# 세미나 및 콜로퀴엄

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2017 제2회 정오의 수학산책

강연자: 이윤원 (인하대)

일시: 2017년 4월 28일(금) 12:00 ~ 13:15

장소: 카이스트 자연과학동 E6-1 3435호

제목: Atiyah-Singer index theorem

내용: The Atiyah-Singer Index Theorem appeared in 1960’s, which is one of great mathematical achievements in 20th century. It is a far reaching generalization of the Gauss-Bonnet-Chern Theorem for the Euler characteristic, the Hirzebruch Signature Theorem for the Signature of 4k-dimensional compact manifold, the Riemann-Roch-Hirzebruch Theorem for the Arithmetic Genus. Hence to understand this magnificent theorem, we need to investigate how the Euler characteristic, Signature and Arithmetic Genus are expressed by the Fredholm Indices for some appropriate geometric operators. In this talk, I will explain briefly the historical background of the Index Theorem, the Fredholm Indices of elliptic operators and discuss how the Index Theorem was motivated from the above classical celebrated theorems. And then, I will go through very briefly the proof of the Index Theorem by using the heat kernel method. If time permits, I will explain the Index Theorem on a compact manifold with boundary, where the eta-invariant appears as a boundary correction term.

등록: 2017년 4월 26일(수) 오후 3시까지

https://goo.gl/forms/0lqSboiW5A9vv9rd2

문의: hskim@kias.re.kr / 내선:8545

Let X be a smooth projective rational surface over an algebraically closed field. One version of a long standing conjecture asks whether the set of self-intersections of reduced curves is bounded below. This question can be reduced to studying singular plane curves. One approach is to measure how singular the curve is using a recently introduced quantity called an H-constant. This has raised some new open problems I will discuss.

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E6-1, ROOM 1409
Discrete Math
Brendan Rooney (KAIST)
Eigenpolytopes, Equitable Partitions, and EKR-type Theorems

_{n}due to Godsil and Meagher. Along the way we will see some useful tools from algebraic graph theory. Namely, a bound on the maximum size of an independent set in a graph, equitable partitions, and eigenpolytopes.

In this talk, I will introduce the distribution of eigenvalues of random normal matrix ensembles. Specifically, I will discuss the existence and universality of scaling limits for the eigenvalues at a bulk singularity, an isolated point in the interior at which the equilibrium density vanishes. I will describe how to find a suitable scale and how the rescaled ward’s identity can be used to prove the universality. This is joint work with Yacin Ameur.

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E6-1, ROOM 1409
Discrete Math
Dieter Spreen (Universität Siegen, Siegen, Germany)
Bi-Topological Spaces and the Continuity Problem

The continuity problem is the question when effective (or Markov computable) maps between effectively given topological spaces are effectively continuous. It will be shown that this is always the case if the the range of the map is effectively bi-regular. As will be shown, such spaces appear quite naturally in the context of the problem.

Evolutionary games of cyclic competitions have been extensively studied

to gain insights into one of the most fundamental phenomena in nature:

biodiversity that seems to be excluded by the principle of natural selection.

The Rock-Paper-Scissors (RPS) game of three species and

its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are

paradigmatic models in this field.

In all previous studies, the intrinsic symmetry associated with the cyclic competitions imposes

a limitation on the resulting coexistence states, leading to only selective types of such states.

We investigate the effect of nonuniform intraspecific competitions on coexistence and

find that a wider spectrum of coexistence states can emerge and persist.

This surprising finding is substantiated using three classes of cyclic game models through stability analysis,

Monte Carlo simulations and patterns of continuous spatiotemporal dynamical evolution.

Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms

can promote biodiversity to a broader extent than previously thought.

Recent development of high throughput sequencing and CRISPR genome editing technology have brought great advances in the understanding of molecular mechanisms underlying diseases. We studied infertility in Drosophila model: deep sequencing analyses of small RNA, mRNA, and RNA immunoprecipitation sequencing (RIP-seq) showed that a conserved mRNA export factor, Thoc5, represses the expression of transposable elements, which are deleterious mobile genetic elements that cause genome instability and germ cell death. We also found that Thoc5 binds nascent RNAs and facilitates biogenesis of small RNAs, which in turn regulated transposable elements. Furthermore, we used CRISPR-Cpf1, a recently found class 2/type V CRISPR RNA-guided endonuclease that is distinct from the common CRISPR-Cas9, to generate mutant mice. Whole genome analyses of the mutant mice showed precise alteration of DNA sequence at the target genomic locus with no detectable off-target mutation. We generated a cell line with a single base change in Isocitrate Dehydrogenase I (IDH1), a gain-of-function mutation widely found in glioma patients. We further applied CRISPR genome editing to pathogenic bacteria to generate bacterial strains with mutations in genes required for quorum sensing, which is an intercellular communication system for detection cell density. The quorum sensing mutant strains showed reduced biofilm formation, and changes in expression profile of genes in the metabolic pathway.

Among the most well-known examples of L-functions are the Riemann zeta

function and the L-functions associated to classical modular forms.

Less well known, but equally important, are the L-functions associated

to Maass forms, which are eigenfunctions of the Laplace-Beltrami

operator on a hyperbolic surface. Named after H. Maass, who discovered

some examples in the 1940s, Maass forms remain largely mysterious.

Fortunately, there are concrete tools to study Maass forms: trace

formulas, which relate the spectrum of the Laplace operator on a

hyperbolic surface to its geometry. After Selberg introduced his

famous trace formula in 1956, his ideas were generalised, and various

trace formulas have been constructed and studied. However, there are

few numerical results from trace formulas, the main obstacle being

their complexity. Various types of trace formulas are investigated,

constructed and used to understand automorphic representations and

their L-functions from a theoretical point of view, but most are not

explicit enough to implement in computer code.

Having explicit computations of trace formulas makes many potential

applications accessible. In this talk, I will explain the

computational aspects of the Selberg trace formula for GL(2) for

general levels and applications towards the Selberg eigenvalue

conjecture and classification of 2-dimensional Artin representations

of small conductor.

This is a joint work with Andrew Booker and Andreas Strömbergsson.

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E6-1, ROOM 1409
Discrete Math
Otfried Cheong (School of Computing, KAIST)
Putting your coin collection on a shelf

Imagine you want to present your collection of n coins on a shelf, taking as little space as possible – how should you arrange the coins?

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: hskim@kias.re.kr / 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.

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자연과학동 E6-1, ROOM 1409
Discrete Math
Bernard Lidický (Iowa State University)
3-coloring triangle-free planar graphs

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.