강현배  Kang, Hyeonbae

• 인하대학교 자연과학대학 수학과 교수
    

Title: Quantitative analysis of field concentration in composites of high contrast

In a composite consisting of inclusions and a matrix with material properties of high contrast, some inclusions are located closely, and huge stress occurs in between them. Stress is a kind of field concentration and another is field enhancement to be used for imaging. In perspective of applications, it is important to understand this field concentration in quantitatively precise manner. It turns out that the mathematical problem for the field concentration is quite challenging since it can not be properly handled using standard elliptic PDE theory. Many significant results have been produced in this field of mathematics in last thirty years or so, and still several outstanding problems are being produced and remain open to be solved. The purpose of this talk is to review some of recent important development and to discuss challenging open problems.

 지동표   Chi, Dong Pyo
    

• 서울대학교 자연과학대학 수리과학부 교수
  

Title: Quantum Computer and Mathematics

양자 세계에서는 소인수분해를 쉽게 하고 건초더미에서 바늘도 빨리 찾을 수 있다. 도대체 양자 세계에서는 어떤 계산 작업을 어디까지 잘 할 수 있을까? 최근 2020년 1월, MIP*=RE 라는 경이스러운 연구결과가 세상에 나왔다. 이것의 의미와 수학과 물리에서의 따름 결과 등에 대하여 이야기 한다.

[과학기술유공자 임덕상 교수 헌정강연] 금종해   Keum, JongHae
    

• 고등과학원 수학부 교수
  

Title: 과학기술유공자 임덕상 교수 헌정강연

임덕상 교수님(1928~1982)은 대수학과 정수론 분야에서 코호몰로지에 관한 개척자적 연구를 포함하여 대수적 K-이론의 초석을 놓으신 업적으로 수학 발전에 크게 기여한 세계적 수학자이시다. 황해도 개성에서 출생하셨고 1954년 서울대학교 수학과 졸업 후 미국에 유학하여 1957년에 인디애나 대학에서 박사학위를 받고 컬럼비아 대학에서 박사 후 연구원/조교수를 역임한 후 1960년 브랜다이스 대학 조교수를 거쳐, 1965년부터 1982년 타계할 때까지 펜실베이니아 대학에서 교수로 재직하셨다.
컬럼비아 대학에서 박사 후 연구원으로 재직 시 Cartan-Eilenberg의 유명한 책 Homological Algebra에 제기된 미해결 문제를 해결하여 (“Modules Over Finite Groups.” Annals of Mathematics, vol. 69, n. 3, 1959, pp. 700–712) 당시 수학계의 큰 주목을 받으셨다. Cartan-Eilenberg 의 Homological Algebra에서 제기된 문제는 “유한 군의 군환(group ring) 상의 투사 모듈(projective module)이 자유 가군(free module)인가?” 이었는데 위 논문에서 임덕상 교수님은 이 질문에 부정적인 답을 내놓으셨다.
1960~1970년대 당대 저명한 수학자들과 교류하며 영향력 있는 논문들을 연속적으로 발표하여 명실공히 최우수 수학자의 한 명으로 인정받으셨다. 1960년대 후반에는 1년간 프랑스 IHES에 머물며 필드상 수상자인 A. Grothendieck이 주관하는 세미나(SGA)에 참여하여 변형이론(deformation theory)을 연구한 세미나 노트 SGA 7에 발표하고 후속 논문으로 이 이론을 더욱 발전시키셨다.
선생님은 주로 미국에서 연구하시고 후진 양성을 하셨지만 1970년대 말 서울대 방문교수로서 강의하셨는데 당시 학문적 뿌리가 약하던 시절에 첨단 학문을 소개하고 학생들에게 첨단 수학 공부에 대한 열망과 동기를 부여하셨다.
또한 펜실베이니아 대학 수학과장 (1975~1978)으로서 행정적 능력도 발휘하셨으며 재미과학기술자협회(KSEA) 창립위원, 초대 본부평의원(1972~1975)과 초대 장학위원(1978~1981)으로 봉사하셨다. 본 강연에서는 선생님의 발자취를 더듬어 보면서 대한민국 과학기술유공자에 선정되신 것을 진심으로 축하드리고 감사하는 시간을 갖고자 한다. 

[2018년 젊은과학자상 수상 기념] 이지운   Lee, Ji Oon

• 한국과학기술원 자연과학대학 수리과학과 교수

Title: Random matrix, spin glass, and signal detection

Large matrices whose entries are random variables, known as random matrices, have been extensively studied in the last a few decades. Many interesting properties of the eigenvalues of random matrices are known, most notably the universality results that asserts the local statistics of eigenvalues exhibit universal behavior.
In this talk, I will explain how the results from random matrix theory can be used in the study of spin glass. Further, I will also demonstration some applications of the theory of spin glass to the research on signal detection problems. 

[2020년 젊은과학자상 수상기념] 서인석   Seo, Insuk

• 서울대학교 자연과학대학 수리과학부 교수
    

Title: Mathematical research of metastability

메타안정성(Metastability)은 물리학, 화학의 여러 시스템 뿐만 아니라 최적화나 딥러닝의 알고리즘 등 다양한 곳에서 공통적으로 발현되는 현상이다. 본 강연에서는 이 현상을 정확하게 이해하기 위해 수학자들, 특히 확률론을 연구하는 사람들은 어떤 접근법을 택했는지 살펴보고 이에 연사의 연구가 기여한 바를 살펴본다.

[대수학 Algebra]-Ⅰ   김명호   Kim, Myungho     

* 10.21.(목) 13:00-13:35

• 경희대학교 이과대학 수학과 교수

Title: Cluster algebras and monoidal categories

Cluster algebras are special commutative rings introduced by Fomin and Zelevinsky in the early 2000s. Specifically, the cluster algebra refers to a subring generated by special elements called cluster variables in the field of rational functions, and the process of creating a new cluster variable from given cluster variables is called a mutation. Cluster algebra is being actively studied as it is observed that the mutation operation appears in various forms in various fields of mathematics.
A monoidal categorification  of a given cluster algebra means that the Grothenieck ring is isomorphic to the cluster algebra and that special elements called cluster monomials correspond to simple objects. If there is such a monoidal categorification, then the given monoidal category and the cluster algebra are closely related and  help understand each other's properties.
In this talk, I will explain that the category of finite-dimensional representations of quiver Hecke algebras and that of quantum affine algebras form monoidal categorifications  of cluster algebras. This is based on several joint works with Seok-Jin Kang, Masaki Kashiwara, Se-jin Oh, and Euiyong Park.

[대수학 Algebra]-Ⅱ   원준영   Won, Joonyeong     

* 10.21.(목) 13:55-14:30

• 고등과학원 수학난제센터 책임연구원

Title: Algebraic stability condition for the existence problem of optimal metrics

Algebraic stability condition, K-stability is one of the most important concept in modern geometry originally due to differential geometers. It was introduced to characterize the existence of Kaehler-Einstein metrics  on Fano manifolds which is defined by signs of an analytic invariant of all possible equivariant degeneration of Fano manifolds.  Later, it settles in completely algebraic terms.

Existence of Kaehler-Einstein metrics on Fano manifolds is detected by K-stability. Moreover,  Kaehler-Einstein metrics on certain  Fano varieties can be lifted to Sasaki-Einstein metrics on Sasakian manifolds that is odd-dimensional analogue Kaehler manifolds.

We discuss how algebraic methods practically show the existence of these optimal metrics.

[해석학 Analysis]-Ⅱ   계승혁   Kye, Seung-Hyeok     

* 10.21.(목) 13:00-13:35

• 서울대학교 자연과학대학 수리과학부 교수

Title: Lattices arising from quantum information theory

A state, a unital positive linear functional on the tensor product of matrix algebras, is called separable if it is the convex sum of product states. A state is called entangled if it is not separable. The notion of entanglement had been originated from Einstein's era, and is now considered as one of the most important resources in current quantum information theory. The separability/entaglement depends on partitions of systems in multi-partite cases. In the tri-qubit sytem which is the simplest case, we have three kinds of partial separability, A-BC, B-CA and C-AB separability. We call those basic partial separability. After it was known that a state needs not to be separable as a tri-partite state even though it satisfies all the basic partial separability, many authors tried to classify partial separability according to intersection and convex hull of the three convex sets consisting of basic partial separable states. In this talk, we will consider the lattice which is generated by those three convex sets with respect to intersection and convex hull. It turns out that this lattice violates the distributive rule and modular identity. For an important subclass including Greenberger-Horne-Zeilinger diagonal states, three generators satisfy a weaker version of the modular identity. The convex sets consisting of GHZ diagonal states turn out to be polytopes, and we exhibit a sequence of GHZ diagonal states to see that this lattice contains infinitely many elements. This talk is based on several co-work with Kil-Chan Ha, Kyung Hoon Han and/or Szilard Szalay. 

[해석학 Analysis]-Ⅰ   강문진   Kang, Moon-Jin     

* 10.21.(목) 13:55-14:30

• 한국과학기술원 자연과학대학 수리과학과 교수
• 2021년도 대한수학회 논문상 수상자

Title: The method of weighted relative entropy with shifts applied to stability estimate of viscous shock waves 

The method of weighted relative entropy with shifts was recently developed to resolve open problems on stability of viscous shocks to the compressible Navier-Stokes equations. First, this method was used to get stability of any large perturbations of viscous shocks. The stability is uniform with respect to the strength of viscosity, which plays a crucial role for a resolution of the long-standing conjecture on the uniqueness of entropy shock to the Euler equations. Secondly, since this method is energy based differently from the anti-derivative method, it can be used to solve the long-standing problem on stability of Navier-Stokes flows slightly perturbed from a Riemann datum generating composite wave of shock and rarefaction waves. In this talk, I will explain about a key idea of the method and a resolution of the open problems.

[기하학 Geometry]  황승수  Hwang, Seungsu     

* 10.21.(목) 13:55-14:30

• 중앙대학교 자연과학대학 수학과 교수

Title: $V$-static mertics with positive isotropic curvature

One of the natural means of finding canonical metrics on smooth manifolds is to look for critical metrics of curvature functionals. Einstein metrics are among them.
In this talk, we briefly review some results on critical metrics of curvature functionals, including $V$-static metrics. We also discuss positive isotropic curvature condition. Then we derive rigidity results on $V$-static metrics under positive isotropic curvature condition. It is a joint work with Gabjin Yun.

[위상수학 Topology]  강성모   Kang, Sungmo     

* 10.21.(목) 13:00-13:35

•전남대학교 사범대학 수학교육과 교수

Title: The classification of primitive/Seifert knot in the 3-sphere, its required theories and consequences

As a long project, Berge and I have worked the classification of primitive/Seifert(P/SF) knots in the 3-sphere. P/SF knots are of interest, because P/SF knots admit Dehn surgeries yielding Seifert-fibered manifolds and knots with Dehn surgeries yielding Seifert-fibered spaces are not well understood.
 The classification of P/SF knots requires the various theories and produces some consequences. In this talk, I will present the classification of P/SF knots in the 3-sphere, its required theories, and its consequences.

[확률·통계학 Probability and Statistics]  이기정   Lee, Kijung

* 10.21.(목) 13:00-13:35

• 아주대학교 자연과학대학 수학과 교수

Title: Stochastic parabolic equation and Dirichlet boundary condition

A stochastic parabolic equation(SPE) describes a heat type diffusion under the random influence that disturbs diffusion in time and space. When we lie our interest on the evolution of the diffusion in a specific domain and control the heat at the boundary of the domain like imposing Dirichlet condition, the diffusion seems losing its averaging power near the boundary. This behavior is more significant with the random influence than the one with a deterministic influence. We then need appropriate tools to have a closer look on this. This sort of quantitative study amounts to a part of regularity theory.
  The related theories are developed over decades and in this talk we briefly introduce the history of them and the recent development. The quite recent one addresses SPE on cone shaped domain while the predecessors have mainly focused on the domains with smooth boundaries. It turns out that the existing tools for smooth domains fall short for the cone shaped domains and we need a more delicate one as the diffusion struggles near the tip of the cone. The nature of the new found tool is of quite different kind compared with the existing ones. But now we have no doubt that it should be a front player of the band. In this talk we explain why.  

[응용수학(AI, Data Science 포함)-Applied Mathematics(including AI, Data Science)]  류경석   Ryu, Ernest K.     

* 10.21.(목) 13:55-14:30

• 서울대학교 자연과학대학 수리과학부 교수

Title: WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.

[수학교육 Mathematical Education]  권오남   Kwon, Oh-Nam     

* 10.21.(목) 13:55-14:30

• 서울대학교 사범대학 수학교육과 교수

Title: 미래세대를 위한 수학교육표준 개발: 기초연구

지식중심 교육을 넘어 미래사회를 살아가는 데 필요한 수학역량을 통합적으로 함양할 수 있도록 미래지향적 수학교육의 목표, 내용, 교수‧학습 및 수학교육환경 등을 담은 미래세대를 위한 한국형 수학교육표준 개발 기초연구의 진행과정을 소개한다.

[이산수학 Discrete Mathematics]  이애자   Yee, Ae Ja     

* 10.21.(목) 13:00-13:35

• The Pennsylvania State University 수학과 교수

Title: Dyson, partition rank and crank

As a combinatorial object, an integer partition carries interesting statistics, one of which is partition rank. In 1944, Freeman Dyson defined rank statistic claiming that it combinatorially accounts for Ramanujan's mod 5 and 7 partition congruences. Dyson's claim was confirmed by Atkin and Swinnerton-Dyer in 1955. In the same paper, Dyson also conjectured the existence of another statistic for the mod 11 congruence, namely crank, and this conjecture was settled by Andrews and Garvan in 1988. Since then, rank and crank have received a lot of attention. In this lecture, I will first survey some results on partition rank and crank presenting their significances in the theory of partitions, and then discuss some recent discoveries on these statistics.

[암호학 Cryptography]  이창민   Lee, Changmin     

 * 10.21.(목) 13:55-14:30

• 고등과학원 계산과학부 연구교수

Title: Overview of NTRU.

Lattice problems restricted to module lattices are most promising security foundations in post-quantum cryptography.
Indeed, five out of the seven final candidates in the NIST PQC standardization have their security that relies on the presummed hardness problems; (module) NTRU, (module) R-LWE problems.
While (module) R-LWE problem have been well studied, there was a lack of research on the (module) NTRU problem.
In this talk, I will provide an overview of NTRU problem including definition, possible solving algorithm, and its hardness.