2017 제1회 정오의 수학산책

강연자: 한종규 (서울대)

일시: 2017년 3월 31일(금) 12:00 ~ 13:15

장소: 카이스트 자연과학동 E6-1 3435호

제목: Symmetry, invariants and conservation laws

내용: The notion of symmetry plays a central role in understanding natural laws and in solving equations.  To be symmetric means to be invariant under a group action.  In this lecture we are mainly concerned with continuous groups of the symmetries of differential equations.  I will explain Sophus Lie's ideas on solvability of an ordinary differential equation in terms of its symmetry group and Emmy Noether's theorem on conservation laws for variational problems.   As time permits I will present other viewpoints on the conservation laws.

등록: 2017년 3월 29일(수) 오후 3시까지

         https://goo.gl/forms/KllQvcZnFjp57sOk1

문의: / 내선:8545

재미로 풀어보는 퀴즈에나 등장할 법한 추상적인 수학적 개념이 기계공학(예, 응용역학) 연구에 도움을 줄 수 있을까? 수학과 역학 사이의 간극이 가장 좁았던 때는 언제였고, 수학과 역학이 만나는 지점에서 두 학문을 두루 섭렵했던 수리과학자는 누구였을까? 이와 같은 질문에 대한 답변의 일환으로, 본 발표의 전반부에서는 수학과 역학(유체역학, 고체역학, 열역학, 파동학)의 역사가 공존했던 시절을 인물 중심으로 살펴보고자 한다. 본 발표의 후반부에서는, 역학적 파동과 메타물질에 관한 발표자의 연구주제(음향 투명망토, 음향 블랙홀, 생물음향학 등)를 간략하게 소개한다.

There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations.  This is a joint work with Tengren Zhang.

Many problems in control and optimization require the treatment

of systems in which continuous dynamics and discrete events

coexist. This talk presents a survey on some of our recent work on such

systems. In the setup, the discrete event is given by a random

process with a finite state space, and the continuous component is the

solution of a stochastic differential equation. Seemingly similar

to diffusions, the processes have a number of salient features

distinctly different from diffusion processes. After providing

motivational examples arising from wireless communications,

identification, finance, singular perturbed Markovian systems,

manufacturing, and consensus controls, we present necessary and

sufficient conditions for the existence of unique invariant

measure, stability, stabilization, and numerical solutions of

control and game problems.

Originated from applications in signal processing, random evolution,
telecommunications, risk management, financial engineering,
and manufacturing systems, two-time-scale Markovian systems have
drawn much attention. This talk discusses asymptotic
expansions of solutions to the forward equations, scaled and unscaled
occupation measures, approximation error bounds, and associated
switching diffusion processes. Controlled dynamic systems will also
be mentioned.

1952년 영국의 수학자 A. Turing은 그 당시 생물학자들 조차 전혀 상상 할수 없었던, 다 같은 종류의 세포들이 각자 다른 세포로 분화할수 있는 메커니즘을  Reaction-Diffusion System(RD system)을 이용하여 수학적으로 제시했습니다.  그 이후로,  RD system은 수리해석학적으로도 많은 발전을 거듭해왔으며,  수리모델링을 통해  생명과학 분야에 있어서도 생명의 메커니즘을 밝히는 도구로서 발전을 거듭해 오고 있습니다. 

이 강연에서는 제가 최근에 연구를 진행하고 있는 다양한 생명현상을 예로 그 메커니즘을 밝히기 위해 개발한 수리모델 및 수리모델링 수법을 간단하게 소개하겠습니다.  여기에는  수학적으로 재미있는 구조를  가지고 있을 지 모르는  문제들이 숨어 있을 수 있습니다. 그런 문제들을  여러분들께서 직접 찾아 보시길 바랍니다.

Keywords: Mathematical modeling, PDE, Phase-field method 

Liquid crystal is a state of matter between isotropic fluid and crystalline solid, which has properties of both liquid and solid. In a liquid crystal phase, molecules tend to align a preferred direction and molecules are described by a symmetric traceless 3x3 matrix which is often called a second order tensor. Equilibrium states are corresponding to minimizers of the governing Landau-de Gennes energy which plays an important role in mathematical theory of liquid crystals. In this talk, I will present a brief introduction to Landau-de Gennes theory and recent development of mathematical theory together with interesting mathematical questions.

A well known theorem of Grötzsch states that every planar graph is 3-colorable. We will show a simple proof based on a recent result of Kostochka and Yancey on the number of edges in 4-critical graphs. Then we show a strengthening of the Grötzsch’s theorem in several different directions. Based on joint works with Ilkyoo Choi, Jan Ekstein, Zdeněk Dvořák, Přemek Holub, Alexandr Kostochka, and Matthew Yancey.

Consider a simple symmetric random walk $S$ and another random walk $S'$ whose $k$th increments are the $k$-fold product of the first $k$ increments of $S$.
The random walks $S$ and $S'$ are strongly dependent. Still the 2-dimensional walk $(S, S')$, properly rescaled, converges to a two dimensional Brownian motion. The goal of this talk is to present the proof of this fact, and its generalizations. Based on joint works with K. Hamza and S. Meng.

We review two examples of interesting interactions between number theory and string compactification and raise some new questions and issues in context of these examples based on review article by Gregory W. Moore (arXiv:hep-th/0401049). The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the "attractor mechanism" of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications.

In this talk, we will survey the book "Vertex operator algebras and the monster- I.Frankel, J. Lepowsky and A. Meurman (1988)".

In this talk, for a fixed positive integer d at least 3, we study upper

There is a curious relationship between two sets of apparently unrelated numbers. On one side one has the coefficients of the cube root of the classical modular function J, and on the other one has the dimensions of irreducible representations of the exceptional Lie algebra E8. In this talk I will describe how this mysterious connection has been explained in terms of certain infinite dimensional algebras called vertex algebras.

The curious observation discussed in the first lecture is just a prelude to an even more mysterious one, this time between the coefficients of the modular J function itself, and the dimensions of irreducible representations of the largest of the sporadic finite simple groups: the monster group. In this talk I will describe how vertex algebras have been used to explain this connection, as well as to derive remarkable identities satisfied by the J function.