2015. 10. 23(금) ~ 25(일), 연세대학교
◎ KIAS Plenary Lecture (10.24(토) 15:30~16:20)
Bruno Buchberger We will report on the current state of automated discovery and proof in mathematics by discussing progress made in the speaker’s Theorema Project: By recent advances in automated proving and algorithm synthesis we were able to automatically synthesize a Gröbner bases algorithm automatically from the problem specification. This seems to be epistemologically interesting since the invention of the first Gröbner bases algorithm in 1965 solved a problem that was 65 years open. We will discuss some perspectives of the increasing power of automated reasoning methods on the future of mathematical research, teaching, and applications. In particular, we will report on a task force that aims at building up a “Global Math Formal Knowledge Base”. This global program was initiated at the International Congress of Mathematicians 2014 in Seoul, and so, the Meeting of the KMS seems to be an appropriate occasion for discussing the current state of the program. |
◎ KIAS Public Lecture (10.23(금)17:00~18:00)
박용문 Park, Yong Moon |
(10.24(토) 10:40~12:10)
Jin Akiyama |
(10.24(토) 16:30~17:20)
Gunnar Carlsson |
◎ Secial Invited Lecture (10.24(토) 13:20~14:00)
Chieh-Yu Chang |
◎ Invited Lectures (10.24(토) 13:20~14:00)
[대수학 Algebra] 이동희 Lee, Donghi • 부산대학교 자연과학대학 수학과 교수 Professor, Department of Mathematics, Pusan National University Title: Construction of a new family of two-generator non-hopfian groups A group $G$ is called hopfian if every epimorphism from $G$ to $G$ is an isomorphism. The simplest non-hopfian groups are Baumslag-Solitar groups, and not many are known, up to now, finitely presented non-hopfian groups which do not essentially use Baumslag-Solitar groups. Motivated by this, we will construct a new family of two-generator non-hofian groups (and hence non-residually finite) using one relator presentations of 2-bridge link groups and Heckoid groups. These groups are not finitely presented, but we will talk about a possibility to turn them into finitely presented non-hopfian groups by adding some further relations, in contrast with B. H. Neumann's two-generator non-hopfian group. This is a joint work with Makoto Sakuma. |
[해석학 Analysis] 김준일 Kim, Joonil • 연세대학교 이과대학 수학과 교수 Professor, Department of Mathematics, Yonsei University • 2015년도 대한수학회상 봄 논문상 수상자 The winner of 2015 Excellent Research Paper Award Title: Oscillatory integrals associated with vector polynomials We discuss about the asymptotic behaviors for the 2D oscillatory integrals with polynomial phase functions. Especially, we study the cases that (1) the domain of integral is unbounded, or (2) the phase function involves a vector polynomial. Similar problem is measuring a sublevel-set which can be an unbounded set, or an intersection of various different sublevel sets. Our results are described in terms of generalized notions of Newton polyhedra. |
[기하학 Geometry] 조종택 Cho, Jong Taek • 전남대학교 자연과학대학 수학과 교수 Professor, Department of Mathematics, Chonnam National University Title: CR geometry of contact manifolds For a contact manifold, we have the two fundamental structures associated with the given contact form. One is Riemannian structure and the other is CR structure. In this talk, we study CR geometry, which preserves transversal almost complex structure. Using the generalized Tanaka-Webster connection as a canonical connection, we define it’s curvature tensor, Ricci tensor, Bochner type pseudo-conformal invariant, and we investigate their geometrical properties. Then we introduce several recent results on the topic. |
[위상수학 Topology] 이대웅 Lee, Dae-Woong |
[확률 및 통계학 Probability and Statistics] 김판기 Kim, Panki • 서울대학교 자연과학대학 수리과학부 교수 Professor, Department of Mathematical Sciences, Seoul National University • 2015년도 대한수학회상 봄 논문상 수상자 The winner of 2015 Excellent Research Paper Award Title: Two-sided Green function estimates for killed subordinate Brownian motions A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-\phi(-\Delta)$, where $\phi$ is the Laplace exponent of the subordinator. In this talk we discuss the sharp two-sided estimates on the Green functions of a large class of subordinate Brownian motions without diffusion component in any bounded $\kappa$-fat open set $D$. When $D$ is a bounded $C^{1,1}$ open set, such estimates have an explicit form in terms of the distance to the boundary and $\phi$. We also discuss consequences of such sharp Green function estimates such as boundary Harnack principle in $C^{1,1}$ open sets with explicit rate of decay, Poisson Kernel estimates, Fatou type theorem. This talk is based on several joint papers with Jaehoon Kang, Yunju Lee, Ante Mimica, Renming Song, Zoran Vondraček. |
[응용수학 Applied Mathematics] 임미경 Lim, Mikyoung Two nearly touching inclusions with extreme conductivities can cause the enhancement of the electric field. In an external electric field of long wavelength compared to the size of conductors, the presence of nearly touching conductors induces a very large electric field confined in the narrow gap region between the conductors. In this talk, we analyze this field enhancement based on the quasi-static approximation and the related conductivity equation for the electric scalar potential. The generic blow-up rate of the gradient of the solution to the conductivity problem in presence of two perfect conductors is $|\epsilon\ln\epsilon|^{-1}$ in three dimensions and $\epsilon^{-1/2}$ in two dimensions, where $\epsilon$ is the distance between the two conductors. I derive asymptotic formulas for the solution to the conductivity problem, which characterize the gradient blow-up of the solution. |
[수학교육학 Mathematical Education] 박경미 Park, Kyungmee • 홍익대학교 사범대학 수학교육과 교수 Professor, Department of Mathematics Education, Hongik University Title: The direction of mathematics curriculum revised in 2015 2015 개정 교육과정 총론의 비전은 창의･융합형 인재를 양성하는 것으로, 대표적인 개정의 방향은 인문학적 상상력과 과학기술 창조력을 갖춘 균형 잡힌 인재의 양성이다. 2015 개정 수학과 교육과정 시안은 교육과정 총론이 추구하는 방향성을 따르고, 수학과 교육과정의 국제적 동향을 반영하는 한편 2014년에 이루어진 ʻ문･이과 통합형 수학과 교육과정 재구조화 연구ʼ의 개정 방향을 이어받아 개정의 방향을 ʻ수학 교과역량의 강조ʼ, ʻ학습 부담 경감 실현ʼ, ʻ학습자의 정의적 측면 강조ʼ, ʻ실생활 중심의 통계 내용 재구성ʼ, ʻ공학적 도구의 활용 강조ʼ의 다섯 가지로 선정하였다. |
[전산수학 Mathematics for Information Sciences] 천기상 Cheon, Gi-Sang • 성균관대학교 자연과학대학 수학과 교수 Professor, Department of Mathematics, SungKyunKwan University Title: Riordan arrays and related topics Analytic combinatorics is a branch of combinatorics that describes combinatorial classes using generating functions and aims at predicting precisely the properties of large structured combinatorial configurations. This approach starts from an exact enumerative description of combinatorial structures by means of generating functions; the next step consists in extracting their coefficients in exact or asymptotic form and thus finding results for the corresponding counting sequences. When we describe a sequence with a bivariate generating function it has a particular form, we call it a Riordan array or Riordan matrix. Riordan arrays strongly simplify computations about this kind of sequences in addition to having an interesting algebraic structure and to provide a powerful and simple method to obtain the solution of combinatorial sums. Also, Riordan arrays can be used as a machine to generate combinatorial identities, given so new proofs to old ones or directly finding new ones. In this talk, we will introduce recent developments for the theory of Riordan arrays and related topics arising in combinatorics, matrix theory, group theory, polynomial ring, analytic number theory and Lie theory. |
[암호학 Cryptography] 이주영 Lee, Jooyoung Recently, card shuffle algorithms has begun to attract renewed interest from a cryptographic point of view with applications to format preserving encryption. In this work, we naturally extend the swap-or-not shuffle by replacing a perfect matching used in each round of the swap-or-not shuffle by a certain uniform keyed partition. By using such a random set partition of blocks of size D, we prove that the number of rounds can be reduced by 1/log D for the same level of security compared to the swap-or-not shuffle, improving the efficiency of the resulting encryption scheme~(in terms of the number of rounds). We also give a couple of examples of the uniform keyed partitions that allow efficient implementation in practice. |