- ⋅ 29th-D-09:00 − 09:40 Chair: Jong-Do Park (Kyung Hee University)
- ⋅ 29th-D-09:00 − 09:20 Continuity of singular K\"ahler-Einstein potentials (Ye-Won Luke Cho, Young-Jun Choi)
- 조예원*((부산대)), 최영준((부산대))
Ye-Won Luke Cho*, Pusan National University, Young-Jun Choi, Pusan National University
In 2009, Eyssidieux-Guedj-Zeriahi showed that any compact normal K\"ahler variety with trivial or ample canonical line bundle admits a singular K\"ahler-Einstein (SKE) metric, generalizing the works of Aubin and Yau. A singular K\"ahler-Einstein potential generating the SKE metric is locally Hölder continuous on the regular locus of the variety but only little is known about its behavior near the singular locus. In this talk, we show that any SKE potential on a compact normal Kähler variety is continuous on the whole variety. This in particular generalizes the work of Guedj-Guenancia-Zeriahi (2023).
2010 Mathematics Subject Classification: 14J17, 32E10, 32Q20, 32U20
Key Words and Phrases: Degenerate complex Monge-Amp\`ere equation, compact Kähler space, pluripotential theory, singular K\"ahler-Einstein metric
- ⋅ 29th-D-09:20 − 09:40 Schwarz lemma at the boundary for the intersection of two balls (Hanjin Lee)
- 이한진((한동대))
Hanjin Lee, Handong Global University
According classic Schwarz lemma, any holomorphic self-map on the unit disc fixing the origin has a certain obstruction on the derivative at the same point. It induces a rigidity result for holomorphic self-maps for the unit disc. What if a holomorphic self-map on the unit disc fixes a boundary point where it is also holomorphic? It was Gaston Julia who asked and answered this question, which leads to Julia-Carathedory theorem, or boundary Schwarz lemma. Recently, this type of boundary Schwarz lemma has been extended to the unit ball, strongly pseudo-convex domains, convex domains of finite type and bounded symmetric domains. All these extensions assume that the boundary points are smooth. We investigate non-smooth boundary point case, in particular the intersection of two balls.
2010 Mathematics Subject Classification: 32A07, 32H02
Key Words and Phrases: Boundary Schwarz lemma, intersection of balls, Kobayashi metric
- ⋅ 29th-D-09:50 − 10:30 Chair: Hanjin Lee (Handong Global University)
- ⋅ 29th-D-09:50 − 10:10 Limit of Bergman kernels on a tower of coverings of compact K{\"a}hler manifolds (Jihun Yum, Sungmin Yoo)
- 염지훈*((기초과학연구원)), 유성민((인천대))
Jihun Yum*, IBS, Sungmin Yoo, Incheon National University
The Bergman kernel $B_X$, which is by the definition the reproducing kernel of the space of $L^2$ holomorphic $n$-forms on a $n$-dimensional complex manifold $X$, is one of the important objects in complex geometry. In this talk, we observe the asymptotics of the Bergman kernels, as well as the Bergman metric, on a tower of coverings.
More precisely, we show that, for a tower of finite Galois coverings $\{ \phi_j : X_j \rightarrow X\}$ of compact K{\"a}hler manifold $X$ converging to an infinite Galois covering $\phi : \widetilde{X} \rightarrow X$, the sequence of push-forward Bergman kernels $\phi_{j*} B_{X_j}$ locally uniformly converges to $\phi_* B_{\widetilde{X}}$.
Also, as an application, we show that sections of canonical line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into some projective space, if so do sections of $K_{\widetilde{X}}$.
This is joint work with S. Yoo at Incheon National University.
2010 Mathematics Subject Classification: 32A36
Key Words and Phrases: Bergman kernel, Bergman metric, tower of coverings
- ⋅ 29th-D-10:10 − 10:30 Holomorphicity of totally geodesic isometric embeddings into bounded symmetric domains (Aeryeong Seo, Sung-Yeon Kim)
- 서애령*((경북대)), 김성연((기초과학연구원 복소기하학연구단))
Aeryeong Seo*, Kyungpook National University, Sung-Yeon Kim, IBS-CCG
We study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon \Omega\to\Omega'$ where $\Omega$, $\Omega'$ are bounded symmetric domains, if $\Omega$ is irreducible and $\text{rank}(\Omega) \geq \text{rank}(\Omega')$ or more generally, $\text{rank}(\Omega) \geq \text{rank}(f_*v)$ for any tangent vector $v$ of $\Omega$, then $f$ is either holomorphic or anti-holomorphic. Secondly we characterize $C^1$ Kobayashi isometries from a reducible bounded symmetric domain to itself.
2010 Mathematics Subject Classification: 32F45, 32Q45, 32M15, 53C35
Key Words and Phrases: Bounded symmetric domain, Bergman metric, Kobayashi metric, totally geodesic isometric embedding, holomorphicity
- Nonlinear Dynamics: Mathematical Models in Natural and Man-made Systems
- ⋅ 28th-A-09:00 − 10:30 Chair: Woojoo Shim (KIAS)
- ⋅ 28th-A-09:00 − 09:25 Local well-posedness for the kinetic Cucker-Smale model with super-Coulombic communication weights (Young-Pil Choi, Jinwook Jung)
- 최영필((연세대)), 정진욱*((전북대))
Young-Pil Choi, Yonsei University, Jinwook Jung*, Jeonbuk National University
We consider the kinetic Cucker--Smale model with super-Coulombic communication \linebreak weights $\phi(r) = r^{-\gamma}$, $\gamma \in (d-1, d+1/4) \setminus \{d\}$. Here $d \in \mathbb{N}$ denotes the dimension of the spatial domain. By taking into account the singular communication weight as the Fourier multiplier, we establish the local-in-time well-posedness for that kinetic equation in a weighted Sobolev space. In the case of hypersingular communication weights, i.e. $\gamma \in (d, d+1/4)$, we additionally make use of the averaging lemma.
2010 Mathematics Subject Classification: 35A01, 35Q70
Key Words and Phrases: Kinetic Cucker-Smale model, singular communication weights, averaging lemma, well-posedness
- ⋅ 28th-A-09:35 − 10:00 On steady states for the Vlasov-Schr\"odinger-Poisson system (Sangdon Jin, Younghun Hong)
- 진상돈*((충북대)), 홍영훈((중앙대))
Sangdon Jin*, Chungbuk National University, Younghun Hong, Chung-Ang university
The Vlasov-Schrödinger-Poisson system is a kinetic-quantum hybrid model describing quasi-lower dimensional electron gases. In this talk, we discuss the derivation of kinetic quantum hybrid models using partial confinement. Also, for this system, we study the construction of a large class of 2D kinetic/1D quantum steady states in a bounded domain as generalized free energy minimizers, and we show their finite subband structure, monotonicity, uniqueness, and conditional dynamical stability. This talk is based on joint work with Younghun Hong (Chung-Ang University).
2010 Mathematics Subject Classification: 35B35
Key Words and Phrases: Vlasov-Schrödinger-Poisson systems, kinetic-quantum hybrid models, steady states
- ⋅ 28th-A-10:10 − 10:30 Mean curvature flow of graphs and its application to obstacle problems (Hyunsuk Kang, Ki-ahm Lee, Taehun Lee)
- 강현석((광주과학기술원)), 이기암((서울대)), 이태훈*((고등과학원))
Hyunsuk Kang, Gwangju Institute of Science and Technology, Ki-ahm Lee, Seoul National University, Taehun Lee*, KIAS
We study the evolution of complete noncompact graphs by mean curvature. The domains of deﬁnition for the graphs then evolve in time, and it turns out that this motion also follows the mean curvature ﬂow in one less dimensional space. In this talk, we develop height independent curvature estimates which allow us to transfer information from graphs to their domains. We apply these estimates to obstacle problems for the mean curvature ﬂow and establish the optimal regularity of solutions. This is joint work with Hyunsuk Kang and Ki-ahm Lee.
2010 Mathematics Subject Classification: 53E10, 53C21, 35K55
Key Words and Phrases: Mean curvature ﬂow, graph, height independent estimate
- ⋅ 28th-B-10:50 − 12:20 Chair: Myeongju Kang (KIAS)
- ⋅ 28th-B-10:50 − 11:15 A conservative semi-Lagrangian scheme for the Boltzmann equation (Sebastiano Boscarino, Seung-Yeon Cho, Giovanni Russo)
- Sebastiano Boscarino((University of Catania)), 조승연*((경상국립대)), Giovanni Russo\linebreak((University of Catania))
Sebastiano Boscarino, University of Catania, Seung-Yeon Cho*, Gyeongsang National University, Giovanni Russo, University of Catania
In this talk, we introduce a high order conservative semi-Lagrangian scheme for the Boltzmann equation. In particular, we aim to describe the rarefied gas flow. To construct an efficient method, we use semi-Lagrangain approach for treating convection term and the fast spectral method to compute the collision operator. We also adopt a conservative reconstruction and $L^2$-minimization techniques to preserve conservative quantities. Contrary to existing numerical schemes, we do not use any time-splitting approach, which enables us to attain high order accuracy with relatively less computational cost. Numerical results will be presented to demonstrate the performance of the proposed scheme. This is the joint work with two UNICT professors, G. Russo and S. Boscarino.
2010 Mathematics Subject Classification: 65L06, 65M25, 76P05
Key Words and Phrases: Boltzmann equation, semi-Lagrangian method, fast spectral method
- ⋅ 28th-B-11:20 − 11:40 Asymptotic dynamics for the Cucker-Smale model with velocity control (Junhyeok Byeon)
- 변준혁((서울대))
Junhyeok Byeon, Seoul National University
We study the collective behaviors of a second-order nonlinear consensus model with velocity control. The proposed model includes the speed-regulated, relativistic, and almost unit-speed models. Using subsystem analysis, we explore various collective behaviors of the proposed model, including mono- or bi-cluster flocking, sticking, collision avoidance, and strict spacing, depending on the regularity and singularity of the communication weight at the origin.
2010 Mathematics Subject Classification: 34D05, 34D09, 82C22
Key Words and Phrases: Consensus model, clustering, emergence
- ⋅ 28th-B-11:40 − 12:00 Relativistic BGK model for chemically reacting gas mixtures (Myeong-Su Lee, Seok-Bae Yun, Byung-Hoon Hwang)
- 이명수*((성균관대)), 윤석배((성균관대)), 황병훈((상명대))
Myeong-Su Lee*, Sungkyunkwan University, Seok-Bae Yun, Sungkyunkwan University, Byung-Hoon Hwang, Sangmyeong University
In this talk, we provide an overview of the relativistic Boltzmann equation for chemically reacting gas mixtures and its application. We discuss the physical properties of this equation, including conservation laws, equilibrium states, and the H-theorem. And then, we derive a BGK-type relaxation model to satisfy the aforementioned properties, which means that our model could be a good replacement of the relativistic Boltzmann equation in various numerical simulations.
2010 Mathematics Subject Classification: 35Q20, 35Q75, 76V05, 82C40, 83A05.
Key Words and Phrases: Kinetic equation, BGK model, relativistic gases, reactive gases, gas mixtures
- ⋅ 28th-B-12:00 − 12:20 Interplay of inertia and adaptive couplings in the emergent dynamics of Kuramoto ensemble (Hangjun Cho, Jiu-Gang Dong, Seung-Yeal Ha)
- 조항준*((서울대)), Jiu-Gang Dong((Dalian University of Technology)), 하승열((서울대))
Hangjun Cho*, Seoul National University, Jiu-Gang Dong, Dalian University of Technology, Seung-Yeal Ha, Seoul National University
In this talk, we consider the emergent dynamics of Kuramoto oscillators under the interplay between inertia and adaptive couplings. For the phase dynamics, we use the inertial Kuramoto model with time-dependent mutual coupling strengths. For the constant and uniform coupling strength, the inertial Kuramoto model can exhibit the slow relaxation dynamics toward phase-locked state under suitable conditions on system parameters and initial data. To model the time-evolution of mutual coupling strengths, we employ two types of coupling functions, namely “Hebbian coupling” and “anti-Hebbian coupling”. With these modeling spirit, the resulting coupled dynamics for the phase and mutual coupling strength becomes the coupled second-order and first-order systems of ordinary differential equations. For the proposed coupled system, we provide several sufficient frameworks for phase and frequency synchronization in terms of system parameters and initial data for Hebbian and anti-Hebbian coupling functions. This talk is based on the joint work with Jiu-Gang Dong(Dalian University of Technology) and Seung-Yeal Ha(Seoul National University).
2010 Mathematics Subject Classification: 34D05, 34D06
Key Words and Phrases: Adaptive coupling, Hebbian learning rate, Inertia, Kuramoto model synchronization
- Harmonic Analysis and Related Topics
- ⋅ 28th-C-13:30 − 15:00 Chair: Youngwoo Koh (Kongju National University)
- ⋅ 28th-C-13:30 − 13:45 A bilinear estimate associated with Dirac equation (Yonggeun Cho, Seokchang Hong, Kiyeon Lee)
- 조용근*((전북대)), 홍석창((Universit\"at Bielefeld)), 이기연((카이스트))
Yonggeun Cho*, Jeonbuk National University, Seokchang Hong, Universit\"at Bielefeld, Kiyeon Lee, KAIST
In this talk we consider a frequency localized bilinear form associated with Dirac equation.
To obtain $L^2$-biliear estimate we exploit null structure and anguar regularity
2010 Mathematics Subject Classification: 35Q55
Key Words and Phrases: Dirac equation, bilinear form, null structure, angular regularity
- ⋅ 28th-C-13:45 − 14:00 Dimension of divergence sets of oscillatory integrals with concave phase (Chu Hee Cho, Shobu Shiraki)
- 조주희*((서울대)), Shobu Shiraki((Tecnico Lisboa))
Chu Hee Cho*, Seoul National University, Shobu Shiraki, Tecnico Lisboa
We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{it(-\Delta)^\frac m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence along a non-tangential curve and a set of lines are also considered, where we find different nature from the case when $m\in(1,\infty)$. In particular, non-tangential curves are no longer regarded as vertical lines.
2010 Mathematics Subject Classification: 35Q41
Key Words and Phrases: Fractional Schr\"odinger equation, pointwise convergence, divergence sets
- ⋅ 28th-C-14:00 − 14:15 Remarks on dimension of unions of curves (Seheon Ham, Hyerim Ko, Sanghyuk Lee, Sewook Oh)
- 함세헌*((서울대)), 고혜림((서울대)), 이상혁((서울대)), 오세욱((고등과학원))
Seheon Ham*, Seoul National University, Hyerim Ko, Seoul National University, Sanghyuk Lee, Seoul National University, Sewook Oh, KIAS
We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.
2010 Mathematics Subject Classification: 42B25
Key Words and Phrases: Average over curves
- ⋅ 28th-C-14:15 − 14:30 Almost everywhere convergence of Bochner-Riesz means for the twisted Laplacian (Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu)
- 정은희*((전북대)), 이상혁((서울대)), 유재현((고등과학원))
Eunhee Jeong*, Jeonbuk National University, Sanghyuk Lee, Seoul National University, Jaehyeon Ryu, KIAS
In this talk, we are concerned with the Bochner--Riesz means $S^\delta_\mu (\mathcal L)$ for the twisted Laplacian on $\mathbb C^d$, $d\ge1$. As in the celebrated work of Carbery, Rubio de Francia, and Vega for the classical Bochner--Riesz mean, we establish some weighted $L^2$ estimate for the maximal operator $\sup_{t>0}|S^\delta_t(\mathcal L)|$. As a result, we determine the sharp range of the summability indices $\delta$ for which a.e. convergence of $S^\delta_t(\mathcal L)f$ holds for every $f\in L^p(\mathbb C^d)$, when $p\ge2$. This talk is based on the joint work with Sanghyuk Lee(Seoul National University) and Jaehyeon Ryu(KIAS).
2010 Mathematics Subject Classification: 42B15, 42B25, 42C10
Key Words and Phrases: Twisted Laplacian, almost everywhere convergence, Bochner--Riesz means
- ⋅ 28th-C-14:30 − 14:45 Carleman inequalities and unique continuation for Schrodinger operators (Eunhee Jeong, Yehyun Kwon, Sanghyuk Lee)
- 정은희((전북대)), 권예현*((창원대)), 이상혁((서울대))
Eunhee Jeong, Jeonbuk National University, Yehyun Kwon*, Changwon National University, Sanghyuk Lee, Seoul National University
We obtain a complete characterization of $L^p-L^q$ Carleman inequalities with weight $e^{v\cdot x}$ for the polyharmonic operators. Oure result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schr\"odinger equations relaxing the integrability assumption on the solution spaces.
2010 Mathematics Subject Classification: 42B15, 35B60
Key Words and Phrases: Carleman inequality, unique continuation
- ⋅ 28th-C-14:45 − 15:00 The operator splitting method for the nonlinear Schr\"odinger equation (Youngwoo Koh, Hyung Jun Choi, Seonghak Kim)
- 고영우*((공주대)), 최형준((한국기술교육대)), 김성학((경북대))
Youngwoo Koh*, Kongju National University, Hyung Jun Choi, Korea University of Technology and Education, Seonghak Kim, Kyungpook National University
In this talk, we introduce the operator splitting method for some space-time nonlinear equations, which gives an approximation solution depending only on the initial data. In this method, we divide the nonlinear equation into computable parts and consist of the approximation solution by composing these computable parts. This method is widely used for numerical computation. However, it remains a fundamental question about a rigorous analysis of the convergence between the approximation with the original solution. We introduce many works which apply harmonic analysis tools to numerical analysis. Especially, we focus on the recent result for the nonlinear Schr\"odinger equation with rough initial data in $L^2$.
2010 Mathematics Subject Classification: 35Q55, 65M15
Key Words and Phrases: Nonlinear Schr\"odinger equations, splitting method
- Elliptic and Parabolic PDE with Applications
- ⋅ 28th-A-09:00 − 09:40 Chair: Injee Jeong (Seoul National University)
- ⋅ 28th-A-09:00 − 09:20 The modified scattering for Dirac equations of scattering-critical nonlinearity (Yonggeun Cho, Soonsik Kwon, Kiyeon Lee, Changhun Yang)
- 조용근((전북대)), 권순식((카이스트)), 이기연((카이스트)), 양창훈*((충북대))
Yonggeun Cho, Jeonbuk National University, Soonsik Kwon, KAIST, Kiyeon Lee, KAIST, Changhun Yang*, Chungbuk National University
In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1}$. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Masmoudi-Shatah, as well as the spinorial null-structure. On the way, we recognize a long range interaction which is responsible for a logarithmic phase correction in the modified scattering statement.
2010 Mathematics Subject Classification: 35Q41, 35Q40, 35Q55
Key Words and Phrases: Dirac equation, modified scattering, spacetime resonance method
- ⋅ 28th-A-09:20 − 09:40 $L_p$ spaces for time-fractional evolutionary equations with non-zero initial value (Doyoon Kim, Kwan Woo)
- 김도윤((고려대)), 우관*((고려대))
Doyoon Kim, Korea University, Kwan Woo*, Korea University
Fractional calculus is a well-established field of research, and the Sobolev space theory of time-fractional equations has gained attention in recent years. Despite notable progress, some aspects of this theory remain ambiguous and need further investigation.
In this presentation, we aim to establish (weighted) $L_p$ spaces for evolutionary equations that incorporate Caputo fractional derivatives, such as time-fractional sub/super-diffusion equations. Additionally, we will discuss several noteworthy properties of the solutions to the equations. These properties significantly depend on the order of the time-fractional derivative $\partial_t^{\alpha}$, i.e., the extent of $\alpha$.
2010 Mathematics Subject Classification: 35B30, 26A33, 35R11
Key Words and Phrases: Fractional integrals and derivatives, initial trace
- ⋅ 28th-A-09:50 − 10:30 Chair: Eunkyung Ko (Keimyung University)
- ⋅ 28th-A-09:50 − 10:10 Existence of weak solutions for porous medium equation with a drift term (Sukjung Hwang)
- 황숙정((충북대))
Sukjung Hwang, Chungbuk National University
We consider degenerate porous medium equations with a divergence type of drift term representing a certain form of reaction-diffusion equation that arises naturally in many applications. The conditions on the drift term may affect the well-posedness of the diffusive parabolic equation. In this talk, we introduce suitable conditions for the drift concerning the existence of weak solutions.
2010 Mathematics Subject Classification: 35A01, 35K55
Key Words and Phrases: PME, existence, weak solution
- ⋅ 28th-A-10:10 − 10:30 Obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions (Sun-Sig Byun, Jeongmin Han, Jehan Oh)
- 변순식((서울대)), 한정민((University of Jyv\"askyl\"a)), 오제한*((경북대))
Sun-Sig Byun, Seoul National University, Jeongmin Han, University of Jyv\"askyl\"a, Jehan Oh*, Kyungpook National University
This talk concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness, and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
2010 Mathematics Subject Classification: 35J25, 35J15, 35J60
Key Words and Phrases: Fully nonlinear equations, oblique boundary data, obstacle problems
- ⋅ 28th-B-10:50 − 11:30 Chair: Sukjung Hwang (Chungbuk National University)
- ⋅ 28th-B-10:50 − 11:10 On vorticity supported on logarithmic spirals (Injee Jeong, Ayman Said)
- 정인지*((서울대)), Ayman Said((Duke University))
Injee Jeong*, Seoul National University, Ayman Said, Duke University
Abstract: We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a non linear transport system on the unit circle. We show that this system is locally well posed in $L^p$ with any $1 \le p \le \infty$ as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. Within symmetry we show that the vortex sheet limit holds locally in time. We give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and do occur.
2010 Mathematics Subject Classification: 35Q35
Key Words and Phrases: Euler equations, logarithmic spiral
- ⋅ 28th-B-11:10 − 11:30 On an iteration structure of the Chern-Simons limit (Jongmin Han, Kyungwoo Song)
- 한종민*((경희대)), 송경우((경희대))
Jongmin Han*, Kyung Hee University, Kyungwoo Song, Kyung Hee University
In this tlak, we suggest a method to show the Chern-Simons limit arising the Maxwell-Chern-Simons gauge field models. We show that solutions of the Maxwell-Chern-Simons equations converge to solutions of the Chern-Simons equations as the coupling parameter $q \to \infty$. We employ an iteration technique based on the maximum principle.
2010 Mathematics Subject Classification: 35J61, 35Q75, 81T13
Key Words and Phrases: Maxwell-Chern-Simons-Higgs model, Chern-Simons limit
- ⋅ 28th-B-11:40 − 12:20 Chair: Jehan Oh (Kyungpook National University)
- ⋅ 28th-B-11:40 − 12:00 Existence of multiple positive solutions of elliptic equations (Eunkyung Ko)
- 고은경((계명대))
Eunkyung Ko, Keimyung University
We discuss a multiplicity result for positive solutions to the Schr\"odinger-type singular problem:
$-\Delta u+V(x)u=\lambda \frac{ f(u)}{u^\beta}$ in $ \Omega, ~u=0 $ on $\partial \Omega,$
where $\Omega$ is a bounded domain in $ \mathbb{R}^N, N > 1$ with a smooth boundary $\partial \Omega$ or $\Omega=(0,1),$ $ 0 \leq \beta <1, \lambda$ is a positive parameter,
$V \in L^\infty(\Omega)$ and $f:[0, \infty) \rightarrow (0, \infty)$ is a continuous function.
In particular, when $g$ is sublinear at $ \infty$ where $g(s):= \frac{f(s)}{s^\beta}$,
we establish the existence of at least three positive solutions for a certain range of $\lambda$. The proofs are mainly based on the sub and supersolution method.
2010 Mathematics Subject Classification: 35J66, 35J20
Key Words and Phrases: Schr\"odinger-type singular problem, positive solution
- ⋅ 28th-B-12:00 − 12:20 A limit of a microscopic dweller-wanderer system to a reaction-diffusion system (Min-Gi Lee)
- 이민기((경북대))
Min-Gi Lee, Kyungpook National University
In this talk, we study a derivation of a chemotaxis model where species exhibits adaptive diffusion depending on the local information of population and food density. More specifically, the flux of population is the negative gradient of $\gamma(m)u$, where $m$ is the food density, $u$ is the population density, and $\gamma(\cdot)$ is a decreasing function. This type of flux law has attracted considerable attention in the reaction-diffusion community. Funaki, Mimura, and Urabe (2012) presented a derivation by hydrodynamic limit of a kinetic model where species may exists either as a fast diffusive or a slow diffusive mode; they convert to each other quickly to meet the quasi-equilibrium ratio between them. One small concern is that there has been an employment of asymptotic power laws $\gamma(m) = \frac{1}{1 + m^p}$ in literature while the derived $\gamma(m)$ is bounded from below by the diffusion constant of slow diffusive mode, away from 0. To fit the power laws in, one must drop the diffusion term of the slow mode, which might cause technical challenges. We show that the limit presented by Funaki, Mimura, and Urabe is still done the same with $0$ diffusive dwelling mode.
This is joint work with Kyunghan Choi.
2010 Mathematics Subject Classification: 35K57
Key Words and Phrases: Reaction-diffusion system
- ⋅ 28th-C-13:30 − 14:10 Chair: Jinhae Park (Chungnam National University)
- ⋅ 28th-C-13:30 − 13:50 Semi-classical limits for white dwarfs (Younghun Hong, Sangdon Jin, Jinmyoung Seok)
- 홍영훈*((중앙대)), 진상돈((충북대)), 석진명((서울대))
Younghun Hong*, Chung-Ang University, Sangdon Jin, Chungbuk National University, Jinmyoung Seok, Seoul National University
White dwarfs can be described as minimizers for the quantum or the kinetic energy depending on the models we use. In this talk, we introduce the mathematical settings of the problem, and show how they are related via the semi-classical limit.
2010 Mathematics Subject Classification: 35J50, 49S05, 70F15
Key Words and Phrases: Semi-classical limit
- ⋅ 28th-C-13:50 − 14:10 Liouville type problem of the steady $p$-Stokes system in the half-space (Kyung Keun Kang, Michael Ruzicka)
- 강경근*((연세대)), Michael Ruzicka((Albert-Ludwigs University))
Kyung Keun Kang*, Yonsei University, Michael Ruzicka, Albert-Ludwigs University
We study the Liouville problem for the steady $p$-Stokes system in the half-space. We prove that a bounded weak solution of the $p$-Stokes system with $p > 1$ vanishes in two dimensions. For the three dimensional case, the same result is concluded, provided that $p > \frac{5}{3}$. This is a joint work with M. Ruzicka.
2010 Mathematics Subject Classification: 35B53, 76A05, 76D07
Key Words and Phrases: Liouville type problem, $p$-Stokes type problem
- Analytical and Numerical Methods for PDEs and Integral Equations
- ⋅ 28th-A-09:20 − 10:30 Chair: Sanghyeon Yu (Korea University)
- ⋅ 28th-A-09:20 − 09:40 Exceptional points in parity-time-symmetric subwavelength metamaterials (Habib Ammari, Bryn Davies, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu)
- Habib Ammari((ETH Zurich)), Bryn Davies((Imperial College London)), Erik Orvehed \linebreak Hiltunen((Yale University)), 이현대*((인하대)), 유상현((고려대))
Habib Ammari, ETH Zurich, Bryn Davies, Imperial College London, Erik Orvehed Hiltunen, Yale University, Hyundae Lee*, Inha University, Sanghyeon Yu, Korea University
When sources of energy gain and loss are introduced to a wave-scattering system, the underlying mathematical formulation will be non-Hermitian. This paves the way for the existence of exceptional points, where eigenmodes are linearly dependent. The primary goal of this work is to study the existence of exceptional points in high-contrast subwavelength metamaterials. We begin by studying a parity-time-symmetric pair of subwavelength resonators and prove that this system supports asymptotic exceptional points. These are points at which the subwavelength eigenvalues and eigenvectors coincide at leading order in the asymptotic parameters. We then investigate further properties of parity-time-symmetric subwavelength metamaterials. First, we study the exotic scattering behaviour of a metascreen composed of repeating parity-time-symmetric pairs of subwavelength resonators. We prove that the non-Hermitian nature of this structure means that it exhibits asymptotic unidirectional reflectionless transmission at certain frequencies and demonstrate extraordinary transmission close to these frequencies. Thereafter, we consider cavities containing many small resonators and use homogenization theory to show that non-Hermitian behaviour can be replicated at the macroscale.
2010 Mathematics Subject Classification: 35J05, 35C20, 35P20
Key Words and Phrases: Exceptional points, parity-time-symmetric
- ⋅ 28th-A-09:50 − 10:10 Random walk in a spatially heterogeneous medium (Jaywan Chung, Yong-Jung Kim, Min-Gi Lee)
- 정재환((한국전기연구원)), 김용정((카이스트)), 이민기*((경북대))
Jaywan Chung, Korea Electrotechnology Research Institute, Yong-Jung Kim, KAIST, Min-Gi Lee*, Kyungpook National University
Diffusion is one of the mechanisms of transporting mass in nature. The random walk model in the microscopic picture gives a good explanation for that; from the computational point of view, an explicit finite difference scheme converges for the continuum equation. Suppose a liquid (or a gas) phase diffuses in a solid medium that is made simply by joining two different materials. Admitting that the law of diffusion is simply the Fick's Law in each side (not incorporating any nonlinear effect at this stage), it is not unrealistic to think two different crystalline structures lead to having two different pairs of walk length and walk frequency, and thus having two possibly different diffusion constants in each side. Engineers well aware of that in such a case the equilibrium state of the concentration after a long time jumps crossing the interface. This, in turn, tells us that a diffusion equation of $\partial_t u = div(D(x)\nabla u)$ fails and is too naive: For this equation, we would expect the constant equilibrium state.
The random walk in an inhomogeneous medium, in fact, abounds in many areas. For example, the celebrated Keller-Segel model assumes micro-organisms who random walk but dependently on a certain chemical concentration and the population density, which makes the system nonlinear but diverse and rich as well.
In this talk, we aim to conclude one fact, at least for the simple configuration described above, that the diffusion law has the walk frequency inside of the gradient. The study of random walk in an inhomogeneous medium is vast in literature and is quite involved, but we only use the elementary finite difference method and its (linear functional analytic) convergence study to show the conclusion.
2010 Mathematics Subject Classification: 60Kxx
Key Words and Phrases: Heterogeneous medium, random walk, diffusion
- ⋅ 28th-A-10:10 − 10:30 Super-convergence analysis of symmetric Poisson solvers with adaptive grids (Jeongho Kim, Chohong Min, Byungjoon Lee)
- 김정호((경희대)), 민조홍((이화여대)), 이병준*((가톨릭대))
Jeongho Kim, Kyung Hee University, Chohong Min, Ewha Womans University, Byungjoon Lee*, The Catholic University of Korea
The Poisson equation plays an important role in solving incompressible Navier-Stokes equations. The solution of the equations, which corresponds to the pressure of the fluid, enables us to implement the Hodge decomposition, that is a crucial feature of incompressible fluid flows. In order to simulate incompressible fluid flows in a stable manner, it is desired to utilize a Poisson solver that attains the orthogonality of the Hodge decomposition in a discrete level.
When a Poisson solver induces the orthogonality, its associated linear system is necessarily symmetric. With this regard, the well-known symmetric Poisson solvers are more advantageous not only to efficiently solving the linear system but also to stably simulating fluid flows than nonsymmetric ones. Their numerical solutions were empirically observed to be first and second order accurate, respectively. One may expect that each of their numerical gradients has convergence order that is one less than that of its numerical solution.
We in this work show that super-convergence holds true with Poisson solvers. Rigorous analysis is presented to prove that the difference is one half, not one between the convergence orders of numerical solution and gradient in both solvers. The analysis is then validated with numerical results. We furthermore show that both Poisson solvers, being symmetric, indeed satisfy the orthogonal property in the discrete level and yield stable implementations of the Hodge decomposition in octree grids.
2010 Mathematics Subject Classification: 65N06
Key Words and Phrases: Incompressible Naiver-Stokes equations, Poisson equation, linear system, octree grids
- ⋅ 28th-B-10:50 − 11:35 Chair: Hyundae Lee (Inha University)
- ⋅ 28th-B-10:50 − 11:05 Spectral properties of the Neumann-Poincar\'e operator on rotationally symmetric domains (Yong-Gwan Ji, Hyeonbae Kang)
- 지용관*((고등과학원)), 강현배((인하대))
Yong-Gwan Ji*, KIAS, Hyeonbae Kang, Inha University
We concern the spectral properties of the Neumann-Poincaré (NP) operator on two- and three-dimensional bounded domains which are invariant under either rotation or reflection. We prove that if the domain has such a symmetry, then the function space on which the NP operator acting is decomposed into invariant subspaces defined as eigenspaces of the unitary transformation corresponding to rotation or reflection. In two dimensions, an $m$-fold rotationally symmetric simply connected domain $D$ can be generated by the $m$th-root transform of a domain, say $\Omega$. We prove that the NP spectrum on $D$ contains the NP spectrum on $\Omega$ counting multiplicities. We also discuss some examples including lemniscates, $m$-star-shaped domains, and Cassini ovals.
2010 Mathematics Subject Classification: 31A10, 35P05, 30B10
Key Words and Phrases: Neumann-Poincar\'e operator, $m$-fold symmetric domains
- ⋅ 28th-B-11:05 − 11:20 Asymptotic behavior of dispersive equations via the space-time resonance argument (Soonsik Kwon, Kiyeon Lee, Changhun Yang)
- 권순식((카이스트)), 이기연*((카이스트)), 양창훈((충북대))
Soonsik Kwon, KAIST, Kiyeon Lee*, KAIST, Changhun Yang, Chungbuk National University
In this talk, we give a survey of asymptotic behavior for the nonlinear dispersive equations which has Hartree-type nonlinearity by exploiting the space-time resonance argument. This argument is first introduced by Germain-Masmoudi-Shatah in 2008. In 2 dimension, since it is difficult to apply to obtaining global well-posedness by using Strichartz estimates, we observe space, time, and space-time resonant sets and their smallness in the oscillatory integral of the semi-relativistic and Dirac equations. Especially, we give the asymptotic behavior of dispersive equations which has scattering critical nonlinearity.
2010 Mathematics Subject Classification: 35Q41, 35Q40, 35Q55
Key Words and Phrases: Space-time resonance argument, Dirac equations, modified scattering, spinorial null-structure
- ⋅ 28th-B-11:20 − 11:35 Subwavelength localized modes for acoustic waves (Sanghyeon Yu, Habib Ammari, Erik Orvehed Hiltunen)
- 유상현*((고려대)), Habib Ammari((ETH Zurich)), Erik Orvehed Hiltunen((Yale University))
Sanghyeon Yu*, Korea University, Habib Ammari, ETH Zurich, Erik Orvehed Hiltunen, Yale University
The recent development of subwavelength photonic and phononic crystals shows the possibility of controlling wave propagation at deep subwavelength scales. Subwavelength bandgap phononic crystals are typically created using a periodic arrangement of subwavelength resonators, for example, small gas bubbles in a liquid. In this talk, we consider various structures generated by modifying periodic phononic crystals by `defects'. Our aim is to prove that the defect acts as a waveguide; waves of certain frequencies will be localized to, and guided. The key result is an original formula for the frequencies of the defect modes. This talk is based on joint works with Habib Ammari (ETH) and Erik Orvehed Hiltunen (Yale).
2010 Mathematics Subject Classification: 35R30, 35C20
Key Words and Phrases: Subwavelength resonance, phononic crystal, line defect, weak localization
- Regularity Theory for Partial Differential Equations and Its Applications
- ⋅ 28th-A-09:00 − 10:30 Chair: Jihoon Ok (Sogang University)
- ⋅ 28th-A-09:00 − 09:20 Regularity of weak solutions for porous medium equation (Sukjung Hwang)
- 황숙정((충북대))
Sukjung Hwang, Chungbuk National University
In this talk, we consider porous medium equations, which are important nonlinear evolution equations describing a number of physical applications: Fluid flow, heat radiations in plasma, mathematical biology, and other fields. In particular, we discuss the H\"{o}lder regularity of both homogeneous porous medium equations and one with a divergence form of drift.
2010 Mathematics Subject Classification: 35A01, 35K55
Key Words and Phrases: PME, regularity, weak solution
- ⋅ 28th-A-09:20 − 09:40 Gradient potential theory for elliptic obstacle problems with measure data (Sun-Sig Byun, Kyeong Song, Yeonghun Youn)
- 변순식((서울대)), 송경((고등과학원)), 윤영훈*((영남대))
Sun-Sig Byun, Seoul National University, Kyeong Song, KIAS, Yeonghun Youn*, Yeungnam University
In this talk, we will study gradient potential theory for elliptic obstacle problems with measure data. Most of the regularity results for such problems assume higher differentiability assumptions to make it possible to treat the effect of the obstacle as another measure. In our recent result, we obtained a similar gradient potential estimate without such higher differentiability assumptions on the obstacle. To obtain such a result, we also established a gradient excess decay estimate from the known $C^{1,\alpha}$ regularity of the solutions to homogeneous $p$-Laplace type equations.
2010 Mathematics Subject Classification: 35B65, 35J87, 35R06, 35D99
Key Words and Phrases: Gradient potential theory, obstacle problems, measure data
- ⋅ 28th-A-09:50 − 10:10 Nonlinear potential estimates for local and nonlocal problems (Sun-Sig Byun, Kyeong Song)
- 변순식((서울대)), 송경*((고등과학원))
Sun-Sig Byun, Seoul National University, Kyeong Song*, KIAS
In this talk, we first review some classical results in nonlinear potential theory for local and nonlocal equations with measure data. We then present their counterparts for mixed local-nonlocal equations. The main results are concerned with pointwise potential estimates and continuity criteria for solutions. We also discuss further topics, such as gradient potential estimates.
2010 Mathematics Subject Classification: 31C45, 35A01, 35B65, 35R06
Key Words and Phrases: Mixed local and nonlocal equations, measure data, potential estimates
- ⋅ 28th-A-10:10 − 10:30 Robust near-diagonal Green function estimates (Minhyun Kim)
- 김민현((한양대))
Minhyun Kim, Hanyang University
We study sharp near-diagonal pointwise bounds for the Green function for nonlocal operators of fractional order $\alpha \in (0,2)$. The novelty of our results it two-fold: the estimates are robust as $\alpha \to 2-$ and we prove the bounds without making use of the Dirichlet heat kernel. In this way we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not.
2010 Mathematics Subject Classification: 35J08, 47G20, 60G52, 31C25
Key Words and Phrases: Green function, nonlocal operator
- ⋅ 28th-B-10:50 − 12:20 Chair: Mikyoung Lee (Pusan National University)
- ⋅ 28th-B-10:50 − 11:10 Some integrability results for a borderline case of double phase problems with measure data (Jung-Tae Park, Pilsoo Shin)
- 박정태*((한국기술교육대)), 신필수((경기대))
Jung-Tae Park*, Korea University of Technology and Education, Pilsoo Shin, Kyonggi University
In this talk, we consider a borderline case of double phase problems when the right-hand side is a signed Radon measure with finite total mass. We discuss an integrability result for the gradient of a solution in terms of the first-order maximal function of the associated measure. We also introduce a notion of a solution that guarantees such a regularity result.
2010 Mathematics Subject Classification: 35J60, 35R06, 35B65
Key Words and Phrases: Double phase problems, measure data, regularity
- ⋅ 28th-B-11:10 − 11:30 Estimates for fundamental solutions of parabolic equations in non-divergence form (Hongjie Dong, Seick Kim, Sungjin Lee)
- Hongjie Dong((Brown University)), 김세익((연세대)), 이성진*((연세대))
Hongjie Dong, Brown University, Seick Kim, Yonsei University, Sungjin Lee*, Yonsei University
We construct the fundamental solution of second order parabolic equations in non-diver\-gence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.
2010 Mathematics Subject Classification: 35K10
Key Words and Phrases: Fundamental solution, parabolic equation in non-divergence form, Dini mean oscillation
- ⋅ 28th-B-11:40 − 12:00 The regularity theory for single and double obstacle problems (Jinwan Park)
- 박진완((공주대))
Jinwan Park, Kongju National University
In this talk, I will introduce the regularity theory for various single and double obstacle problems.
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, regularity of free boundary
- ⋅ 28th-B-12:00 − 12:20 Gradient estimates for Stokes systems with piecewise DMO coefficients (Jongkeun Choi, Hongjie Dong, Longjuan Xu)
- 최종근*((부산대)), Hongjie Dong((Brown University)), Longjuan Xu((National University of Singapore))
Jongkeun Choi*, Pusan National University, Hongjie Dong, Brown University, Longjuan Xu, National University of Singapore
The regularity theory for Stokes systems with irregular coefficients has important applications in mathematical fluid dynamics, for instance, Stokes flow over composite materials. In this talk, I will present recent work on partial regularity of weak solutions to stationary Stokes systems with piecewise DMO coefficients in a domain which consists of a finite number of disjoint subdomains, and discuss an application to stationary Navier-Stokes systems.
2010 Mathematics Subject Classification: 76D07, 35B65, 35J47
Key Words and Phrases: Stokes system, piecewise Dini mean oscillation, gradient estimate
- Recent Developments in Parabolic Partial Differential Equations
- ⋅ 28th-C-13:00 − 14:55 Chair: Hantaek Bae (UNIST)
- ⋅ 28th-C-13:00 − 13:25 Instability of rolls in the $2$-dimensional generalized Swift-Hohenberg equation (Myeongju Chae, Soyeun Jung)
- 채명주 ((한경대)), 정소연*((공주대))
Myeongju Chae, Hankyong National University, Soyeun Jung*, Kongju National University
In this talk we investigate the instability of roll waves bifurcating from an equilibrium in the $2$-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist. Also, we show that the spectral instability of the rolls leads to linear and nolinear instability.
2010 Mathematics Subject Classification: 35B35
Key Words and Phrases: Instability, Roll waves, Generalized Swift-Hohenberg equation
- ⋅ 28th-C-13:30 − 13:55 Global existence of the mild solution of Hall-MHD (TongKeun Chang)
- 장통근((연세대))
TongKeun Chang, Yonsei University
In this talk, I introduce the Hall-MHD and show the global existence of the mild solution of Hall-MHD in homogeneous anisotropic Besov space.
2010 Mathematics Subject Classification: 35C05
Key Words and Phrases: Hall-MHD, mid solution, global existence, homogeneous anisotropic Besov space
- ⋅ 28th-C-14:00 − 14:25 Global solutions and decay rates for Oldroyd type models in hybrid Besov spaces (Hantaek Bae, Jaeyong Shin)
- 배한택((울산과학기술원)), 신재용*((울산과학기술원))
Hantaek Bae, UNIST, Jaeyong Shin*, UNIST
We consider viscoelastic fluids which have a variety of different properties from Newtonian fluids. One of the most well-known models for viscoelastic fluids is described by Oldroyd (1950). Oldroyd type models do not have scaling invariance and more interestingly, behave differently in different frequencies (low or high frequencies). Keeping this in mind, we discuss the global existence of solutions for Oldroyd type models in hybrid Besov spaces by splitting them into low frequency part and high frequency part. Furthermore, we shall also present the temporal decay rates of the solutions. To the best of our knowledge, this is the first result for decay rates of solutions ever in this framework.
2010 Mathematics Subject Classification: 35Q35, 35A01, 35B40
Key Words and Phrases: Viscoelastic fluids, Oldroyd type models, global existence, temporal decay rates, hybrid Besov spaces
- ⋅ 28th-C-14:30 − 14:55 Global solutions to a chemotaxis-fluid system with singular chemotactic sensitivity (Dongkwang kim)
- 김동광((연세대))
Dongkwang kim, Yonsei university
In this talk, a system of equations that models chemotaxis in a fluid environment, known as the chemotaxis-Navier-Stokes system
\begin{equation*}
(\star)\hfill\hfill\qquad\qquad\qquad \hfill\left\{\begin{aligned}
&n_t+u\cdot\nabla n = \Delta n - \nabla\cdot(n\chi(c)\nabla c),\\
&c_t+u\cdot\nabla c = \Delta c -nf(c),\\
&u_t+\kappa (u\cdot\nabla) u = \Delta u +\nabla P +n\nabla\Phi,
\quad\nabla\cdot u=0,\qquad\qquad \quad
\end{aligned}\right.\qquad\qquad
\end{equation*}
is considered in a smoothly bounded domain subject to Neumann/Neumann/Dirichlet boundary conditions.
Previous research has shown that $(\star)$ possesses the global solution under some mild assumptions on $\chi$ and $f$ (M. Winkler '12, '16). It is shown that although $\chi$ admits some singularity (e.g., $\chi(c)=c^{-\alpha}$ for some $\alpha>0$), appropriate use of some quantity involving $\chi$ and $f$ guarantees the existence of the global solution to $(\star)$ for sufficiently smooth initial data.
2010 Mathematics Subject Classification: 35Q92, 35Q35
Key Words and Phrases: Chemotaxis, Keller-Segel system, global existence
- Functional Analysis and Mathematical Understanding of Quantum Phenomena
- ⋅ 28th-C-13:00 − 15:00 Chair: Ja A Jeong (Seoul National University)
- ⋅ 28th-C-13:00 − 13:30 Quantum detailed balance and entropy production for quantum Markov semigroups (Chul Ki Ko, Hyun Jae Yoo)
- 고철기*((연세대)), 유현재((한경대))
Chul Ki Ko*, Yonsei University, Hyun Jae Yoo, Hankyong National University
We discuss the quantum detailed balance conditions and entropy production for quantum Markov semigroups with an faith normal invariant state.
The quantum detailed balance condition with respect to an invariant state is a condition that makes the semigroup symmetric.
The entropy production is to measure a deviation from the symmetry between a Markov semigroup and the semigroup of its time reversed process.
There have been various versions of the detailed balance condition and entropy production. In this talk, we adopt the concept of quantum detailed balance condition and entropy production for quantum Markov semigroups with main references of F. Fagnola and V. Umanita, and F. Fagnola and R. Rebolledo and introduce their results.
We also consider quantum Markov semigroups associated with open quantum walks on the periodic graphs (the cycle and torus for one dimensional and two dimensional spaces, and the crystal torus). We compute the entropy productions in these models and find the relationship between the detailed balance condition and noexistence of entropy production in these models. Theses models serve as good examples to study the quantum detailed balance condition and the entropy production.
2010 Mathematics Subject Classification: 81P17, 81R15, 82C41
Key Words and Phrases: Quantum detailed balance condition, entropy production, open quantum walks, quamtum Markov semigroups
- ⋅ 28th-C-13:40 − 14:00 Schmidt number detection under group symmetry (Sang-Jun Park, Sang-Gyun Youn)
- 박상준*((서울대)), 윤상균((서울대))
Sang-Jun Park*, Seoul National University, Sang-Gyun Youn, Seoul National University
Quantum entanglement is a fundamental phenomenon and serves as an essential resource in Quantum Computation and Quantum Information Processing. A vast amount of research has been conducted to quantify quantum entanglement, and Schmidt number is one of the most important discrete entanglement measure. A natural approach to analyze the Schmidt number is to use k-positive maps as Schmidt number witnesses. In this talk, we propose how to efficiently extract Schmidt number witnesses when the states have compact group symmetry. As an application, we completely characterize the Schmidt number of orthogonally invariant bipartite states.
2010 Mathematics Subject Classification: 46N50, 81P45, 43A65
Key Words and Phrases: Mapping cone, duality, Schmidt number, orthogonally invariant states
- ⋅ 28th-C-14:00 − 14:20 Bundle theoretic descriptions of single-particle state spaces: An introduction to relativistic quantum information theory (Heon Lee)
- 이헌((서울대))
Heon Lee, Seoul National University
Bundle theoretic descriptions of single-particle state spaces have been introduced to resolve some perplexities of Relativistic Quantun Information Theory (RQI). In this talk, we present this description and its implications.
2010 Mathematics Subject Classification: 81P16, 81P45, 81Q99
Key Words and Phrases: Relativistic quantum information, relativistic perception, single-particle state space, spin
- ⋅ 28th-C-14:30 − 15:00 Duality of group actions: Rokhlin action and approximate representable action (Hyun Ho Lee)
- 이현호((울산대))
Hyun Ho Lee, University of Ulsan
We suggest an improved notions of group actions on unital $C^*$-algebras, namely the weak tracial Rokhlin property and the weak tracial approximate representability. Originally,
M. Izumi defined the Rokhlin property and the approximate representability for finite group actions and N. C. Phillips extended them with respect to the trace or Cuntz comparison. Both notions heavily depend on the existence of abundant projections in the algebra. In this talk, we introduce the weak tracial approximate representability based on the new framework developed by H. Osaka and me and claim duality between the weak tracial Rokhlin property and the weak approx. representability
2010 Mathematics Subject Classification: 46L35, 19K14
Key Words and Phrases: Rokhlin property, approximate representability, duality
- Theory and Application on Spaces of Matrices and Operators
- ⋅ 28th-A-09:20 − 10:30 Chair: Sun Kwang Kim (Chungbuk National University)
- ⋅ 28th-A-09:20 − 09:40 Weak log-majorization between the geometric and Wasserstein means (Sejong Kim)
- 김세정((충북대))
Sejong Kim, Chungbuk National University
There exist lots of distinct geometric means on the cone of positive definite Hermitian matrices such as the metric geometric mean, spectral geometric mean, log-Euclidean mean and Wasserstein mean. In this talk, we prove the log-majorization relation on the singular values of the product of given positive definite matrices and their (metric and spectral) geometric means. We also establish the weak log-majorization between the spectra of the two-variable Wasserstein mean and spectral geometric mean. In particular, for a specific range of the parameter, the two-variable Wasserstein mean converges decreasingly to the log-Euclidean mean with respect to the weak log-majorization.
2010 Mathematics Subject Classification: 15A42, 15B48, 47A64
Key Words and Phrases: Positive definite matrix, metric geometric mean, spectral geometric mean, Wasserstein mean, weak log-majorization
- ⋅ 28th-A-09:45 − 10:05 A binomial expansion formula for weighted geometric means of unipotent matrices (Hayoung Choi, Sejong Kim, Yongdo Lim)
- 최하영*((경북대)), 김세정((충북대)), 임용도((성균관대))
Hayoung Choi*, Kyungpook National University, Sejong Kim, Chungbuk National University, Yongdo Lim, Sungkyunkwan University
Following the Kubo-Ando theory of operator means we consider the weighted geometric mean $A \#_t B$ of $n \times n$ upper triangular matrices $A$ and $B$ whose main diagonals are all $1$, named the upper unipotent matrices. We also present its binomial expansion
$$ A \#_t B = \sum_{k=0}^{n-1}
\begin{pmatrix}
t\\
k\\
\end{pmatrix}A(A^{-1} B - I)^{k}, \ t \in \mathbb{R}.$$
Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent
(resp. the Lie algebra of nilpotent) matrices.
2010 Mathematics Subject Classification: 15B30, 22E25, 65F45
Key Words and Phrases: Unipotent matrix, weighted geometric mean, log-Euclidean mean, binomial expansion, Loos symmetric space
- ⋅ 28th-A-10:10 − 10:30 Multi-variable Wasserstein means of positive definite operators (Vatsalkumar Nandkishor, Sejong Kim)
- Vatsalkumar Nandkishor*((충북대)), 김세정((충북대))
Vatsalkumar Nandkishor*, Chungbuk National University, Sejong Kim, Chungbuk National University
In this talk, I will discuss two forms of least squares mean: The Karcher
mean and the Wasserstein mean on the cone of positive definite Hermitian
matrices. Then, I will talk about the extension of Karcher mean in the infinite
dimensional setting of positive operators on a Hilbert space and its important
properties. Lastly, the Wasserstein mean in the infinite dimensional setting
of positive operators on a Hilbert space and its attractive properties will be
explored.
2010 Mathematics Subject Classification: 47B65, 15B48
Key Words and Phrases: Positive definite operator, Wasserstein mean
- ⋅ 28th-B-10:50 − 12:20 Chair: Sejong Kim (Chungbuk National University)
- ⋅ 28th-B-10:50 − 11:10 Urysohn type lemma in function algebras (Han Ju Lee)
- 이한주((동국대))
Han Ju Lee, Dongguk University
Urysohn lemma plays an important role in many areas in Mathematics as well as in topology. Recently, a version of Urysohn lemma was introduced in a uniform algebra which is a unital closed separating subalgebra of the space of complex-valued continuous functions on a compact Hausdorff space.
The Gelfand transform shows that any function algebra on a topological space which is a closed subalgebra of the space of bounded. Complex-valued continuous functions is, in fact, isometrically isomorphic to a uniform algebra of its maximal ideal space. However, since the domain is changed, it is difficult to find a Urysohn type lemma.
We suggest some types of function algebras which allows a version of Urysohn lemma and its application to the denseness of norm-attaining operators.
2010 Mathematics Subject Classification: 46B04
Key Words and Phrases: Urysohn lemma, uniform algebra, norm-attaining operators
- ⋅ 28th-B-11:10 − 11:30 On characterizations of the Radon-Nikodym property (Geunsu Choi, Yun Sung Choi, Mingu Jung, Sun Kwang Kim, Miguel Martin)
- 최근수*((동국대)), 최윤성((포항공대)), 정민구((고등과학원)), 김선광((충북대)), Miguel Martin((University of Granada))
Geunsu Choi*, Dongguk University, Yun Sung Choi, POSTECH, Mingu Jung, KIAS, Sun Kwang Kim, Chungbuk National University, Miguel Martin, University of Granada
The Radon-Nikodym property plays a significant role in the study of geometric aspects of Banach spaces, and nowadays there are many characterizations of the property in terms of different terminologies in functional analysis. In this talk, we study the characterization of the Radon-Nikodym property in terms of norm attaining operators. Namely, we introduce the quasi norm attainment of operators, which complements the equivalence on the denseness of classical norm attainment. A related recent result is also presented.
2010 Mathematics Subject Classification: 46B04, 46B20, 46B22
Key Words and Phrases: Banach space, Radon-Nikodym property, norm attainment
- ⋅ 28th-B-11:40 − 12:00 Rank-one perturbation of linear operators (Mingu Jung, Gonzalo Martinez-Cervantes, Abraham Rueda Zoca)
- 정민구*((고등과학원)), Gonzalo Martinez-Cervantes((Universidad de Alicante)), Abraham Rueda Zoca((Universidad de Granada))
Mingu Jung*, KIAS, Gonzalo Martinez-Cervantes, Universidad de Alicante, Abraham Rueda Zoca, Universidad de Granada
In this talk, we observe that for every (reflexive) infinite-dimensional Banach space $X$ there exist a reflexive Banach space $Y$ and $T, R \in \mathcal{L}(X,Y)$ such that $R$ is a rank-one operator, $\|T+R\|>\|T\|$, but $T+R$ does not attain its norm. Furthermore, motivated by the parallelism exhibited in the literature between the $V$-property introduced by V. A. Khatskevich, M. I. Ostrovskii and V. S. Shulman and the weak maximizing property introduced by R. M. Aron, D. Garcia, D. Pellegrino, and E. V. Teixeira, we also study the relationship between these two properties and the so-called compact perturbation property.
2010 Mathematics Subject Classification: 46B10, 46B20, 46B28
Key Words and Phrases: Compact perturbation, reflexivity, weak maximizing property, V-pairs
- ⋅ 28th-B-12:00 − 12:20 Norm attaining compact operators on some function spaces (Sun Kwang Kim, Han Ju Lee)
- 김선광*((충북대)), 이한주((동국대))
Sun Kwang Kim*, Chungbuk National University, Han Ju Lee, Dongguk University
It is well known that every compact operator from Banach spaces to a uniform algebra can be approximated by norm attaining ones. We extended this result to operators to more general function spaces.
2010 Mathematics Subject Classification: 46B20
Key Words and Phrases: Banach space, norm attaining operator
- Operator Theory on Reproducing Kernel Hilbert Spaces
- ⋅ 28th-A-09:50 − 10:30 Chair: In Hyoun Kim (Incheon National University)
- ⋅ 28th-A-09:50 − 10:10 Mixed de Branges--Rovnyak and sub-Bergman spaces (Jaehui Park)
- 박재휘((서울대))
Jaehui Park, Seoul National University
For an analytic function $b$ in the closed unit ball of $H^{\infty}$, if both $T_{b}$ and $T_{b}^{*}$ are $2$-hyper\-contractions on the weighted Bergman space $A_{\alpha}^{2}$ ($\alpha\geq0$), there are four second order sub-Bergman spaces.
In this talk, we study some properties of these mixed sub-Bergman spaces.
2010 Mathematics Subject Classification: 46B22, 47B35
Key Words and Phrases: Higher-order sub-Bergman spaces, reproducing kernel Hilbert spaces
- ⋅ 28th-A-10:10 − 10:30 Composition operators on the vector-valued Hardy spaces (Sumin Kim)
- 김수민((성균관대))
Sumin Kim, Sungkyunkwan University
Let $\varphi$ be an analytic map of the unit disk $\mathbb D $ into itself, the Composition operator $C_{\varphi}$ on the vector-valued Hardy spaces $H^p(\mathbb D, X)$ is defined by
$$
C_{\varphi}h:=h \circ \varphi \quad(h \in H^p(\mathbb D, X)).
$$
In this talk, we consider boundedness of these operators and gives an estimate for their norms. Moreover, for inner functions $\theta$, we get specific information about the norms of composition operators.
This talk is based on a joint work with In Sung Hwang.
2010 Mathematics Subject Classification: 47B33, 42B30
Key Words and Phrases: Composition operators, Hardy spaces
- ⋅ 28th-B-10:50 − 11:50 Chair: In Sung Hwang (Sungkyunkwan University)
- ⋅ 28th-B-10:50 − 11:10 On the structure of conditionally positive definite algebraic operators (Il Bong Jung)
- 정일봉((경북대))
Il Bong Jung, Kyungpook National University
Recently, we have introduced and intensively studied a class of bounded Hilbert space operators called conditionally positive definite. In this talk we give a complete description of algebraic conditionally positive definite operators on inner product spaces; we do not assume that the operators under consideration are bounded.
This is a joint work with Z. Jablonski and J. Stochel.
2010 Mathematics Subject Classification: 47B20, 47B90
Key Words and Phrases: Algebraic operator, conditional positive definiteness, conditionally positive definite operator, similarity
- ⋅ 28th-B-11:10 − 11:30 On properties of ${\mathcal C}$-normal weighted composition operators (Ji Eun Lee, Eungil Ko, Yoenha Kim)
- 이지은*((세종대)), 고응일((이화여대)), 김연하((아주대))
Ji Eun Lee*, Sejong University, Eungil Ko, Ewha Womans University, Yoenha Kim, Ajou University
An operator $T\in{\mathcal L}(\mathcal H)$ is said to be {\it ${\mathcal C}$-normal} if ${\mathcal C}T$ and $({\mathcal C}T)^{\#}$ commute for a conjugation ${\mathcal C}$ on ${\mathcal H}$, where $({\mathcal C}T)^{\#}$ denotes the adjoint of ${\mathcal C}T.$
In this talk, we focus on properties of ${\mathcal C}$-normal weighted composition operators on the Hardy space. Especially, we show that if $W_{\psi, \varphi}$ is ${\mathcal C}$-normal, then $\psi$ is outer, $\varphi$ is univalent, and $W_{\psi, \varphi}^{\ast}$ is cyclic.
Moreover, if the conjugation ${\mathcal C}$ is given by ${\mathcal C}f(z)=\overline{f(\overline{z})}$ for $f\in H^2({\mathbb D})$, then we investigate the symbol functions $\psi$ and $\varphi$ for $W_{\psi,\varphi}$ to be ${\mathcal C}$-normal.
2010 Mathematics Subject Classification: Primary 47A05; Secondary 47B38, 47B33
Key Words and Phrases: ${\mathcal C}$-normal operators, weighted composition operators, complex symmetric operator
- ⋅ 28th-B-11:30 − 11:50 Two problems in Agler's model theory and Toeplitz operator theory (Woo Young Lee)
- 이우영((서울대))
Woo Young Lee, Seoul National University
In this talk, I present two problems emerging from a recent development in operator theory: Agler's abstract model theory and Toeplitz operator theory: more concretely, (1) What is the boundary of the family of $p$-hyponormal operators? (2) Characterize the hyponormality of Toeplitz operators on the bidisk.
2010 Mathematics Subject Classification: 47B35, 47B20
Key Words and Phrases: Agler's model theory, Toeplitz operators on the bidisk
- Geometric Structures and Submanifolds
- ⋅ 28th-A-09:00 − 10:30 Chair: Chang Hwa Woo (Pukyong National University)
- ⋅ 28th-A-09:00 − 09:20 Recent results on weakly eta-Einstein manifolds (Sun Hyang Chun)
- 전선향((조선대))
Sun Hyang Chun, Chosun University
Recently, the notion of a weakly eta-Einstein structure was introduced as a correspondencs of a weakly Einstein structure.
In this talk, we introduce the concept of weakly eta-Einstein and investigate the characteristic of various spaces with weakly eta-Einstein structures.
2010 Mathematics Subject Classification: 53C25, 53D10
Key Words and Phrases: weakly eta-Einstein structure, almost contact metric manifold
- ⋅ 28th-A-09:20 − 09:40 Realizations of CR-symmetric contact manifolds as real hypersurfaces in Hermitian symmetric spaces (Jong Taek Cho)
- 조종택((전남대))
Jong Taek Cho, Chonnam National University
Contact manifolds have the two fundamental associated structures, that is, Riemannian metrics and almost CR-structures. In this talk, we concentrate on contact manifolds from the latter point of view. In particular, we give a classification of CR-symmetric contact manifolds, which are realized as real hypersurfaces in Hermitian symmetric spaces.
2010 Mathematics Subject Classification: 53C15, 53C40
Key Words and Phrases: Contact manifold, CR-cymmetry, real hypersurface, Hermitian symmetric space
- ⋅ 28th-A-09:50 − 10:10 Isometric Reeb flow in Hermitian symmetric spaces (Young Jin Suh)
- 서영진((경북대))
Young Jin Suh, Kyungpook National University
The study of geometric flows has recently aroused the interest of the mathematical community. In this talk, we want to introduce recent research activities on isometric Reeb flow of real hypersurfaces in the complex hyperboilc space and the complex hyperbolic quadric, which are Hermitian symmetric spaces of non-compact type with rank 1 and rank 2, respectively. Besides them, related to our expertise, we will give some motivations and research backgrounds for other Hermitian symmetric spaces like complex two-plane Grassmannians, complex hyperbolic two-plane Grassmannians and complex quadrics.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: Isometric Reeb flow, real hypersurface, complex hyperbolic space, complex hyperbolic quadric, Hermitian symmetric spaces
- ⋅ 28th-A-10:10 − 10:30 Symplectic Dirac operators on transversely symplectic foliations (Seoung Dal Jung)
- 정승달((제주대))
Seoung Dal Jung, Jeju National University
In this talk, we give the properties of the metaplectic structure and transversely symplectic Dirac operators on transversely symplectic foliations.
2010 Mathematics Subject Classification: 53C12, 53C27
Key Words and Phrases: Transversely symplectic foliation, transversely metaplectic structure, transversely symplectic Dirac operator
- ⋅ 28th-B-10:50 − 12:20 Chair: Jong Taek Cho (Chonnam National University)
- ⋅ 28th-B-10:50 − 11:10 Geometry of warped product Legendrian submanifolds via differential equations (Akram Ali, Jae Won Lee, Ali H. Alkhaldi)
- Akram Ali((King Khlid University)), 이재원*((경상대)), Ali H. Alkhaldi((King Khlid University))
Akram Ali, King Khlid University, Jae Won Lee*, Gyeongsang National University, Ali H. Alkhaldi, King Khlid University
The goal of this paper is to investigate the geometry of warped product Legendrian submanifolds in a Sasakian space form without boundary from the extrinsic point of view, subject to the following conditions: topology and geometry of a base manifold, geometric analysis of both a warping function and Ricci curvature of the base manifold. We establish sharp estimates of the relationship between the second fundamental form and the warping function, and also provide some trivial results for warped product Legendrian submanifolds
by using the Ricci curvature along the gradient of the warping function. Taking the clue from the Bochner formula and the second-order ordinary differential equation, we find the characterization for topology of the base space of warped product Legendrian submanifolds via the first non-zero eigenvalue of the warping function, and prove that the base space is isometric to the Euclidean space or the Euclidean sphere under some extrinsic conditions.
2010 Mathematics Subject Classification: 53C42, 53B25
Key Words and Phrases: Warped products, Legendrian, Sasakian space form, Ricci curvature
- ⋅ 28th-B-11:10 − 11:30 Conformal Killing forms in Kaehler geometry (Paul-Andi Nagy)
- 28th-B-11:10 − 11:30
Paul-Andi Nagy, IBS-CCG
For Kaehler manifolds we explicitly determine the solution to the conformal Killing form equation in middle degree. In particular, we complete the classification of conformal Killing forms on compact Kaehler manifolds. We give the first examples of conformal Killing forms on Kaehler manifolds not coming from Hamiltonian 2-forms. These are supported by Calabi-type manifolds over a Kaehler Einstein base. In this set up we also give structure results and examples for the closely related class of Hermitian Killing forms.
2010 Mathematics Subject Classification: 53C25, 53C26, 58C40, 35H10
Key Words and Phrases: Conformal and Hermitian Killing form, conformal foliation, Calabi-type metric
- ⋅ 28th-B-11:40 − 12:00 A characterization of the unit ball by a K\"ahler-Einstein potential (Young-Jun Choi, Kang-Hyurk Lee, Aeryeong Seo)
- 최영준*((부산대)), 이강혁((경상대)), 서애령((경북대))
Young-Jun Choi*, Pusan National University, Kang-Hyurk Lee, Gyeongsang National University, Aeryeong Seo, Kyungpook National University
In this talk, we will show that a universal covering of a compact K\"ahler manifold with ample canonical bundle is the unit ball if it admits a global potential function of the K\"ahler-Einstein metric whose gradient length is a minimal constant. As an application, we will extend the Wong-Rosay theorem to a complex manifold without a boundary.
This is a joint work with Kang-Hyurk Lee and Aeryeong Seo.
2010 Mathematics Subject Classification: 32Q20, 32M05, 53C55
Key Words and Phrases: K\"ahler-Einstein metric, complete holomorphic vector field, the unit ball, automorphism groups
- ⋅ 28th-B-12:00 − 12:20 The classification of cyclic Ricci semi-symmetric hypersurfaces in complex two-plane Grassmannians (Chang Hwa Woo, Imsoon Jeong)
- 우창화*((부경대)), 정임순((청주대))
Chang Hwa Woo*, Pukyong National University, Imsoon Jeong, Cheongju University
In this paper, we have introduced a new notion of cyclic Ricci semi-symmetric real hypersurfaces in complex two-plane Grassmannians. Next, we show a non-existence property of real hypersurfaces in complex two-plane Grassmannians satisfying such a condition.
2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C55
Key Words and Phrases: Complex two-plane Grassmannians, Hopf hypersurface, Ricci tensor, cyclic Ricci semi-symmetric
- Geometric Analysis
- ⋅ 28th-A-09:00 − 10:15 Chair: Kyeongsu Choi (KIAS)
- ⋅ 28th-A-09:00 − 09:25 Additive eigenvalues of superquadratic Hamilton-Jacobi equations with changing domains (Dohyun Kwon, Son N. T. Tu, Farid Bozorgnia)
- 권도현*((서울시립대)), Son N. T. Tu((Michigan State University)), Farid Bozorgnia((Instituto Superior T\'ecnico))
Dohyun Kwon*, University of Seoul, Son N. T. Tu, Michigan State University, Farid Bozorgnia, Instituto Superior Técnico
In this talk, we will explore the additive eigenvalues of state-constraint superquadratic Hamilton-Jacobi equations. By treating the additive eigenvalues as a map with respect to domain perturbation via scaling, we demonstrate that this map exhibits regularity. We also establish that the derivative exists if and only if a new invariant on viscosity Mather measures is present. Furthermore, we connect these properties to the convergence of a vanishing discount problem with respect to changing domains, as well as the parametrization of solutions to the corresponding ergodic problem. This work provides an analogy to the principal eigenvalue of the elliptic problem in the linear case, with the connection being a one-to-one correspondence for the quadratic case through the Hopf-Cole transform.
2010 Mathematics Subject Classification: 35B40, 35D40, 49J20, 49L25, 70H20
Key Words and Phrases: First-order Hamilton–Jacobi equations, second-order Hamilton–Jacobi equations, state-constraint problems, optimal control theory, rate of convergence, viscosity solutions, semiconcavity, boundary lay
- ⋅ 28th-A-09:25 − 09:50 Scalar curvature rigidity of rotationally symmetric sets (Chai Xiaoxiang, Gaoming Wang)
- Chai Xiaoxiang*((포항공대)), Gaoming Wang((Cornell University))
Chai Xiaoxiang*, POSTECH, Gaoming Wang, Cornell University
Scalar curvature rigidity has attracted a lot of attention recently due to the work of Gromov, among them are the Gromov dihedral rigidity conjecture which addresses the scalar curvature rigidity of convex polytopes. I am going to present a scalar curvature rigidity result of weakly convex, rotationally symmetric sets in the Euclidean 3-space. This is joint work with Gaoming Wang (Cornell).
2010 Mathematics Subject Classification: 53C24
Key Words and Phrases: Scalar curvature, minimal surface, rigidity, mean curvature, capillary
- ⋅ 28th-A-09:50 − 10:15 On slowly converging evolution equations (Beomjun Choi, Pei-Ken Hung)
- 최범준*((포항공대)), Pek-Ken Hung((University of Minnesota))
Beomjun Choi*, POSTECH, Pei-Ken Hung, University of Minnesota
We present higher order asymptotics for non-linear evolution equations arising in the study of isolated singularities of geometric variational problems. The main novelty is to characterize the rate and the direction of convergence (the secant at the limit) for solutions which do not converge exponentially. We show slowly converging solutions satisfy the Adams-Simon non-negativity condition and this dictates possible higher asymptotics.
2010 Mathematics Subject Classification: 53E10, 53A10, 35J93, 35K93
Key Words and Phrases: Geometric PDEs, Lojasiewics-Simon inequality, Minimal Surface, Mean Curvature Flow
- ⋅ 28th-B-10:50 − 12:05 Chair: Beomjun Choi (POSTECH)
- ⋅ 28th-B-10:50 − 11:15 A warped product metric, Hilbert-Einstein functional and Weyl’s embedding problem (Huang Jiuzhou)
- 28th-B-10:50 − 11:15
Huang Jiuzhou, KIAS
Izmestiev introduced a warped metric and used it to study the infinitesimal rigidity of Weyl's embedding problem in his PhD thesis. In this talk, I will first explain how this warped product metric can be used to give a simple proof of the closeness of Weyl's embedding problem. Then I will derive the variational property of the Hilbert-Einstein functional for this metric and use it to study the stability of the problem.
2010 Mathematics Subject Classification: 35J15, 53C42
Key Words and Phrases: Isometric embedding, elliptic PDE, stability
- ⋅ 28th-B-11:15 − 11:40 The weighted Yamabe problem (Jinwoo Shin, Pak Tung Ho, Zetian Yan)
- 신진우*((고등과학원)), Pak Tung Ho((Tamkang University)), Zetian Yan((Penn State University))
Jinwoo Shin*, KIAS, Pak Tung Ho, Tamkang University, Zetian Yan, Penn State University
The classical Yamabe problem is to find a conformal Riemannian metric on a compact manifold such that its scalar curvature is constant. In this talk, we consider several Yamabe-type problems on a compact smooth metric measure space with or without boundary. This is a joint work with Pak Tung Ho and Zetian Yan.
2010 Mathematics Subject Classification: 53C21
Key Words and Phrases: Yamabe problem, smooth metric measure space
- ⋅ 28th-B-11:40 − 12:05 Smooth metric measure spaces induced by Poincare-Einstein manifolds (Seunghyeok Kim)
- 김승혁((한양대))
Seunghyeok Kim, Hanyang University
From a conformal point of view, we consider smooth metric measure spaces whose structures are induced by Poincare-Einstein manifolds. If time permits, we will discuss some recent results for them obtained in collaboration with S. Jin (Chungbuk National University).
2010 Mathematics Subject Classification: 53C18
Key Words and Phrases: Smooth metric measure spaces, Poincare-Einstein manifolds
- ⋅ 28th-C-13:45 − 15:00 Chair: Seunghyeok Kim (Hanyang University)
- ⋅ 28th-C-13:45 − 14:10 Recent development of differential Harnack inequality for extrinsic curvature flows (Hyunsuk Kang, Ki-ahm Lee)
- 강현석*((광주과학기술원)), 이기암((서울대))
Hyunsuk Kang*, GIST, Ki-ahm Lee, Seoul National University
The Harnack inequality of Li-Yau type has been a topic of huge interests for decades. In this talk, we focus on its recent development in the extrinsic curvature flows. A natural choice of such flows is the mean curvature flow which has been studies in various settings and we give a survey on other general flows including ones with perturbation on curved spaces. This is a joint work with Ki-ahm Lee.
2010 Mathematics Subject Classification: 53E10
Key Words and Phrases: Harnack inequality, extrinsic curvature flow
- ⋅ 28th-C-14:10 − 14:35 Wiener criterion for nonlocal Dirichlet problems (Minhyun Kim)
- 김민현((한양대))
Minhyun Kim, Hanyang University
It is well known that the solvability of the Dirichlet problem relies on the geometry of the boundary of a given domain, and it was first characterized by Wiener in 1924. In this talk, we establish a nonlocal counterpart of the Wiener criterion for nonlocal Dirichlet problems. This talk is based on a joint work with Ki-ahm Lee and Se-Chan Lee.
2010 Mathematics Subject Classification: 31B25, 31B15, 35R11
Key Words and Phrases: Wiener criterion, harmonic function, nonlocal equation
- ⋅ 28th-C-14:35 − 15:00 Regular solutions to the $L_p$ Minkowski problem (Kyeongsu Choi, Minhyun Kim, Taehun Lee)
- 최경수((고등과학원)), 김민현((한양대)), 이태훈*((고등과학원))
Kyeongsu Choi, KIAS, Minhyun Kim, Hanyang University, Taehun Lee*, KIAS
A cornerstone of the Brunn--Minkowski theory is the Minkowski problem initiated by Minkowski himself over a century ago. This problem characterizes measures generated by convex bodies and has been generalized to the $L_p$ Minkowski problem. In recent years, much of the interest in the $L_p$ Minkowski problem has migrated to the study of regular solutions. In this talk, we discuss sharp regularity result for $p\in(-n+1,-n+2]$ which is optimal if $n=2$ and $p=0$, i.e., the logarithmic Minkowski problem in $\mathbb R^3$. This talk is based on a joint work with Kyeongsu Choi and Minhyun Kim.
2010 Mathematics Subject Classification: 35K96, 35B65, 35C06, 53E99, 53A05
Key Words and Phrases: Minkowski problem, curvature ﬂow, regularity estimates
- Geometric Structures and Representation Spaces
- ⋅ 28th-C-13:30 − 13:55 Chair: Joonhyung Kim (Chungnam National University)
- ⋅ 28th-C-13:30 − 13:55 Optimal independent generating system for the Hecke congruence subgroups (Nhat Minh Doan, Sang-hyun Kim, Mong Lung Lang, Ser Peow Tan)
- Nhat Minh Doan((Vietnam Academy of Science and Technology)), 김상현*((고등과학원)), Mong Lung Lang((National University of Singapore)), Ser Peow Tan((National University of Singapore))
Nhat Minh Doan, Vietnam Academy of Science and Technology, Sang-hyun Kim*, KIAS, Mong Lung Lang, National University of Singapore, Ser Peow Tan, National University of Singapore
We prove that if $n = p$ or $n = p^2$ for a prime $p$, then the Hecke congruence subgroup $\Gamma_0(n)$ admits a freely independent set of generators whose $(2,1)$ components are exactly $0$ or $n$. For the case when $n = pq$ for sufficiently near primes $p$ and $q$, we can require that such components are $0$, $n$ or $2n$.
2010 Mathematics Subject Classification: 11F06, 11B57, 30F35
Key Words and Phrases: Congruence subgroup, Euler totient function, Farey sequence, hyperbolic geometry
- ⋅ 28th-C-13:55 − 14:20 Chair: Sang-hyun Kim (KIAS)
- ⋅ 28th-C-13:55 − 14:20 Structural stability of meandering hyperbolic group actions (Sungwoon Kim, Misha Kapovich, Jaejeong Lee)
- 김성운*((제주대)), Misha Kapovich((U. C. Davis)), 이재정((서울대))
Sungwoon Kim*, Jeju National University, Misha Kapovich, U. C. Davis, Jaejeong Lee, Seoul National University
Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. We relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on \linebreak meandering-hyperbolic actions and give other examples of such actions.
2010 Mathematics Subject Classification: 37F15
Key Words and Phrases: Structural stability, meandering hyperbolic group action, uniform lattice
- ⋅ 28th-C-14:30 − 14:55 Chair: Sungwoon Kim (Jeju National University)
- ⋅ 28th-C-14:30 − 14:55 PCR-Kahler equivalent metrics in the Siegel domain (Joonhyung Kim, Ioannis D. Platis, Lijie Sun)
- 김준형*((충남대)), Ioannis D. Platis((University of Crete)), Lijie Sun((Yamaguchi University))
Joonhyung Kim*, Chungnam National University, Ioannis D. Platis, University of Crete, Lijie Sun, Yamaguchi University
In this talk, we consider the Lie group $H\times \mathbb{R}_{>0}$, where $H$ is the Heisenberg group.
From the standard CR structure of $H$, we construct the complex hyperbolic structure of the
Siegel domain. Additionally, using the same minimal data for $H$, we provide the Siegel domain
with yet another Kahler structure: this structure is of unbounded negative sectional curvature,
and its complex structure does not commute with the standard complex structure. However,
we show that those two Kahler structures are essentially the same when restricted to $H$
2010 Mathematics Subject Classification: 53C17, 53C25, 32C16
Key Words and Phrases: Heisenberg group, Sasakian manifolds, complex hyperbolic plane, horospherical model, PCR-mappings
- ⋅ 29th-D-09:00 − 09:25 Chair: Hongtaek Jung (Seoul National University)
- ⋅ 29th-D-09:00 − 09:25 Tubular neighborhoods in complex hyperbolic manifolds (Youngju Kim)
- 김영주((건국대))
Youngju Kim, Konkuk University
The collar lemma says that a closed geodesic in a real hyperbolic 2-manifold has an embedded tubular neighborhood whose width only depends on the length of the geodesic. The width of the collar does not depend on the underlying hyperbolic 2-manifold. On the other hand, a totally geodesic surface with codimension bigger than 1 in a hyperbolic manifold can be arbitrary closed to itself. Here, we prove that an embedded complex totally geodesic surface in a complex hyperbolic 2-manifold has a tubular neighborhood whose size depends only on its area. This is a joint work with A. Basmajian.
2010 Mathematics Subject Classification: 30F40, 53C20
Key Words and Phrases: Complex hyperbolic manifold, a tubular neighborhood, an embedded complex totally geodesic surface, collar lemma
- ⋅ 29th-D-09:25 − 09:50 Chair: Youngju Kim (Konkuk University)
- ⋅ 29th-D-09:25 − 09:50 Cluster variety approach to higher Teichm\"uller spaces (Hyun Kyu Kim)
- 김현규((고등과학원))
Hyun Kyu Kim, KIAS
I will give an introduction to the cluster variety approach to higher Teichmüller spaces. I will review Fock-Goncharov's construction of moduli spaces parametrizing G-local systems on a punctured surface with certain decoration data at punctures, where G is an algebraic group. I will explain the cluster variety structures of these moduli spaces, and how they lead to higher Teichm\"uller spaces. Some problems posed in the theory of cluster varieties, known results, as well as some open problems, will be discussed.
2010 Mathematics Subject Classification: 57K20, 13F60, 14D20, 57K31
Key Words and Phrases: Cluster variety, higher Teichm\"uller space, skein algebra, canonical basis
- ⋅ 29th-D-10:00 − 10:25 Chair: Hyun Kyu Kim (KIAS)
- ⋅ 29th-D-10:00 − 10:25 Symplectic coordinates on the deformation space of real convex structures on orbifolds (Hongtaek Jung, Suhyoung Choi)
- 정홍택*((서울대)), 최서영((카이스트))
Hongtaek Jung*, Seoul National University, Suhyoung Choi, KAIST
The deformation space $D(O)$ of real convex projective structures on closed orientable 2-orbifolds $O$ carries a natural symplectic structure, generalizing the Atiyah-Bott-Goldman form. We will show that $D(O)$ admits global Darboux coordinates. To this end, we will discuss how to construct a natural symplectic form on the deformation space of real convex projective structures on compact orientable orbifolds and prove that $D(O)$ can be decomposed into smaller deformation spaces.
2010 Mathematics Subject Classification: 57K20
Key Words and Phrases: Real convex projective geometry, Hitchin component, Darboux coordinates
- Low-Dimensional Topology and Knot Theory
- ⋅ 28th-A-09:20 − 09:40 Chair: JungHwan Park (KAIST)
- ⋅ 28th-A-09:20 − 09:40 Double slice genus and von Neumann rho invariants of knots (Se-Goo Kim, Taehee Kim)
- 김세구*((경희대)), 김태희((건국대))
Se-Goo Kim*, Kyung Hee University, Taehee Kim, Konkuk University
The double slice genus of a knot $K$ is the minimum of genera of unknotted closed surfaces in the 4--sphere whose intersection with the 3--sphere is $K$. Wenzhao Chen found a lower bound for the double slice genus using Casson--Gordon invariants and showed that there exist ribbon knots whose double slice genera can be arbitrarily large. We show that von Neumann rho invariant obstructs the double slice genus and find ribbon knots with vanishing Casson--Gordon invariants whose double slice genera can be arbitrarily large.
2010 Mathematics Subject Classification: 57K10
Key Words and Phrases: Double slice genus, von Neumann rho invariant
- ⋅ 28th-A-09:40 − 10:00 Chair: Se-Goo Kim (Kyung Hee University)
- ⋅ 28th-A-09:40 − 10:00 On the algebraic structure for long flat virtual knots (Sera Kim)
- 김세라((해군사관학교))
Sera Kim, Republic of Korea Naval Academy
Im, I and Lee introduced two embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we constructed invariants for long flat virtual knots via invariants for long virtual knots. We investigate the one of image sets of these embeddings in the set of long virtual knots and show an example that the connected-sum is a non-commutative operation in the set of flat long virtual knots.
2010 Mathematics Subject Classification: 57K10, 57K12
Key Words and Phrases: Flat long virtual knot, embeddings, Gauss diagram, intersection index
- ⋅ 28th-A-10:10 − 10:30 Chair: Sera Kim (Korea Naval Academy)
- ⋅ 28th-A-10:10 − 10:30 On the Alexander polynomial of twisted torus knots (Kyungbae Park, Adnan)
- 박경배*((강원대)), Adnan((강원대))
Kyungbae Park*, Kangwon National University, Adnan, Kangwon National University
The twisted torus knot is a generalization of the torus knot by adding extra full twists on some adjacent strings of a torus knot. In this talk, we give an explicit computation of the Alexander polynomial of twisted torus knots. We use a presentation of the knot group of twisted torus knots and Fox’s free differential calculus. We also discuss applications of our computations.
2010 Mathematics Subject Classification: 57K10, 57K14
Key Words and Phrases: Alexander polynomial, torus knots, twisted torus knots, knot thoery
- ⋅ 28th-B-10:50 − 11:10 Chair: Kyungbae Park (Kangwon National University)
- ⋅ 28th-B-10:50 − 11:10 Involutive bordered techniques in knot concordance problems (Sungkyung Kang)
- 강성경((기초과학연구원 기하학수리물리연구단))
Sungkyung Kang, IBS-CGP
Involutive bordered techniques, introduced last year by the speaker, provide a new way to compute involutive actions on knot Floer homology. This has several far-reaching consequences, including linear independence of (odd,1)-cables of ``half" of torsion knots, ``one is not enough" for exotic structures on contractible 4-manifolds, and the existence of an infinite-rank free subgroup of the smooth concordance group of topologically and rationally slice knots. In this talk, we review this techniques and explain how it were explain to solve those questions. This talk contains a joint work with JungHwan Park, and also an ongoing joint work with Jennifer Hom and JungHwan Park.
2010 Mathematics Subject Classification: 57K10, 57K18
Key Words and Phrases: Involutive knot Floer homology, bordered Floer homology, smooth concordance group
- ⋅ 28th-B-11:10 − 11:30 Chair: Sungkyung Kang (IBS-CGP)
- ⋅ 28th-B-11:10 − 11:30 On the group of homology $S^1\times S^2$'s (Dongsoo Lee)
- 이동수((서울대 모듈과 공간의 양자구조 연구센터))
Dongsoo Lee, QSMS, Seoul National University
Kawauchi defined a group $\Omega(S^1\times S^2)$ on the set of homology $S^1\times S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. We will talk about the kernel of the zero-surgery homomorphism and the 2-torsion subgroup of $\Omega(S^1\times S^2)$.
2010 Mathematics Subject Classification: 57K10, 57K18, 57N70
Key Words and Phrases: Knot concordance, $\widetilde{H}$-cobordism
- ⋅ 28th-B-11:40 − 12:00 Chair: Dongsoo Lee (Seoul National University)
- ⋅ 28th-B-11:40 − 12:00 Knotted handlebodies in the 4-sphere and 5-ball (Mark Hughes, Seungwon Kim, Maggie Miller)
- Mark Hughes((Brigham Young University)), 김승원*((성균관대)), Maggie Miller((Stanford University))
Mark Hughes, Brigham Young University, Seungwon Kim*, Sungkyunkwan University, Maggie Miller, Stanford University
For every integer $g\geq 2$ we construct 3–dimensional genus–$g$ 1–handlebodies smoothly embedded in $S^4$ with the same boundary, and which are defined by the same cut systems of their boundary, yet which are not isotopic rel. boundary via any locally flat isotopy even when their interiors are pushed into $B^5$. This proves a conjecture of Budney–Gabai for genus at least 2.
2010 Mathematics Subject Classification: 57K99, 57K45
Key Words and Phrases: Knotted handlebodies
- ⋅ 28th-B-12:00 − 12:20 Chair: Seungwon Kim (Sungkyunkwan University)
- ⋅ 28th-B-12:00 − 12:20 Rational concordance and knot reversal (Taehee Kim)
- 김태희((건국대))
Taehee Kim, Konkuk University
For an oriented knot $K$ in the 3-sphere, the reverse of $K$ is the knot obtained by reversing the orientation of $K$. In this talk, we show that there exist knots which are not rationally concordant to their reverses.
2010 Mathematics Subject Classification: 57N70
Key Words and Phrases: Knot reversal, rational concordance
- Random Matrix Theory and Related Topics
- ⋅ 28th-C-13:30 − 14:30 Chair: Sung-Soo Byun (KIAS)
- ⋅ 28th-C-13:30 − 14:00 Spectral large deviations for sparse random matrices (Kyeongsik Nam, Shirshendu Ganguly, Ella Hiesmayr)
- 남경식*((카이스트)), Shirshendu Ganguly((University of California, Berkeley)), Ella Hiesmayr((University of California, Berkeley))
Kyeongsik Nam*, KAIST, Shirshendu Ganguly, University of California, Berkeley, Ella Hiesmayr, University of California, Berkeley
It has been an active research topic to understand the spectral statistics of random matrices. An important and interesting question is investigating large deviation properties of spectral observables, such as the empirical spectral measure and extreme eigenvalues. In the absence of exact formulas such as in Gaussian ensembles, these questions are quite hard to analyze. In particular, for sparse random matrices, even investigating the typical behavior of eigenvalues is nontrivial. In this talk, we will survey recent progress in understanding spectral large deviations of such sparse random networks. Based on joint works with Shirshendu Ganguly and Ella Hiesmayr.
2010 Mathematics Subject Classification: 60B20, 05C80
Key Words and Phrases: Random matrices, spectrum, large deviations
- ⋅ 28th-C-14:00 − 14:30 Universality of the cokernels of random $p$-adic matrices (Jungin Lee)
- 이정인((고등과학원))
Jungin Lee, KIAS
The moment problem is to determine whether a probability distribution is uniquely determined by its moments. Recently, the moment problem for random groups has been applied to prove universality results for the cokernels of random integral and $p$-adic matrices. In this talk, we introduce the moments of random groups and several universality results on the distribution of the cokernels of random $p$-adic matrices. In particular, we explain the universality of the cokernels of random $\varepsilon$-balanced Hermitian matrices over the ring of integers of a quadratic extension of $\mathbb{Q}_p$.
2010 Mathematics Subject Classification: 11E39, 15B52, 60B20
Key Words and Phrases: Universality, moments, random $p$-adic matrix
- ⋅ 29th-D-09:00 − 10:30 Chair: Seong-Mi Seo (Chungnam National University)
- ⋅ 29th-D-09:00 − 09:30 Large deviations and fluctuations for real elliptic random matrices (Sung-Soo Byun, Leslie Molag, Nick Simm)
- 변성수*((고등과학원)), Leslie Molag((University of Sussex)), Nick Simm((University of Sussex))
Sung-Soo Byun*, KIAS, Leslie Molag, University of Sussex, Nick Simm, University of Sussex
In this talk, I will discuss real eigenvalues of real elliptic random matrices, exploring both the strong and weak non-Hermiticity regimes. Specifically, I will present the central limit theorem governing the number of real eigenvalues, as well as the large deviation probabilities associated with the absence of real eigenvalues.
2010 Mathematics Subject Classification: 60B20, 33C45
Key Words and Phrases: Random matrices, real elliptic matrices, real eigenvalues, central limit theorem, large deviation probabilities
- ⋅ 29th-D-09:30 − 10:00 Extremal spectral behavior of weighted random d-regular graphs (Jaehun Lee, Kyeongsik Nam)
- 이재훈*((HKUST)), 남경식((카이스트))
Jaehun Lee*, HKUST, Kyeongsik Nam, KAIST
The large deviation of extremal eigenvalues was shown for Erdos-Renyi graphs with random edge-weights very recently. As a natural extension, we study random networks supported on random dependent graphs, especially random regular graphs. In this talk, we will look into the typical location of the largest eigenvalue for a random regular graph with random edge-weights. Interestingly, we find out some relation between the largest eigenvalue and a tree structure inside regular graphs. In addition, there is a phase transition of the largest eigenvalue with respect to the tail of edge-weights. We will also talk about the delocalization property of the top eigenvector as an application.
2010 Mathematics Subject Classification: 60B20, 05C80
Key Words and Phrases: Random regular graph, extremal eigenvalue, delocalization
- ⋅ 29th-D-10:00 − 10:30 Detection problems in the spiked random matrix models (Ji Hyung Jung, Hye Won Chung, Ji Oon Lee)
- 정지형*((카이스트)), 정혜원((카이스트)), 이지운((카이스트))
Ji Hyung Jung*, KAIST, Hye Won Chung, KAIST, Ji Oon Lee, KAIST
The spiked random matrix is one of the simplest models for modeling noisy data, where the signal is a fixed rank matrix, and the noise is a structured random matrix. One prominent phenomenon in this model is that its principal components (extremal eigenvalues and the corresponding eigenvectors) exhibit the sharp phase transition, the so-called BBP transition, depending on the signal-to-noise ratio (SNR). Due to this, the principal component analysis (PCA) can reliably detect the signal when the SNR is above a certain threshold. In contrast, when the SNR is small, it is natural to consider the hypothesis test between the null and alternative hypotheses. In this talk, we review some relevant results and then introduce our results.
2010 Mathematics Subject Classification: 62H25, 62H15, 60B20
Key Words and Phrases: Spiked Wigner/rectangular matrix, principal component analysis, likelihood ratio, linear spectral statistics
- Scientific Computing and Machine Learning
- ⋅ 28th-A-09:00 − 10:30 Chair: Seungchan Ko (Inha University)
- ⋅ 28th-A-09:00 − 09:20 A discrete Leray projection for solving a semi-implicit immersed boundary method in a staggered grid (Sangbeom Park, Soyoon Bak, Philsu Kim, Yunchang Seol)
- 박상범((경북대)), 박소윤((경북대)), 김필수((경북대)), 설윤창*((성균관대))
Sangbeom Park, Kyungpook National University, Soyoon Bak, Kyungpook National University, Philsu Kim, Kyungpook National University, Yunchang Seol*, Sungkyunkwan University
In many real-world applications involved with interaction problems between a fluid and an immersed interface, the effects of high inertia and elasticity are very important. In this talk, we present an efficient and stable immersed boundary (IB) method for solving the motion of elastic interface, in particular, when the fluid has high inertia and the interface has strong elasticity.
Our contributions are three folds. First, an iteration-free semi-Lagrangian approach is used in Navier-Stokes equations so that the high inertial effect of fluid is more stably simulated. Second, the elastic interfacial force is treated semi-implicitly allowing to handle strong elasticity and also to construct a resulting linear system. Last but not least, in solving the original linear system, we transform the 3-by-3 block matrix to a reduced 2-by-2 block matrix after eliminating the fluid pressure term. Such rank reduction is possible by applying the discrete Leray projection operator in a staggered MAC grid. By virtue of this feature, we call our method as a reduced immersed boundary method (rIBM). The equivalence between the rIBM and the original problem is proved in a theorem. Then an efficient algorithm for finding two unknowns is suggested using the Schur complement. We provide numerical results which show that the proposed rIBM improves both the numerical stability and the computational speed when solving more realistic problems.
2010 Mathematics Subject Classification: 76Rxx
Key Words and Phrases: Immersed boundary method, projection approach, fluid-structure interaction
- ⋅ 28th-A-09:20 − 09:40 Mathematical and computational modeling of ephaptic coupling between the myelinated nerve fibers (Sehun Chun)
- 천세훈((연세대))
Sehun Chun, Yonsei University
The classical deterministic methods of tractography hypothesize the streamline of white matter fiber. The neural fiber is assumed to be not much different from the structure of the cable, but the neural fiber has a more complex structure, such as the Ranvier node. This tiny node mainly provides additional ions to the action potential through the membrane channels to retrieve the original signal strength, which is leaked during the propagation in myelin. In recent years, there have been more conjectures on the roles of the Ranvier nodes, particularly in synchronization and interference by neighboring fibers’ neural spikes, known as ephaptic coupling. The current theoretical studies with oversimplification on the fiber configuration fail to reveal many critical functions of the Ranvier node in information propagation. This study aims to confirm and shed new light on the role of the Ranvier nodes in various configurations of the fiber bundle, which may lead to a better understanding of the meaning of fiber bundle geometry in the brain. Mathematical modeling and computational simulations of ephaptic coupling in multidimensional dimensional space are proposed to reveal the role of ephaptic coupling by observing the geometry of the fiber bundle and understanding how the information is delivered and stored by the neural fiber bundle.
2010 Mathematics Subject Classification: 92C20, 92C05, 65N30
Key Words and Phrases: Neural spike, ephaptic coupling, diffusion-reaction equation
- ⋅ 28th-A-09:50 − 10:10 A postprocessing for the hybrid difference method for elliptic problems (Dongwook Shin, Youngmok Jeon, Eun-Jae Park)
- 신동욱*((아주대)), 전영목((아주대)), 박은재((연세대))
Dongwook Shin*, Ajou University, Youngmok Jeon, Ajou University, Eun-Jae Park, Yonsei University
In this talk, we consider a postprocessing for the hybrid difference method (Shin--Jeon--Park, Appl. Math. Comput., 2022) for elliptic problems. The hybrid difference (HD) method can be seen a finite difference method based on the hybrid discontinuous Galerkin method introduced by Jeon and Park (SIAM J. Numer. Anal., 2010). The HD method allows arbitrarily high-order approximations, and the local conservation property holds. Also, the method has great reduction in global degrees of freedom. In the previous work (Jeon-Park-Shin, Comput. Methods Appl. Math., 2017), due to the missing information, it is hard to calculate the approximation value at any point without loosing accuracy. Here, we developed and generalized the HD method by introducing a simple postprocessing. Also the method can be extended to more complex domain geometry with a simple modification. In this case the local conservation property may not hold, so we need a postprocessing for that. Several numerical experiments are presented to show the performance of the proposed method, which support our theoretical findings.
2010 Mathematics Subject Classification: 65N06, 65N30
Key Words and Phrases: Hybrid discontinuous Galerkin, finite difference, postprocessing
- ⋅ 28th-A-10:10 − 10:30 Flow computations in unbounded domain using Axial Green function Method (Junhong Jo, Wanho Lee, Do Wan Kim)
- 조준홍((인하대)), 이완호((국가수리과학연구소)), 김도완*((인하대))
Junhong Jo, Inha University, Wanho Lee, NIMS, Do Wan Kim*, Inha University
We consider the flow problems in unbounded domains. The unboundedness of a domain makes trouble in calculating numerical solutions of partial differential equations. Particularly, flow problems have important information on the far-field behavior of velocity, for instance, the circulation of inviscid flows, the stress in viscous flows, etc. The axial Green function methods have been developed for numerical computations in complicated domains. Using another type of axial Green function, we efficiently handle the far-field boundary condition, which we call the informative boundary condition. As a result, the consistency of the discretization is validated through numerical experiments. Particularly, the one-dimensional problem is theoretically analyzed.
2010 Mathematics Subject Classification: 65N99
Key Words and Phrases: Unbounded domain, informative boundary condition, numerical solutions
- ⋅ 28th-B-10:50 − 12:20 Chair: Youngjoon Hong (Sungkyunkwan University)
- ⋅ 28th-B-10:50 − 11:10 Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs (Seungchan Ko, Seok-Bae Yun, Youngjoon Hong)
- 고승찬*((인하대)), 윤석배((성균관대)), 홍영준((성균관대))
Seungchan Ko*, Inha University, Seok-Bae Yun, Sungkyunkwan University, Youngjoon Hong, Sungkyunkwan University
In this talk, I will discuss the convergence analysis of unsupervised Legendre-Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based numerical methods for PDEs, the ULGNet expresses the solution as a spectral expansion with respect to the Legendre basis and predicts the coefficients with deep neural networks by solving a variational residual minimization problem. Using the fact that the corresponding loss function is equivalent to the residual induced by the linear algebraic system depending on the choice of basis functions, we prove that the minimizer of the discrete loss function converges to the weak solution of the PDEs. Numerical evidence will also be provided to support the theoretical result. Key technical tools include the variant of the universal approximation theorem for bounded neural networks, the analysis of the stiffness and mass matrices, and the uniform law of large numbers in terms of the Rademacher complexity.
2010 Mathematics Subject Classification: 68T07, 65N35, 65K10, 65N12, 35J25
Key Words and Phrases: Deep neural network, unsupervised learning, elliptic partial differential equations, Legendre-Galerkin approximation, spectral element method, Rademacher complexity
- ⋅ 28th-B-11:10 − 11:30 Learning solution operators of partial differential equations and their application (Jae Yong Lee, SungWoong Cho, Hyung Ju Hwang)
- 이재용*((고등과학원)), 조성웅((포항공대)), 황형주((포항공대))
Jae Yong Lee*, KIAS, SungWoong Cho, POSTECH, Hyung Ju Hwang, POSTECH
Many physical phenomena in nature are modeled using mathematical expressions through differential equations and partial differential equations (PDEs). Recently, there has been a growing interest in using neural networks for operator learning to approximate PDE solution operators, which refers to an operator that maps the parameters of the PDE to its solution. In this talk, I will introduce several recently proposed neural network architectures for approximating PDE solution operators and my research related to them.
2010 Mathematics Subject Classification: 68T07, 65M99, 65N99
Key Words and Phrases: PDE, solution operator, operator learning, deep operator network
- ⋅ 28th-B-11:40 − 12:00 Bayesian deep learning framework for uncertainty quantification in stochastic systems (Minseok Choi, Jeahan Jung)
- 최민석*((포항공대)), 정재한((포항공대))
Minseok Choi*, POSTECH, Jeahan Jung, POSTECH
Despite of great progress over the last decades in simulating complex problems
with the numerical discretization of (stochastic) partial differential equations
(PDEs), solving high-dimensional problems governed by parameterized PDEs remains
challenging. Machine learning has emerged as a promising alternative in
scientific computing community by enforcing the physical laws. We present a
novel Bayesian deep learning method for uncertainty quantification. The Bayesian
neural network is employed to learn the distribution of the solution by encoding
the governing physical law of the given stochastic partial differential
equations into the posterior distribution of the network parameters. We present
some efficient methodology for sampling the posterior distribution. Numerial
examples are provided to illustrate the efficiency of the proposed algorithm.
2010 Mathematics Subject Classification: 68T07
Key Words and Phrases: Bayesian neural network, uncertainty quantification, physics-informed machine learning
- ⋅ 28th-B-12:00 − 12:20 Two-timescale extragradient converges to local minimax points (Jiseok Chae, Kyuwon Kim, Donghwan Kim)
- 채지석((카이스트)), 김규원((카이스트)), 김동환*((카이스트))
Jiseok Chae, KAIST, Kyuwon Kim, KAIST, Donghwan Kim*, KAIST
We show that two-timescale extragradient converges to local minimax points without a nondegeneracy condition on the Hessian with respect to the maximization variable. We then prove that it avoids strict non-minimax points, almost surely with random initialization.
2010 Mathematics Subject Classification: 90C47, 90C26, 90C46
Key Words and Phrases: Sequential game, local minimax points, two-timescale, extragradient
- Data Science for Modeling Biochemical Systems
- ⋅ 28th-C-13:30 − 15:00 Chair: Jinsu Kim (POSTECH)
- ⋅ 28th-C-13:30 − 13:50 General model-based inference for causality in biochemical reaction systems (Jae Kyoung Kim)
- 김재경((카이스트))
Jae Kyoung Kim, KAIST
To identify causation, model-free inference methods, such as Granger Causality, have been widely used due to their flexibility. However, they have difficulty distinguishing synchrony and indirect effects from direct causation, leading to false predictions. To overcome this, model-based inference methods were developed that test the reproducibility of data with a specific mechanistic model to infer causality. However, they can only be applied to systems described by a specific model, greatly limiting their applicability. Here, we address this limitation by deriving an easily-testable condition for a general ODE model to reproduce time-series data. We built a user-friendly computational package, GOBI (General ODE-Based Inference), which is applicable to nearly any system described by ODE. GOBI successfully inferred positive and negative regulations in various networks at both molecular and population levels, unlike existing model-free methods. Thus, this accurate and broadly-applicable inference method is a powerful tool for understanding complex dynamical systems.
2010 Mathematics Subject Classification: 92C42
Key Words and Phrases: Causality inference
- ⋅ 28th-C-13:50 − 14:10 Density physics-informed neural network infers an arbitrary density distribution for non-Markovian system (Hyeontae Jo, Hyukpyo Hong, Hyung Ju Hwang, Won Chang, Jae Kyoung Kim)
- 조현태*((기초과학연구원 의생명수학그룹)), 홍혁표((카이스트)), 황형주((포항공대)), 장원((University of Cincinnati)), 김재경((카이스트))
Hyeontae Jo*, IBS-BIMAG, Hyukpyo Hong, KAIST, Hyung Ju Hwang, POSTECH, Won Chang, University of Cincinnati, Jae Kyoung Kim, KAIST
The signal transduction time between signal initiation and final response provides valuable information on the underlying signaling pathway, including its speed and precision. Furthermore, multimodality in transduction-time distribution informs that the response is regulated by multiple pathways with different transduction speeds. Here, we developed Density physics-informed neural network (Density-PINN) to infer the transduction-time distribution, challenging to measure, from measurable final stress response time traces. We applied Density-PINN to single-cell gene expression data from 16 promoters regulated by unknown pathways in response to antibiotic stresses. We found that promoters with slower signaling initiation and transduction exhibit larger cell-to-cell heterogeneity in response intensity. However, this heterogeneity was greatly reduced when the response was regulated by slow and fast pathways together. This suggests a way to identify effective signaling pathways for consistent cellular responses to disease treatments. Density-PINN can also be applied to understand infectious diseases and climate dynamics with time delay.
2010 Mathematics Subject Classification: 68T07, 92B05, 62F30
Key Words and Phrases: Physics-informed, density distribution inference, non-Markovian, cell-to-cell heterogeneity
- ⋅ 28th-C-14:20 − 14:40 Combining data assimilation and deep learning to identify underlying physiological states from wearable data (Dae Wook Kim, Caleb Mayer, Minki P. Lee, Sung Won Choi, Muneesh Tewari, Daniel B. Forger)
- 김대욱*((University of Michigan)), Caleb Mayer((University of Michigan)), 이민기((University of Michigan)), 최성원((University of Michigan)), Muneesh Tewari((University of Michigan)), Daniel B. Forger((University of Michigan))
Dae Wook Kim*, University of Michigan, Caleb Mayer, University of Michigan, Minki P. Lee, University of Michigan, Sung Won Choi, University of Michigan, Muneesh Tewari, University of Michigan, Daniel B. Forger, University of Michigan
Daily (24 h) rhythms of physiological processes such as sleep and hormone secretion are coordinated by an endogenous biochemical timer, the circadian clock. Depending on the internal clock state, the efficacy of diverse drugs largely changes. Despite the benefits of considering the clock state, current clinical practice guidelines have largely ignored it due to the lack of reliable methods to identify the internal state in the real world. In this talk, I will present a multi-time scale data assimilation approach to estimate the internal state from wearable data (e.g., Apple Watch activity measures) based on a new extension of Kalman filtering that reformulates the underlying Fokker-Planck equation for the internal system as a system of ordinary differential equations. I will also introduce a Kalman filter-assisted neural network approach for early detection of aberrant changes in circadian physiology related to disease progression from wearable measurements. The mathematics of the wearables can pave the way toward precision medicine in the real world.
2010 Mathematics Subject Classification: 37N25, 68T07, 92B25, 92C30, 92-08, 92-10
Key Words and Phrases: Data assimilation, Kalman filter, neural network, nonlinear estimation, mathematical models, wearables, circadian rhythms
- ⋅ 28th-C-14:40 − 15:00 Feature engineering for abnormal detection of three-dimensional gait analysis (Seunggyu Lee, Kwonhee Lee, Singi Park)
- 이승규*((고려대)), 이권희((현대모비스)), 박신기((에이트스튜디오))
Seunggyu Lee*, Korea University, Kwonhee Lee, Hyundai MOBIS, Singi Park, AIT Studio
The study of gait analysis in human biomechanics is essential for patients suffering from incurable diseases such as Parkinson's and Alzheimer's. However, the complexity and non-uniqueness of correlations between individuals make it a challenging problem. Although statistical tools and machine learning techniques are widely used in gait analysis, they still lack the predictive power to detect disorders in a timely and accurate manner. In this talk, we present a simple and efficient feature engineering method for detecting abnormalities in three-dimensional gait data. We demonstrate classification results using this method, which is based on the difference in a topological property of data generated from both vision- and sensor-based sources.
2010 Mathematics Subject Classification: 62R40
Key Words and Phrases: Feature engineering, gait analysis
- ⋅ 29th-D-09:00 − 10:30 Chair: Jae Kyoung Kim (KAIST)
- ⋅ 29th-D-09:00 − 09:25 Neural network implementation via biochemical reaction systems (Jinsu Kim, Sunghwa Kang)
- 김진수*((포항공대)), 강성화((포항공대))
Jinsu Kim*, POSTECH, Sunghwa Kang, POSTECH
Recently, Anderson, Joshi, and Deshpande constructed reaction networks implementing neural networks. By using enzymes as inputs, these biology-based machine learning algorithms create a function as an output, that is made with compositions of smoothed ReLU functions as the original neural networks do. However, they did not implement the part of training the neural networks, which is the critical part of machine learning, by using reaction networks. In this work, as a milestone of `training neural networks by reaction networks', we designed a simple biochemical reaction network that can compute derivatives of smoothed ReLU functions and their
2010 Mathematics Subject Classification: 92B05, 68T07
Key Words and Phrases: Reaction network, neural networks, training networks, gradient descent algorithm
- ⋅ 29th-D-09:30 − 09:55 Physics-informed variational inference for uncertainty quantification of stochastic differential equations (Hyomin Shin, Minseok Choi)
- 신효민*((포항공대)), 최민석((포항공대))
Hyomin Shin*, POSTECH, Minseok Choi, POSTECH
We propose physics-informed learning based on variational autoencoder (VAE) to solve data-driven stochastic differential equations when the governing equation is known and a limited number of measurements are available. Our model integrates VAE with the given random differential equations so that the encoder of our model infers the randomness of the solution. The decoder consists of two networks; one network learns the spatial behavior and the other networks learns the random behavior of the solution. For the training step, we derive an evidence lower bound (ELBO) that includes the given physical law by using automatic differentiation to compute the differential operators. The proposed model presents a unified framework to solve data-driven forward and inverse stochastic partial differential equations. We provide the application of our model for learning stochastic processes and solving stochastic partial differential equations.
2010 Mathematics Subject Classification: 68T07
Key Words and Phrases: Physics-informed learning, generative model, uncertainty quantification, data-driven modeling
- ⋅ 29th-D-10:00 − 10:30 Enhancing downstream analysis of scRNA-seq data through automatic signal distortion correction and dimensionality reduction using random matrix theory (Hyun Kim, JongEun Park, MinSeok Seo, Jae Kyoung Kim)
- 김현*((기초과학연구원)), 박종은((카이스트)), 서민석((고려대)), 김재경((기초과학연구원))
Hyun Kim*, IBS, JongEun Park, KAIST, MinSeok Seo, Korea University, Jae Kyoung Kim, IBS
Single-cell sequencing technology has been utilized since its inception in 2009 to carry out various analyses, including cellular phenotyping via clustering analysis and reconstruction of gene regulatory networks using gene-directed clustering analysis by leveraging the abundant information in scRNA-seq data. However, the sparsity, high dimensionality, bias, skewness in gene distribution, and technical noise in scRNA-seq data pose challenges to downstream analyses like clustering analysis. To address these issues, most packages rely on log-normalization for preprocessing and PCA for dimensionality reduction. Nonetheless, standard log normalization causes signal distortion due to the high number of zeros in scRNA-seq, while the dimension of the reduced data after PCA is left up to the user, resulting in subjectivity and reduced reliability of the outcomes. To mitigate these concerns, we have developed a software package that includes a step to correct signal distortion during preprocessing and automatically calculates the dimension of the reduced data using the universality of random matrix theory to enhance the reliability of the results. Additionally, our package utilizes post-noise-filtering to select only signals from patterns with biologically meaningful zeros, capturing only signals that signify biologically significant differences between cells. The performance of our package demonstrated to surpass those of 11 other tools, including Seurat and Monocle3, using 53 real and simulated data sets.
2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: scRNA-seq data, random matrix theory, dimensionality reduction, noise filtering, low-dimensional embedding, cell clustering
- Applied Algebra and Optimization
- ⋅ 28th-A-09:00 − 10:30 Chair: Soonhak Kwon (Sungkyunkwan University)
- ⋅ 28th-A-09:00 − 09:30 Diagonal quinary quadratic forms with a strong regularity property (Kyoungmin Kim)
- 김경민((한남대))
Kyoungmin Kim, Hannam University
Let $f$ be a positive definite integral quinary quadratic form. We say $f$ is {\it strongly $s$-regular} if it satisfies a strong regularity property on the number of representations of squares of integers by $f$. In this talk, we show that there exist exactly $19$ strongly $s$-regular diagonal quinary quadratic forms representing $1$. In particular, we use eta-quotients of weight $\frac52$ to prove the strongly $s$-regularity of the quinary quadratic form $x^2+2y^2+2z^2+3w^2+3t^2$, which is, in fact, of class number $4$.
2010 Mathematics Subject Classification: 11E12, 11E20, 11E45
Key Words and Phrases: Representations of quinary quadratic forms, squares, eta-quotients
- ⋅ 28th-A-09:30 − 10:00 Transfer principles and a subdifferential formula in Fan-Theobald-von Neumann systems (Juyoung Jeong, Muddappa Seetharama Gowda)
- 정주영*((성균관대)), Muddappa Seetharama Gowda((University of Maryland Baltimore County))
Juyoung Jeong*, Sungkyunkwan University, Muddappa Seetharama Gowda, University of Maryland Baltimore County
A Fan-Theobald-von Neumann system is a triple $(\mathcal{V},\, \mathcal{W},\, \lambda)$, where $\mathcal{V}$ and $\mathcal{W}$ are real inner product spaces and $\lambda : \mathcal{V} \to \mathcal{W}$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). In this talk, we describe transfer principles for spectral sets and functions. Specifically, we study, in an appropriate setting, which properties of spectral sets and functions on $\mathcal{W}$ can be transferred to their corresponding spectral sets and functions on $\mathcal{V}$. Also, we derive the subdifferential formulas for spectral functions. These results extends the existing results in Euclidean Jordan algebras, normal decomposition systems, and some other familiar settings.
2010 Mathematics Subject Classification: 17C20, 46N10, 90C25
Key Words and Phrases: Fan-Theobald-von Neumann system, eigenvalue map, spectral set and function, transfer principle
- ⋅ 28th-A-10:00 − 10:30 Riordan group posets and their combinatorics (Minho Song, Louis W. Shapiro)
- 송민호*((성균관대 응용대수 및 최적화 연구센터)), Louis W. Shapiro((Howard University))
Minho Song*, AORC, Sungkyunkwan University, Louis W. Shapiro, Howard University
For two matrices $A$ and $B$ of the same size, let $T$ be the (right) transit matrix from $A$ to $B$ if $AT=B$.
If $AT=B$ and the transit matrix T consists of nonnegative integers, then we say $A<B$. In this talk, we present various posets with the order defined by the transit matrix. Each of them is related to well-known lattice paths such as generalized Dyck paths, Schroeder paths, and Motzkin paths. In particular, for the poset related to the Motzkin paths, we illustrate a bijection between generalized $(k+1)$-Motzkin paths and a pair of $k$-Motzkin paths and the peakless Motzkin paths.
2010 Mathematics Subject Classification: 05A19
Key Words and Phrases: Poset, Riordan group, bijection, lattice paths
- ⋅ 28th-B-10:50 − 12:20 Chair: Sukmoon Huh (Sungkyunkwan University)
- ⋅ 28th-B-10:50 − 11:20 Structural matrix rings as rudimentary rings (Gangyong Lee, Mauricio Medina-B\'arcenas, Nguyen Khanh Tung)
- 이강용*((충남대)), Mauricio Medina-B\'arcenas((Benem\'erita Universidad Aut\'onoma de \linebreak Puebla)), Nguyen Khanh Tung((Vietnam National University))
Gangyong Lee*, Chungnam National University, Mauricio Medina-B\'arcenas, Benem\'erita Universidad Aut\'onoma de Puebla, Nguyen Khanh Tung, Vietnam National University
In 1945, Jacobson introduced the notion of primitive rings
and obtained structure theorem for primitive rings as analogues of the fundamental Wedderburn-Artin structure theorem for semisimple artinian rings.
The existence of a faithful simple module plays a crucial role in studying primitive rings.
In particular, Schur showed that the endomorphism ring of a simple module is a division ring, but the converse does not hold true, in general. In 1949, T. Szele obtained the result for abelian groups: The endomorphism ring of an abelian group $G$ is a division ring iff $G$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Z}_p$ where $p$ is prime. That is, there is no noncommutative division ring which can serve as the endomorphism ring of an abelian group. In 1970, Ware and Zelmanowitz extended Szele's result that there is no noncommutative division ring as the endomorphism ring of a module over a commutative ring. Recently, Lee, Roman, and Zhang defined a partial matrix ring in 2014 as an example of a rudimentary ring.
We call an $n\times n$ \emph{partial matrix ring over a ring $A$}, denoted
by $\mathsf{PM}_n(A)$, a subring of a full $n\times n$ matrix ring over $A$, with elements matrices whose entries are either elements of $A$ or $0$, such that nonzero entries are independent of each other.
Note that it is the same as the notion of a structural matrix ring.
In this talk, we briefly introduce the notion of a rudimentary ring.
we will provide rudimentary rings as subrings of $n\times n$ full matrix rings.
Also, by using the concept of preordered sets, transitive directed graphs, and Hasse diagrams,
we discuss how to get structural matrix rings as subrings of $n\times n$ full matrix rings.
This talk is based on joint work with Mauricio Medina-B\'arcenas and Khanh Tung Nguyen.
2010 Mathematics Subject Classification: 05C25, 15B99, 06A06
Key Words and Phrases: Rudimentary rings, structural matrix rings, preordered sets, transitive directed graphs
- ⋅ 28th-B-11:20 − 11:50 On ideals associated to Gaussian graphical models (Kangjin Han)
- 한강진((대구경북과학기술원))
Kangjin Han, DGIST
For any graph $G$, one can associate $G$ with a statistical model called 'Gaussian graphical model' which parametrizes a set of multivariate Gaussian distributions. This set can be seen as an algebraic variety in a suitable embedding, so in principle, geometric viewpoints and algebraic techniques can be adopted to answer questions arising from statistics. Many fundamental questions for the geometry of this variety and its algebra are widely open. In this talk, we consider the prime ideal defining this variety when the given $G$ is a cycle graph. This is a joint work with A. Conner and M. Michalek.
2010 Mathematics Subject Classification: 62R01, 14M17, 14N10
Key Words and Phrases: Multivariate Gaussian distributions, Gaussian Graphical model, Reciprocal varieties, Cycle graph, Groebner basis, Minimal generating set
- ⋅ 28th-B-11:50 − 12:20 On some homogeneous contact metric spaces (Yunhee Euh)
- 어윤희((성균관대))
Yunhee Euh, Sungkyunkwan University
In this talk, we focus on a 3-dimensional homogeneous contact metric manifiold. We explain the classification of 3-dimensional homogeneous contact metric manifolds based on results of Milnor and Perrrone, respectively. Here, we introduce a new geometric structure and give a classificaion of 3-dimensional homogeneous contact metric manifolds.
2010 Mathematics Subject Classification: 53C25
Key Words and Phrases: Homogeneous metric space, contact metric manifold
- Extremal Combinatorics: Methods and Applications
- ⋅ 28th-A-09:00 − 10:30 Chair: Jinha Kim (IBS-DIMAG)
- ⋅ 28th-A-09:00 − 09:20 Domination inequalities and dominating graphs (David Conlon, Joonkyung Lee)
- David Conlon((Caltech)), 이준경*((연세대))
David Conlon, Caltech, Joonkyung Lee*, Yonsei University
We say that a graph $H$ dominates another graph $H'$ if the number of homomorphisms from $H'$ to any graph $G$ is dominated, in an appropriate sense, by the number of homomorphisms from $H$ to $G$. We study the family of dominating graphs, those graphs with the property that they dominate all of their subgraphs. It has long been known that even-length paths are dominating in this sense and a result of Hatami implies that all weakly norming graphs are dominating. In a previous paper, we showed that every finite reflection group gives rise to a family of weakly norming, and hence dominating, graphs. Here we revisit this connection to show that there is a much broader class of dominating graphs.
2010 Mathematics Subject Classification: 05C35, 05A20
Key Words and Phrases: Graph homomorphism inequalities
- ⋅ 28th-A-09:20 − 09:40 $C_5$-critical series parallel graphs (Eun-Kyung Cho, Ilkyoo Choi, Boram Park, Mark H. Siggers)
- 조은경((한국외대)), 최일규((한국외대)), 박보람((아주대)), Mark H. Siggers* ((경북대))
Eun-Kyung Cho, Hankuk University of Foreign Studies, Ilkyoo Choi, Hankuk University of Foreign Studies, Boram Park, Ajou University, Mark H. Siggers*, Kyungpook National Univeristy
An open case of Jaeger's conjecture applied to planar graphs can be stated by saying that all planar graphs of girth at least 8 admit a homomorphism to $C_5$ (a circular $(5,2)$-colouring). Restricting further to series parallel graphs ($K_4$-minor-free graphs), Pan and Zhu showed that all series parallel graphs of girth at least 7 admit a homomorphims to $C_5$, but that there exist series parallel graphs of girth $5$ that do not.
A graph is $C_5$-critical if it does not admit a homomorphism to $C_5$, but every proper subgraph does. Characterising the $C_5$-critical graphs, we show that any series parallel graph of girth $5$, in which no two $5$-cycles share a vertex, has a homomorphism to $C_5$.
2010 Mathematics Subject Classification: 05C15, 05C10, 05C35
Key Words and Phrases: Jaeger's conjecture, circular colourings, series parallel, graph homomorphisms
- ⋅ 28th-A-09:50 − 10:10 On the extremal problems related to Szemeredi's theorem (Younjin Kim)
- 김연진((기초과학연구원 극단 조합 및 확률 그룹))
Younjin Kim, IBS-ECOPRO
In 1975, Szemeredi proved that for every real number $\delta > 0 $ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set $\{1, 2, \dots, N \}$ with $|A| \geq \delta N$ contains an arithmetic progression of length $k$. There has been a plethora of research related to Szemeredi's theorem in many areas of mathematics. In 1990, Cameron and Erdos proposed a conjecture about counting the number of subsets of the set $\{1,2, \dots, N\}$ which do not contain an arithmetic progression of length $k$. In the talk, we study a natural higher dimensional version of this conjecture, and also introduce recent extremal problems related to Szemeredi's theorem.
2010 Mathematics Subject Classification: 05D05
Key Words and Phrases: Szemeredi's theorem, arithmetic progression
- ⋅ 28th-A-10:10 − 10:30 Bounds on maximum directed cut (Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong zhou)
- Jiangdong Ai*((The State University of New York, Korea)), Stefanie Gerke((Royal Holloway, University of London)), Gregory Gutin((Royal Holloway, University of London)), Anders Yeo((University of Southern Denmark)), Yacong zhou((Royal Holloway, University of London))
Jiangdong Ai*, The State University of New York, Korea, Stefanie Gerke, Royal Holloway, University of London, Gregory Gutin, Royal Holloway, University of London, Anders Yeo, University of Southern Denmark, Yacong zhou, Royal Holloway, University of London
We obtain lower and upper bounds for the maximum weight of a directed cut in the classes of weighted digraphs and weighted acyclic digraphs as well as in some of their subclasses. We compare our results with those obtained for the maximum size of a directed cut in simple digraphs and state some open problems.
2010 Mathematics Subject Classification: 05c20
Key Words and Phrases: Digraph, cut, weights
- ⋅ 28th-B-10:50 − 12:20 Chair: Zixiang Xu (IBS)
- ⋅ 28th-B-10:50 − 11:10 Rainbow cycles in edge-colored graphs (Joonkyung Lee, Jaehoon Kim, Hong Liu, Tuan Tran)
- 이준경((연세대)), 김재훈*((카이스트)), Hong Liu((기초과학연구원)), Tuan Tran((University of Science and Technology of China))
Joonkyung Lee, Yonsei University, Jaehoon Kim*, KAIST, Hong Liu, IBS, Tuan Tran, University of Science and Technology of China
We prove that every properly edge-colored $n$-vertex graph with average degree at least $32(\log 5n)^2$ contains a rainbow cycle, improving upon
$(\log n)^{2+o(1)}$ bound due to Tomon.
We also prove that every properly colored $n$-vertex graph with at least $10^5 k^3 n^{1+1/k}$ edges contains a rainbow $2k$-cycle, which improves the previous bound $2^{ck^2}n^{1+1/k}$ obtained by Janzer.
Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erd\H{o}s--Simonovits supersaturation theorem for even cycles, which may be of independent interest.
2010 Mathematics Subject Classification: 05C35
Key Words and Phrases: Transversal, Dirac theorem, bandwidth theorem
- ⋅ 28th-B-11:10 − 11:30 How connectivity affects the extremal number of trees (Suyun Jiang, Hong Liu, Nika Salia)
- Suyun Jiang*((Jianghan University \& IBS)), Hong Liu((IBS)), Nika Salia((IBS))
Suyun Jiang*, Jianghan University \& IBS, Hong Liu, IBS, Nika Salia, IBS
The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$.
Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.
2010 Mathematics Subject Classification: 05C35
Key Words and Phrases: Connected Tur\'an number, tree, broom
- ⋅ 28th-B-11:40 − 12:00 Many Hamiltonian subsets in large graphs with given density (Stijn Cambie, Jun Gao, Hong Liu)
- Stijn Cambie((IBS-ECOPRO)), Jun Gao*((IBS-ECOPRO)), Hong Liu((IBS-ECOPRO))
Stijn Cambie, IBS-ECOPRO, Jun Gao*, IBS-ECOPRO, Hong Liu, IBS-ECOPRO
A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ($\approx 2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graphs with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.
2010 Mathematics Subject Classification: 05C35, 05C38, 05C48
Key Words and Phrases: Hamiltonian subset, expander, optimal lower bound
- ⋅ 28th-B-12:00 − 12:20 Maximum total distance of hypergraphs (Stijn Cambie, Ervin Gy\H{o}ri, Nika Salia, Casey Tompkins, James Tuite)
- Stijn Cambie*((IBS)), Ervin Gy\H{o}ri((Alfr\'ed R\'enyi Institute of Mathematics, Budapest)), Nika Salia((IBS)), Casey Tompkins((Alfr\'ed R\'enyi Institute of Mathematics, Budapest)), James Tuite((Open University, Milton Keynes))
Stijn Cambie*, IBS, Ervin Gy\H{o}ri, Alfr\'ed R\'enyi Institute of Mathematics, Budapest, Nika Salia, IBS, Casey Tompkins, Alfr\'ed R\'enyi Institute of Mathematics, Budapest, James Tuite, Open University, Milton Keynes
The total distance of a (hyper)graph is the sum of distances between all pairs of vertices.
It is an elementary and folklore result that the minimum and maximum of the total distance among connected graphs of order n is unique attained by the clique and path.
We present the extension to uniform hypergraphs, which is a particularly intriguing result, as it is both more involved and more surprising.
2010 Mathematics Subject Classification: 05C65, 05C09, 05C12, 05C35
Key Words and Phrases: Total distance, hypergraphs
- ⋅ 28th-C-13:30 − 15:00 Chair: Minho Cho (IBS)
- ⋅ 28th-C-13:30 − 13:50 Intersection patterns and incidence theorems (Thang Pham, Semin Yoo)
- Thang Pham((Vietnam National University)), 유세민*((고등과학원))
Thang Pham, Vietnam National University, Semin Yoo*, KIAS
Let $A$ and $B$ be compact sets in $\mathbb{R}^d$. One of the fundamental problems in Geometric Measure Theory is to study the relations between the Hausdorff dimensions of $A$, $B$, and $A\cap f(B)$, where $f$ runs through a set of transformations. This type of question has been studied intensively by many mathematicians such as Bishop, Elekes, Falconer, Mattila, Kahane, Keleti, and Peres. In this talk, I discuss recent results on the discrete analog of this topic. Our approach is based on a number of techniques including algebraic methods and discrete Fourier Analysis. This is joint work with Thang Pham.
2010 Mathematics Subject Classification: 39A12
Key Words and Phrases: Intersection, rigid motions, incidences
- ⋅ 28th-C-13:50 − 14:10 Note on the quotient set of the quadratic distance set over finite fields (Doowon Koh)
- 고두원((충북대))
Doowon Koh, Chungbuk National University
Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\mathbb F_q$ and $E\subset \mathbb F_q^d$, we define $W(r)$ as the number of quadruples $(x,y,z,w)\in E^4$ such that $
Q(x-y)/Q(z-w)=r,$ where $Q$ is a non-degenerate quadratic form in $d$ variables over $\mathbb F_q.$ In this talk we introduce a new method for estimating an lower bound of $W(r).$ As a consequence, we obtain the optimal result on the Falconer type problem for the quotient set of distance set over finite fields. The discrete Fourier analysis and the Gauss sum estimates play an important role in deducing the sharp results.
2010 Mathematics Subject Classification: 05D05
Key Words and Phrases: Quotient set, distance problem
- ⋅ 28th-C-14:20 − 14:40 Convexity and chi-boundedness (Andreas Holmsen)
- 28th-C-14:20 − 14:40
Andreas Holmsen, KAIST
In this talk we introduce some interesting chi-bounded classes of k-uniform hypergraphs. The proof that these classes are chi-bounded is based on extremal results from the theory of abstract convexity spaces.
2010 Mathematics Subject Classification: 52A35
Key Words and Phrases: Helly-type theorems, abstract convexity
- ⋅ 28th-C-14:40 − 15:00 Exceptional projections in finite vector spaces (Ben Lund)
- 28th-C-14:40 − 15:00
Ben Lund, IBS-DIMAG
A general phenomenon in is that most projections are nearly as large as possible. I will discuss recent results bounding the number of very small projections of sets of points in finite dimensional vector spaces over finite fields. I will mention some connections to fractal geometry. This talk will be based on joint works with Paige Bright, Thang Pham, and Vu Thi Huong Thu.
2010 Mathematics Subject Classification: 05D99
Key Words and Phrases: Incidences, projections, finite geometry
- ⋅ 29th-D-09:00 − 10:30 Chair: Alexander Clifton (IBS-DIMAG)
- ⋅ 29th-D-09:00 − 09:20 Rainbow bandwidth theorem (Debsoumya Chakraborti, Seonghyuk Im, Jaehoon Kim, Hong Liu)
- Debsoumya Chakraborti((IBS-DIMAG)), 임성혁*((카이스트)), 김재훈((카이스트)), Hong Liu ((IBS-ECOPRO))
Debsoumya Chakraborti, IBS-DIMAG, Seonghyuk Im*, KAIST, Jaehoon Kim, KAIST, Hong Liu, IBS-ECOPRO
For a collection of graphs $\mathcal{G}=(G_1, \ldots, G_m)$ on the common vertex set $V=V(\mathcal{G})$, we say that a graph $H \subseteq K_{V}$ is called \emph{rainbow} if there exists an injection $\varphi:E(H) \rightarrow [m]$ such that $e \in E(G_{\varphi(e)})$ for every $e \in E(H)$. When $\varphi$ is a bijection, we say $H$ is a \emph{$\mathcal{G}$-transversal}.
The conditions on the minimum degree $\delta(\mathcal{G})=\min_{i\in[h]}\{ \delta(G_i)\}$ for finding a spanning $\mathcal{G}$-transversal isomorphic to a graph $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a spanning tree with maximum degree $o(n/\log n)$ and power of a Hamilton cycle, etc.
This talk will introduce the transversal generalization of the bandwidth theorem by B\"ottcher, Schacht, and Taraz. More precisely, for a graph $H$ and collection of graphs $\mathcal{G}$ on $n$ vertices, if $\delta(\mathcal{G}) \geq (1-1/(\chi(H)-1)+o(1))n$, $H$ has bounded maximum degree, and bandwidth $o(n)$, then there is a $\mathcal{G}$-transversal isomorphic to $H$. Furthermore, the number $1-1/(\chi(H)-1)$ is tight.
This is joint work with Debsoumya Chakraborti, Jaehoon Kim, and Hong Liu.
2010 Mathematics Subject Classification: 05C35
Key Words and Phrases: Bandwidth theorem, rainbow subgraph
- ⋅ 29th-D-09:20 − 09:40 Covering multigraphs with bipartite graphs (Hyunwoo Lee)
- 이현우((카이스트))
Hyunwoo Lee, KAIST
Hansel's lemma states that $\sum_{H\in \mathcal{H}}|H| \geq n \log_2 n$ holds where $\mathcal{H}$ is a collection of bipartite graphs covering all edge of $K_n$. We generalize this statement to obtain statements about a complete multigraph covering problem and a graphon covering problem. We also prove an upper bound for the complete multigraph covering problem which is asymptotically tight for many values of the multiplicity $\lambda$ of the multigraph.
2010 Mathematics Subject Classification: 05C35, 05D40
Key Words and Phrases: Probabilistic method, graph covering, graphon
- ⋅ 29th-D-09:50 − 10:10 Rainbow oriented Hamiltonian paths and cycles in tournaments (Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo)
- Debsoumya Chakraborti((IBS-DIMAG)), 김재훈((카이스트)), 이현우((카이스트)), 서재현*((연세대))
Debsoumya Chakraborti, IBS-DIMAG, Jaehoon Kim, KAIST, Hyunwoo Lee, KAIST, Jaehyeon Seo*, Yonsei University
It is well known that any tournament contains a directed Hamiltonian path, and any strongly connected tournament contains a directed Hamiltonian cycle. Rosenfeld (1974) conjectured that every sufficiently large tournament contains every non-strongly oriented Hamiltonian cycle. This was first proved by Thomason (1986) with the lower bound \(2^{128}\) on the order, and recently Ayman (2023) improved this by explicitly listing 30 exceptions, all of order less than \(9\). Rosenfeld also proposed a similar conjecture for Hamiltonian paths, which was proved by Havet and Thomass\'{e} (2000).
We proved rainbow generalizations of these results. For a collection \(\mathbf{T}=\{T_1,\dots,T_m\}\) of non-necessarily distinct tournaments on the same vertex set \(V\), a directed graph \(\mathcal{D}\) with vertices in \(V\) is called rainbow if there exists an injection \(\phi\colon E(\mathcal{D})\to [m]\) such that \(e\in E(T_{\phi(e)})\) for all \(e\in E(\mathcal{D})\). We proved that for sufficiently large \(m\) and \(|V|=m\), there exists a rainbow copy of every non-strongly oriented Hamiltonian cycle. Moreover, if all the \(T_i\)'s are strongly connected, then there is a rainbow copy of a directed Hamiltonian cycle. We also obtained similar results for rainbow oriented Hamiltonian paths.
2010 Mathematics Subject Classification: 05C35, 05C38
Key Words and Phrases: Tournament, Rosenfeld's conjecture, Hamiltonian path, Hamiltonian cycle, rainbow subgraph
- ⋅ 29th-D-10:10 − 10:30 Colorful Hamilton cycles in random graphs (Debsoumya Chakraborti, Alan Frieze, Mihir Hasabnis)
- Debsoumya Chakraborti*((IBS)), Alan Frieze((Carnegie Mellon University)), Mihir Hasabnis((Carnegie Mellon University))
Debsoumya Chakraborti*, IBS, Alan Frieze, Carnegie Mellon University, Mihir Hasabnis, Carnegie Mellon University
Given an $n$ vertex graph $G$ whose edges are colored from one of $r$ colors $C\!=\!\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)$ $\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges of color $c_i$. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r:m_1+m_2+\cdots+m_r=n\}$.
2010 Mathematics Subject Classification: 05C80
Key Words and Phrases: Random graph, Hamilton cycle, edge-colored graph
- Combinatorics of Symmetric Functions
- ⋅ 28th-B-10:50 − 12:00 Chair: Jang Soo Kim (Sungkyunkwan University)
- ⋅ 28th-B-10:50 − 11:10 GKM graphs of Hessenberg varieties for the double lollipop case (Soojin Cho, JiSun Huh, Seonjeong Park)
- 조수진((아주대)), 허지선*((아주대)), 박선정((전주대))
Soojin Cho, Ajou University, JiSun Huh*, Ajou University, Seonjeong Park, Jeonju University
In this talk, we consider the GKM graph of the Hessenberg varieties determined by the corresponding Hessenberg functions of the double lollipop graphs. We give a criterion for the regularity of the induced subgraph of the GKM graph determined by a permutation w, which provides a condition for the (rationally) smoothness of the corresponding Hessenberg Schubert variety.
This talk is based on the joint work with Soojin Cho and Seonjeong Park.
2010 Mathematics Subject Classification: 05E05
Key Words and Phrases: GKM graph, Hessenberg function, regularity
- ⋅ 28th-B-11:10 − 11:30 Toward Butler's conjecture (Donghyun Kim, Seung Jin Lee, Jaeseong Oh)
- 김동현((서울대)), 이승진((서울대)), 오재성*((고등과학원))
Donghyun Kim, Seoul National University, Seung Jin Lee, Seoul National University, Jaeseong Oh*, KIAS
For a partition $\nu$, let $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Butler conjectured that the divided difference ${\rm I}_{\lambda,\mu}[X;q,t]=(T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t])/$ $(T_\lambda-T_\mu)$ of modified Macdonald polynomials of two partitions $\lambda$ and $\mu$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for ${\rm I}_{\lambda,\mu}[X;q,t]$. This is based on joint work with Donghyun Kim and Seung Jin Lee.
2010 Mathematics Subject Classification: 05E05, 05E10, 05A05
Key Words and Phrases: Butler's conjecture, Macdonald polynomials, LLT equivalence, column exchange rule, Butler permutations
- ⋅ 28th-B-11:40 − 12:00 New proof of the Shuffle conjecture (Donghyun Kim, Jaeseong Oh, Seung Jin Lee)
- 김동현*((서울대)), 오재성((고등과학원)), 이승진((서울대))
Donghyun Kim*, Seoul National University, Jaeseong Oh, KIAS, Seung Jin Lee, Seoul National University
We present a new proof of the Shuffle conjecture.
2010 Mathematics Subject Classification: 05E05
Key Words and Phrases: Shuffle conjecture
- ⋅ 28th-C-13:30 − 14:40 Chair: Euiyong Park (University of Seoul)
- ⋅ 28th-C-13:30 − 13:50 Posets with linear extensions forming a plactic-closed set and their $0$-Hecke modules (Young-Hun Kim, So-Yeon Lee, Young-Tak Oh)
- 김영훈((서울대 모듈과 공간의 양자구조 연구센터)), 이소연*((서강대)), 오영탁((서강대))
Young-Hun Kim, QSMS, Seoul National University, So-Yeon Lee*, Sogang University, Young-Tak Oh, Sogang University
In 1972, Stanley proposed a conjecture that provides a necessary and sufficient condition for the generating function of a labeled poset $(P, \omega)$ to be symmetric. Specifically, the conjecture states that the generating function is symmetric if and only if the poset is isomorphic to a Schur-labeled skew shape poset.
Let us consider the posets of the form $([n], \leq_P)$, where $[n]=\{1,2,\ldots, n\}$ and $\le_P$ is some partial order on $[n]$. These posets can be viewed as labeled posets in a natural way.
In 1993, Malvenuto proved that if the set of linear extensions is
plactic-closed then it is isomorphic to a Schur labeled skew shape poset.
In this talk, we will explore the fascinating properties of posets that satisfy Malvenuto's condition. One notable finding is that posets satisfying Malvenuto's condition precisely appear as regular Schur-labeled skew shape posets. This result has an important implication that the set of linear extensions of $P$ appears as a weak Bruhat interval of $S_n$.
The explicit form of the interval will also be introduced.
Furthermore, we consider the $0$-Hecke modules associated with these posets and study their structural properties.
This work is joint with Young-Hun Kim and Young-Tak Oh.
2010 Mathematics Subject Classification: 05E05, 05E10, 06A07
Key Words and Phrases: Plactic-closed, Schur-labeled skew shape poset, symmetric function, $0$-Hecke algebra
- ⋅ 28th-C-13:50 − 14:10 Poset modules of the 0-Hecke algebras and quasisymmetric power sums (Seung-Il Choi, Young-Hun Kim, Young-Tak Oh)
- 최승일*((서울대)), 김영훈((서울대)), 오영탁((서강대))
Seung-Il Choi*, Seoul National University, Young-Hun Kim, Seoul National University, Young-Tak Oh, Sogang University
In this talk, we investigate a class of mathematical objects called right 0-Hecke modules, which are related to partially ordered sets. Duchamp--Hivert--Thibon introduced right 0-Hecke modules, called poset modules, arising from partially ordered sets. We study a category called $P(n)$, which contains objects made up of multiple copies of these modules put together. Our research leads us to discover that a Hopf algebraic structure is isomorphic to the ring of quasisymmetric functions within $\oplus_{n \geq 0} G_0(P(n))$. We also explore how these modules interact with (anti-)automorphism twists.
We find that the right Bruhat interval modules, a different type of module, can be viewed as poset modules. This insight leads us to discover that the tableau-cyclic modules we investigate are also poset modules. We apply a previous result by Liu-Weselcouch to provide a combinatorial interpretation for the coefficients of the characters of the tableau-cyclic modules, expressing them in terms of quasisymmetric power sums.
This work is joint with Young-Hun Kim and Young-Tak Oh.
2010 Mathematics Subject Classification: 20C08, 05E05, 05E10
Key Words and Phrases: 0-Hecke algebra, poset, quasisymmetric Schur function, quasisymmetric power sum
- ⋅ 28th-C-14:20 − 14:40 Modules of the $0$-Hecke algebra for genomic Schur functions (Young-Hun Kim, Semin Yoo)
- 김영훈((서울대 모듈과 공간의 양자구조 연구센터)), 유세민*((고등과학원))
Young-Hun Kim, QSMS, Seoul National University, Semin Yoo*, KIAS
In 2017, Yong and Pechenik introduced the \textit{genomic Schur function} as a natural deformation of the ordinary Schur function in the context of the K-theory of Grassmannians. Unlike ordinary Schur functions, few of properties of the genomic Schur function are known since the study of the genomic Schur function has just started. Recently, Pechenik (2020) showed that genomic Schur functions are expanded positively in terms of fundamental quasisymmetric functions. Motivated by the Pechenik's work, we construct $0$-Hecke modules whose quasisymmetric characteristics are genomic Schur functions, and study some structural properties of these $0$-Hecke modules.
2010 Mathematics Subject Classification: 20C08, 05E10, 05E05, 14M15
Key Words and Phrases: $0$-Hecke algebra, genomic Schur function, quasisymmetric characteristic
- Algebra
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ The identical cubic equations solving quartic equation (Sangseok Park)
- 박상석
Sangseok Park
In order to solve quartic equation, we need three auxiliary variables. Ferrari's solution on quartic eqation uses one variable to cubic equation, two other variables to quadratic equation. In this paper, we can see the way all three variables to identical cubic eqation.
2010 Mathematics Subject Classification: 11R16
Key Words and Phrases: Quartic equation, cubic equation
- ⋅ Degree of the 3-secant variety (Do Young Choi)
- 최도영((카이스트))
Do Young Choi, KAIST
In this paper, we present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles.
2010 Mathematics Subject Classification: 14N07, 14C25, 14N10
Key Words and Phrases: Double point formula, higher very ampleness, secant bundle, Segre class, secant variety
- Analysis
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ The specular derivative and its application to the wave equation (Kiyuob Jung, Jehan Oh)
- 정기엽*((경북대)), 오제한((경북대))
Kiyuob Jung*, Kyungpook National University, Jehan Oh, Kyungpook National University
In this poster, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives and apply these properties to several differential equations centering around the one-dimensional wave equation with specular derivatives. To address these equations, we introduce new function spaces, which we term specular spaces, consisting of certain specularly differentiable functions.
2010 Mathematics Subject Classification: 26A06, 26A24, 26A27, 26B05, 26B12, 34A36, 34K39, 35L05
Key Words and Phrases: Generalization of derivatives, fundamental theorem of calculus, quasi-mean value theorem, tangent hyperplanes, differential equations
- ⋅ Regularity for double phase functionals with two modulating coefficients (Bogi Kim, Jehan Oh)
- 김보기*((경북대)), 오제한((경북대))
Bogi Kim*, Kyungpook National University, Jehan Oh, Kyungpook National University
We establish regularity results for local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. The model case is defined by the double phase integral with two modulating coefficients
$$
w\mapsto \int [a(x)|Dw|^p+b(x)|Dw|^q] dx, \qquad 1<p<q, \qquad a(\cdot),b(\cdot)\geq 0,
$$
with $0<\mu\leq a(\cdot)+b(\cdot)$. Here, the coefficient $b(\cdot)$ is assumed to be Hölder continuous and the coefficient $a(\cdot)$ is assumed to be uniformly continuous.
2010 Mathematics Subject Classification: 35B65, 35J70, 49N60, 35A15
Key Words and Phrases: Double phase problem, minimizer, nonstandard growth, regularity
- ⋅ The ES-BGK model for the polyatomic molecules with infinite energy (Sungjun Son, Seok-Bae Yun)
- 손성준*((성균관대)), 윤석배((성균관대))
Sungjun Son*, Sungkyunkwan University, Seok-Bae Yun, Sungkyunkwan University
The ES-BGK model covered in this talk is a generalized version of the BGK model of the Boltzmann equation designed to reproduce the correct Prandtl number in the Navier-Stokes limit. In addition, the term ``Polyatomic" refers to molecules that are formed by the combination of two or more atoms, it includes the internal energy $I^{\frac{2}{\delta}} \in \mathbb{R}_+$ resulting from motions such as rotation and vibration. We prove the existence and uniqueness of mild solutions to the ES-BGK model for Polyatomic molecules when has a weaker condition in place of the total energy conservation.
2010 Mathematics Subject Classification: 35Q20, 35F25
Key Words and Phrases: BGK model, elipsoidal BGK model, Boltzmann equation, Kinetic theory of gases, Cauchy problem
- ⋅ Dispersive decay bound of small data solutions to higher order KdV-type equations with low-order nonliearity (Jongwon Lee)
- 이종원((카이스트))
Jongwon Lee, KAIST
We consider the higher order KdV-type equation
\[
\begin{cases}\partial_tu-(-\partial_x^2)^{\frac{n-1}{2}}\partial_xu=\pm u^{p-1}\partial_xu,\\u(0,x)=u_0(x)\end{cases}\leqno{(1)}
\]
with $n\geq 5$ being an odd integer and $p<n.$ It is known by many authors that if $p\geq n$, the small data solutions to (1) shows the linear scattering behavior. The rest case is expected to show the linear dispersive decay in only a finite time scale, and the author proves that this is possibly the case, showing that the linear dispersive decay holds for the time scale of $|t|\ll \epsilon^{-\frac{n(p-1)}{n-p}}$ when $\epsilon\ll 1$ is the size of the initial datum. The proof is done by analyzing the oscillatory integrals on the Fourier side.
2010 Mathematics Subject Classification: 35Q53, 35B40
Key Words and Phrases: Higher order KdV-type equation, lower-order nonlinearity, linear dispersive decay, space-time resonance
- ⋅ Large-time behavior of composite waves of viscous shocks for the barotropic Navier-Stokes equations (Sungho Han, Moon-Jin Kang, Jeongho Kim)
- 한성호*((카이스트)), 강문진((카이스트)), 김정호((경희대))
Sungho Han*, KAIST, Moon-Jin Kang, KAIST, Jeongho Kim, Kyung See University
In this poster, we present results on the large-time behavior of the one-dimensional baro\-tropic compressible Navier-Stokes equations, subject to initial data perturbed from a Riemann data generating a composite of two shock waves with small amplitudes. Specifically, we establish that as time goes to infinity, the perturbed Navier-Stokes flow converges, uniformly in space, to a composition of two viscous shock waves, up to dynamical shifts. Moreover, we introduce the key concept of the ``method of a-contraction with shift," which serves as the fundamental tool utilized throughout this work.
2010 Mathematics Subject Classification: 76N15, 35B35, 35Q30
Key Words and Phrases: Barotropic Navier-Stokes equations, composition of two viscous shock waves, long-time stability, a-contraction with shifts
- Geometry
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ Rational singularities and $q$-birational morphism (Dong Hyeon Kim)
- 김동현((연세대))
Dong Hyeon Kim, Yonsei University
In this paper, we introduce a generalization of the notion of rational singularities. For a normal variety $X$ over a field of arbitrary characteristic, $X$ is said to have \emph{rational singularities} if for all resolutions $f:X'\to X$, $R^if_*\mathcal O_{X'}=0$ and $R^if_*\omega_{X'}=0$ for all $i>0$. We generalize the notion as follows: For a normal variety $X$ and a reflexive sheaf $\mathcal{F}$ on $X$, $\mathcal{F}$ is said to have \emph{rational singularities} if for all resolutions $f:X'\to X$, $R^if_*(f^*\mathcal{F})^{\vee\vee}=0$ and $R^if_*(f^*\mathcal{F})^{\vee}=0$ for all $i>0$. We investigate the properties of rationality of a reflexive sheaf, such as rationality of reflexive sheaves coming from $\mathbb Q$-Cartier Weil divisors on a klt or a strongly $F$-regular variety, and behaving of rationality under finite quotient. Moreover, in this paper, we defined the notion of $(B_{q+1})$ as a dual of well-known Serre's notion $(S_{q+1})$ and proved a duality theorem under a mild condition on that reflexive sheaf. Finally, over a characteristic $0$ field, we defined $q$-rationality of a proper birational morphism as follows: For a proper birational morphism $f:X'\to X$, $f$ is $q$-birational if the center of $f$ has codimension $\ge q+1$ in $X$. We proved that for a resolution $f:X'\to X$ which is $q$-birational and an anti $f$-nef Cartier divisor $D$ on $X'$ such that $f_*D$ is $\mathbb Q$-Cartier, $R^if_*\mathcal O_{X'}(D)=0$ for $1\le i<q$.
2010 Mathematics Subject Classification: 14B05, 14E05, 14G17
Key Words and Phrases: Rational singularities, $(B_{q+1})$, $q$-birational morphism
- ⋅ Weakly Einstein equivalence and golden space forms (Jihun Kim, JeongHyeong Park, Bayram Sahin)
- 김지훈*((성균관대)), 박정형((성균관대)), Bayram Sahin((Ege University))
Jihun Kim*, Sungkyunkwan University, JeongHyeong Park, Sungkyunkwan University, Bayram Sahin, Ege University
In the spirit of the Golden ratio, Crasmareanu and Hretcanu introduced the Golden structure on a manifold. Since then many papers have appeared on such manifolds and their submanifolds. Recently, Sahin et al. defined the concept of Golden sectional curvature and obtained the expression of the curvature tensor field of a Golden space form whose Golden sectional curvature is constant. Unlike space form, a Golden space form may not be an Einstein manifold. On the other hand, there is no implication between Einstein condition and weakly Einstein condition except for 3 and 4-dimensional manifolds. We show that the condition to be Einstein for a Golden space form is equivalent to being weakly Einstein. Moreover, for CR-submanifolds in a Golden space form, we investigate some properties such as a partial geodesic, cyclic parallel, and semi-flat normal connection. Furthermore, we give a characterization of a totally umbilical CR-submanifold of a Golden Riemannian manifold, and we prove the nonexistence of an extrinsic sphere in a nonflat Golden space form.
2010 Mathematics Subject Classification: 53C15, 53C25, 53C40
Key Words and Phrases: Golden structure, golden Riemannian manifold, CR-submanifold, space form, Einstein manifold
- ⋅ Legendrian contact instanton cohomology and its spectral invariants on the one-jet bundles (Yong-Geun Oh, Seungook Yu)
- 오용근((POSTECH \& 기초과학연구원 기하학 수리물리 연구단)), 유승욱*((포항공대))
Yong-Geun Oh, POSTECH \& IBS-CGP, Seungook Yu*, POSTECH
Pseudoholomorphic curves on symplectization has been used to study contact topology and dynamics. The first-named author has developed the analysis of the contact instantons, which replaces that of pseudoholomorphic curves on symplectization for the study of contact topology and dynamics. In particular, we apply this theory to the case of one-jet bundles and associate the Legendrian contact instanton cohomology $HI^*(J^1B,H; R)$ to each Legendrian submanifold $R$ contact isotopic to the zero section of one-jet bundle. Then we give a Floer theoretic construction of Legendrian spectral invariants and establish their basic properties.
2010 Mathematics Subject Classification: 53D42
Key Words and Phrases: One-jet bundles, contact instantons, Legendrian contact instanton cohomology, Legendrian spectral invariants
- Topology
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ Non-relative-hyperbolicity of automorphism and outer automorphism groups of \linebreak right-angled Artin groups (Junseok Kim, Sangrok Oh, Philippe Tranchida)
- 김준석*((카이스트)), 오상록((UPV/EHU)), Philippe Tranchida((Universit\'e Libre de Bruxelles))
Junseok Kim*, KAIST, Sangrok Oh, UPV/EHU, Philippe Tranchida, Universit\'e Libre de Bruxelles
Behrstock, Druţu, and Mosher showed in their paper that the general linear group $\mathrm{GL}_n(\mathbb{Z})$ with $n\geq 3$, the automorphism group and the outer automorphism group of a free group of finite rank at least 3 are not relatively hyperbolic. In this talk, we show a generalized version of this theorem using a criterion of non-relative hyperbolicity made by Anderson, Aramayona, and Shackleton. Our result is as follows: The automorphism groups of right-angled Artin groups whose defining graphs have at least 3 vertices are not relatively hyperbolic. We also show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, they are virtually isomorphic to one of the following: a finite group, an infinite cyclic group, or $\mathrm{GL}_2(\mathbb{Z})$. (Joint with Sangrok Oh and Philippe Tranchida)
2010 Mathematics Subject Classification: 20E36, 20F65
Key Words and Phrases: Relative hyperbolicity, right-angled Artin group, automorphism group, outer automorphism group
- ⋅ On the freeness of groups generated by two parabolic matrices (Wonyong Jang, KyeongRo Kim)
- 장원용*((카이스트)), 김경로((서울대))
Wonyong Jang*, KAIST, KyeongRo Kim, Seoul National University
For a complex number $\alpha$, let $A=\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ and $B_{\alpha}=\left[\begin{smallmatrix}
1 & 0 \\ \alpha & 1 \end{smallmatrix}\right]$ be two parabolic matrices and $G_{\alpha}:= \left< A, B_{\alpha} \right>$ be the subgroup generated by these two matrices, of $\mathbf{SL}_2 (\mathbb{C})$.
Determining whether $G_{\alpha}$ is the free group rank 2 or not, is one of the important problems. Actually, almost all complex numbers make $G_{\alpha}$ to be a free group rank 2 but it is known that the set of all $\alpha$s satisfying $G_{\alpha}$ is not a free group rank 2 is dense in the interval $(-4,4) \cap \mathbb{R}$. From this direction, one of the most outstanding conjectures is that $G_{\alpha}$ is not a free group for all rational numbers between -4 and 4.
Recently, due to Kim and Koberda, many rational numbers have been known to be relation numbers which means the associated group $G_{\alpha}$ is not free but these rationals are still far from 4.
We first assume that $\alpha$ is real, and using the action on the circle, construct a graph called the generalized Farey graph. We prove that $\alpha$ is a relation number if and only if the corresponding generalized Farey graph is not a tree graph.
By using the properties of the generalized Farey graph, we find a new criterion for searching a relation number. With this criterion, we also find new relation numbers so far unknown. In addition, we can construct a sequence of algebraic integers which are relation numbers, and the sequence converges to 4. This work is joint with KyeongRo Kim from SNU.
2010 Mathematics Subject Classification: 20F65, 37C85, 37E10, 57M60, 20E05
Key Words and Phrases: Parabolic isometry, relation number, generalized Farey graph, circle actions
- ⋅ Extreme Khovanov homology of a pretzel links (Hongdae Yun, Mark Halvor Siggers, Jinseok Oh, Seung Yeop Yang)
- 윤홍대*((경북대)), Mark Halvor Siggers((경북대)), 오진석((경북대)), 양승엽((경북대))
Hongdae Yun*, Kyungpook National University, Mark Halvor Siggers, 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 (reduced) homology of independence simplicial complex of a Lando graph at the link. Also, Józef H. Przytycki and M. Silvero propose several formula for the homotopy type of this geometric realization.
We summarize the extreme Khovanov homology of pretzel links and their geometric realizations. Furthemore we give a general concrete formula of extreme Khovanov homology of pretzel links. This is joint work with Mark Halvor Siggers, Jinseok Oh and Seung Yeop Yang.
2010 Mathematics Subject Classification: 57K10, 57M15, 57K18
Key Words and Phrases: Khovanov homology, pretzel knot, geometric realization
- ⋅ On set-theoretic Yang-Baxter cohomology of Alexander biquandles (Jinseok Oh, Xiao Wang, Seung Yeop Yang, Hongdae Yun)
- 오진석*((경북대)), Xiao Wang((Jilin University)), 양승엽((경북대)), 윤홍대((경북대))
Jinseok Oh*, Kyungpook National University, Xiao Wang, Jilin University, Seung Yeop Yang, Kyungpook National University, Hongdae Yun, Kyungpook National University
A homology theory of set-theoretic Yang-Baxter operators was established by J. S. Carter, M. Elhamdadi, and M. Saito.
It was generalized for pre-Yang-Baxter operators independently by V. Lebbed and J. H. Przytycki.
Biquandles, a generalization of quandles, are special cases of set-theoretic Yang-Baxter operators.
In this talk, we determine upper and lower bounds of the Betti numbers for the set-theoretic Yang-Baxter (co)homology of finite Alexander biquandles.
This is joint work with Xiao Wang, Seung Yeop Yang and Hongdae Yun.
2010 Mathematics Subject Classification: 57K12, 57K10, 55N35
Key Words and Phrases: Alexander biquandle, Betti number
- ⋅ Thurston unit ball of a family of $n$-chained links and their fibered face (Juhun Baik, Philippe Tranchida)
- 백주헌*((카이스트)), Philippe Tranchida((Universi\'e Libre de Bruxelles))
Juhun Baik*, KAIST, Philippe Tranchida, Universi\'e Libre de Bruxelles
We determine the Thurston unit ball of a family of $n$-chained link, denoted by $C(n,p)$, where $n$ is the number of link components and $p$ is the number of twists.
When $p$ is strictly positive, we prove that the Thurston unit ball for $C(n,p)$ is an $n$-dimensional cocube, for arbitrary $n$.
Moreover, we clarify the condition for which $C(n,p)$ is fibered and find at least one fibered face for any $p$.
Finally we provide the Teichm\"uller polynomial for the face of Thurston unit ball of $C(n, -2)$ with $n\geq 3$.
2010 Mathematics Subject Classification: Primary 57M25; Secondary 37E30
Key Words and Phrases: Thurston norm, fibered faces, Teichm\"uller polynomial, polytopes
- Probability and Statistics
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ Stochastic aggregation models to describe the qualitative differences in 2D and 3D spaces (Jinyoung Kim)
- 김진영((포항공대))
Jinyoung Kim, POSTECH
Many phenomena in biology are understood using mathematical models of PDEs and stochastic processes. In this talk, we focus on the aggregation phenomenon caused by molecules merging in cells. We use chemical reaction networks described as continuous-time Markov chains to stochastically model aggregation, and the Gillespie algorithm, a computer simulation of stochastic processes, to obtain a visual understanding. Furthermore, we will try to find the qualitative differences in aggregation between 2D and 3D spaces and analyze them based on the underlying network structures, the so-called deficiency. Our results about molecular aggregations in 2D and 3D spaces will give us a starting point for a better understanding of the liquid-liquid phase separation that is our eventual goal to analyze.
2010 Mathematics Subject Classification: 60J28
Key Words and Phrases: Continuous time Markov chain with chemical reaction network
- ⋅ Asymptotic behavior of random quantum channels under group symmetry (Sang-Jun Park, Ion Nechita)
- 박상준*((서울대)), Ion Nechita((Laboratoire de physique th\'eorique, Universit\'e Paul \linebreak Sabatier))
Sang-Jun Park*, Seoul National University, Ion Nechita, Universit\'e Paul Sabatier
Random matrix techniques have become one of the main tools in Quantum Information Theory. These techniques are used to study how ``typical" quantum objects behave and then yield answers to many kinds of existence problems. For example, one of the recent approach to the PPT-squared conjecture is to consider general random channels and investigate their asymptotic PPT and entanglement-breaking behavior. In this project, we suggest how to give a natural randomness on the quantum channels with compact group symmetry. Then we discuss the behavior of these random channels, specifically how they approach PPT and entanglement-breaking as dimension becomes large.
2010 Mathematics Subject Classification: 81R12, 60B20, 43A65
Key Words and Phrases: Quantum information theory, random matrix, Covariant channels, PPT, entanglement-breaking
- Applied Mathematics(including AI, Data Science)
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ Cost-effectiveness of human papillomavirus vaccination including 9-valent HPV vaccine for girls in Japan considering cross-protection (Wongyeong Choi, Eunha Shim)
- 최원경*((숭실대)), 심은하((숭실대))
Wongyeong Choi*, Soongsil University, Eunha Shim, Soongsil University
The Japanese health ministry panel approved the plan of including the 9-valent human papillomavirus (9vHPV) vaccine into the National Immunization Program (NIP) from April 2023. This study aimed to evaluate and compare the economic impact of immunization with 4-valent HPV (4vHPV) and 9vHPV vaccines in Japanese girls aged 12–16 years, presuming vaccine-induced cross-protection. To quantify the economic and epidemiological impacts of including 9vHPV vaccine in Japan, an age-structured mathematical model of HPV transmission was constructed. The cost-effectiveness of including 9vHPV vaccine compared to current 4vHPV vaccination program in Japan was assessed using the incremental cost-effectiveness ratios (ICERs) in various vaccination coverages and market share ratio of both 4vHPV and 9vHPV vaccines. Compared to the 4vHPV vaccination, the 9vHPV vaccination was projected to possess earlier and greater reductions in the incidence of HPV-related diseases. Including 9vHPV vaccine in the HPV vaccination program in Japan was highly cost-effective. Appropriate utilization of two vaccines could be considered economical and reduce the disease burden comparable to switching 9vHPV completely.
2010 Mathematics Subject Classification: 91-10
Key Words and Phrases: Human papillomavirus vaccine, cross-iprotection, cost-effectiveness, Japan
- ⋅ Inversions of the exponential Radon transform in view of the Herglotz wave function (Sunghwan Moon, Jiyoung Oh)
- 문성환((경북대)), 오지영*((경북대))
Sunghwan Moon, Kyungpook National University, Jiyoung Oh*, Kyungpook National University
When X-ray passes through a material, it is absorbed or scattered to varying degrees per millimeter, depending on the material's composition and thickness. This absorption ratio can be represented by a function on a plane, which is transformed using the Radon transform during a tomographic scan to produce output data. By inverting the Radon transform, the original function can be reconstructed from the output data.
The Radon transform is widely used in Computed Tomography (CT) for image reconstruction, and several inversion formulas have been studied, including orthonormal function series formulas. The exponential Radon transform, a generalization of the Radon transform, is used in Positron Emission Tomography (PET) to represent data obtained by measuring radiation emitted from a radioactive material injected into the human body. The function to be obtained by its inversion means the distribution of radioactive material injected into the human body. We provide intensively othonormal function series formula for inversion of the exponential Radon transform, which is not yet well studied. In addition, our study focuses on determining the orthonormal basis required to form the formula for the inversion of the exponential Radon transform. Specifically, we investigate the use of Herglotz wave functions as the basis for this formula.
2010 Mathematics Subject Classification: 44A12
Key Words and Phrases: Radon transform
- ⋅ Korean audio speech separation by using non-negative matrix factorization (June Ho Lee)
- 이준호((부산대))
June Ho Lee, Pusan National University
Speech separation is a technique for classifying speakers in Audio processing. After analyzing people's conversations, in the process of converting the conversations into spectrograms, we can obtain a non-negative matrix of amplitudes. And non-negative matrix factorization (NMF) has been widely used for challenging single-channel audio source separation tasks. In this study, we compare various matrix decompositions to separate speakers in audio and confirm the efficiency of each method in speech separation.
2010 Mathematics Subject Classification: 60G35, 94A12, 65F55
Key Words and Phrases: Speech separation, non-negative matrix factorization, phonemic analysis
- ⋅ Predicting the direction of crude oil markets with the Google Trend data (Sangmin Park)
- 박상민((부산대))
Sangmin Park, Pusan National University
The fluctuations in oil prices are one of the critical issues for South Korea as an energy-importing country. Due to the sensitivity of oil-related events, it is important to predict the impact of oil price changes on the Korean economy for stable operation. This poster aims to analyze the factors that affect fluctuations in crude oil prices using Google Trends data. We explored the predictability of crude oil prices by adding changes in search trend data, in addition to variables of oil supply and demand, to the WTI crude oil price prediction model. To compare the predictability of the WTI crude oil price prediction model, we used LSTM and ARIMA, and showed that incorporating Google Trends data improves the model's predictability.
2010 Mathematics Subject Classification: 62P25
Key Words and Phrases: Google Trends data, WTI crude oil, forecast
- ⋅ Introduction to ensemble Kalman filter and application to epidemic model (Jihyeon Kim)
- 김지현((부산대))
Jihyeon Kim, Pusan National University
The purpose of this poster is to understand the ensemble Kalman filter, and apply it to the epidemic model.
EnKF was developed by Evensen in 1994 to handle strongly non-linear models. and it can be seen as a Monte Carlo approximate version of the Kalman filter. EnKF uses a random sample called an ensemble to represent the distribution of the system and uses a sample covariance matrix calculated by the ensemble to predict and update the ensemble state.
In the experiment, we predicted the number of patients using the SEIR model and COVID-19 patient data.
2010 Mathematics Subject Classification: 65C20
Key Words and Phrases: Ensemble Kalman filter, EnKF, data assimilation, epidemic model
- ⋅ Monotony of a modified newton's method for solving a matrix polynomial equation (Taehyeong Kim, Jeong-Hoon Ju, Sang-Hyup Seo, Hyun-Min Kim)
- 김태형*((부산대)), 주정훈((부산대)), 서상협((((주))스마트소셜)), 김현민((부산대))
Taehyeong Kim*, Pusan National University, Jeong-Hoon Ju, Pusan National University, Sang-Hyup Seo, Smartsocial Co., Ltd., Hyun-Min Kim, Pusan National University
In this study, we consider the matrix equation $\mathcal{F}(X)=X+\left(\sum_{k=1}^{n}A_{k}X^{k-1}\right)^{-1}A_{0}=0$ transformed from the $n$-th matrix polynomial equation $P(X)=\sum_{k=0}^{n}A_{k}X^{k}=0$ (MPE). To find a solution of the equation numerically, we exploit the Newton's method. We suggest some sufficient conditions for monotony of our iteration sequence, and prove the monotone convergence. We finally give some examples related to the benefit of this method.
2010 Mathematics Subject Classification: 65H10
Key Words and Phrases: Matrix polynomial equation, M-matrix, Minimal nonnegative solution, Newton’s method
- ⋅ Isometry property and Inversion of a spherical mean Radon transform with centers on a hyperplane (Jungmin Kim, Sunghwan Moon)
- 김정민*((경북대)), 문성환((경북대))
Jungmin Kim*, Kyungpook National University, Sunghwan Moon, Kyungpook National University
The spherical mean Radon transform maps a suitable function $f$ to the average of the integral of $f$ over the unit sphere. This transform is used in photoacoustic tomography, based on the wave equation. There have been studies on the spherical mean Radon transform, including a method that uses an isometry property. In this paper, we demonstrate that the spherical mean Radon transform can be expressed as the convolution with specific functions. Additionally, we provide an inversion formula for the spherical mean Radon transform, which is derived from the isometry property of the spherical mean Radon transform.
2010 Mathematics Subject Classification: 65R32
Key Words and Phrases: Spherical mean, radon transform, isometry, inversion
- ⋅ New quantum divergence and barycenter with some non Kubo-Ando means (Miran Jeong, Sejong Kim)
- 정미란*((충북대)), 김세정((충북대))
Miran Jeong*, Chungbuk National University, Sejong Kim, Chungbuk National University
Many researchers studied the quantum divergences associated with Kubo-Ando means, which are given by
\begin{displaymath}
\Phi(A,B) = {\rm Tr} [(1-t)A + tB - A \sigma_t B)],
\end{displaymath}
where $A \sigma_t B$ is a Kubo-Ando mean with the weight $0< t <1$.
On the other hand, there are many non Kubo-Ando means such as the generalized mean, the log-Euclidean mean and the Wasserstein mean.\\
In this talk, we consider a new quantum divergence for $0 < t \leq 1/2$
\begin{displaymath}
\Phi_t (A, B) = {\rm Tr} [(1-t)A + tB - A \natural_t B]
\end{displaymath}
associated with the spectral geometric mean $A \natural_t B = (A^{-1} \# B)^t A (A^{-1} \# B)^t$, which is a non Kubo-Ando mean.
We also consider the $t-z$ Bures-Wasserstein quantum divergence
\begin{displaymath}
\Phi_{t, z}(A, B) = {\rm Tr} \left[ (1 - t) A + t B - \left( A^{\frac{1 - t}{2z}} B^{\frac{t}{z}} A^{\frac{1 - t}{2z}} \right)^{z} \right]
\end{displaymath}
for $0 < t \leq z < 1$.
Then we study the barycenter of given quantum divergence, minimizing the weighted sum of divergences.
2010 Mathematics Subject Classification: 81P17, 15B48
Key Words and Phrases: Spectral geometric mean, R\'{e}nyi relative entropy, quantum divergence, barycenter
- ⋅ Portfolio optimization based on the adaptive data selection and LSTM prediction (Jiyeun Park, Huiwon Lee, Hong-joong Kim)
- 박지연*((고려대)), 이희원((고려대)), 김홍중((고려대))
Jiyeun Park*, Korea University, Huiwon Lee, Korea University, Hong-joong Kim, Korea University
The objective of this paper is to enhance the performance of prediction-based portfolio optimization models by integrating feature selection and deep learning (DL) techniques. When using the LSTM model to predict stock prices, it was observed that the dynamic and noisy nature of stock price data degrades the performance of the model. To overcome this, we use two clustering techniques, K-means and Adaptive Data Selection (ADS), which optimize hyperparameters for each cluster and improve the accuracy of stock price predictions. The main difference between them is that K-means uses the concept of partition to divide clusters, whereas ADS uses the concept of covering that allow for overlapping elements within each cluster, addressing boundary issues in K-means. Next, the prediction results were used to simulate two portfolio optimization models, Markowitz Mean-variance (MV) and Semi-Absolute Deviation (SAD), which focus on profit and risk, respectively. To evaluate the performance of purposed models under various circumstances, this paper used historical data of nine representative stock indices in the USA, Europe, and Asia from 2011 to 2016. The experimental results demonstrated that the feature selection technique helped improve the performance of portfolio optimization models, with the ADS+LSTM-based models outperform the other models for investment.
2010 Mathematics Subject Classification: 91B28
Key Words and Phrases: K-means, ADS, LSTM, Markowitz mean-variance, semi-absolute deviation, portfolio optimization, stock price prediction
- Discrete Mathematics
- ⋅ 28th-C-10:00 − 15:00 Display Time(게시)\ \ / 14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ Packing list-colourings (Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Rimma Hamalainen, Ross J. Kang)
- Stijn Cambie*((IBS-ECOPRO)), Wouter Cames van Batenburg((Delft University of Technology)), Ewan Davies((Colorado State University)), Rimma Hamalainen((Technische Universit\"at Berlin)), Ross J. Kang((University of Amsterdam))
Stijn Cambie*, IBS-ECOPRO, Wouter Cames van Batenburg, Delft University of Technology, Ewan Davies, Colorado State University, Rimma Hamalainen, Technische Universit\"at Berlin, Ross J. Kang, University of Amsterdam
Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder.
We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices, and the correspondence packing number (the analogue for correspondence colouring).
We summarize results for bounded degree cases, sketch that every $z \in \mathbb Z^+ \setminus {3}$ can be equal to the correspondence packing number of a graph,
give comments related to the analogue for a Brooks'-type theorem, and remark that a recent conjecture that relates the list packing number and the list flexibility number is false.
2010 Mathematics Subject Classification: 05C07, 05C15, 05C35, 05C69, 05C70
Key Words and Phrases: Graph colouring, partition, perfect matching, list packing, packing chromatic number
- ⋅ Brooks-type theorems for relaxations of square coloring (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
The following relaxation of a proper coloring of the square of a graph was recently introduced: for a positive integer $h$, the \textit{proper $h$-conflict-free $k$-coloring} of a graph $G$ is a proper $k$-coloring of $G$ such that for every vertex $v$ has $\min\{\deg_G(v),h\}$ colors uniquely appearing on its neighborhood. The \textit{proper $h$-conflict-free chromatic number} of a graph $G$, denoted $\chi_{pcf}^{h}(G)$, is the minimum $k$ such that $G$ has a proper $h$-conflict-free $k$-coloring. A Brooks-type conjecture was proposed by Caro, Petru\v sevski, and \v Skrekovski, and its content is as follows: if $G$ is a graph with $\Delta(G)\ge 3$, then $\chi_{pcf}^{1}(G)\le \Delta(G)+1$. Regarding the conjecture, Pach and Tardos proved $\chi_{pcf}^{1}\le 2\Delta(G)+1$. We improve the result for all $h$: if $G$ is a graph with $\Delta(G)\ge h+2$, then $\chi_{pcf}^{h}(G)\le (h+1)\Delta(G)-1$. Also, we show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. This talk is based on joint work with Eun-Kyung Cho, Ilkyoo Choi, and Boram Park.
2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: Proper h-conflict-free coloring, odd coloring