Department Seminars & Colloquia
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The law of iterated logarithm (LIL) is a crowning achievement in classical probability theory that gives the sharp upper bound for the magnitude of fluctuations of a random walk. If each step has mean zero and variance one, then the upper bound (in certain sense) is given by \sqrt{2n\log\log n}, hence the name “iterated logarithm.” Despite being considered the “third fundamental limit theorem in probability” by some probabilists after the law of large numbers and the central limit theorem, its proof is not so accessible to non-experts. For instance, most textbooks either only consider special cases or use sophisticated machineries in their proofs. The purpose of this talk is to provide a relatively simple and elementary proof of the so-called Hartman—Wintner LIL. The idea is to generalize a proof of the central limit theorem (CLT), which will be also presented, to obtain a result on the rate of convergence in the CLT. First principles in probability (e.g. the second Borel—Cantelli lemma) are the only technical prerequisites.
Inside living cells, chemical reactions form a large web of networks
and they are responsible for physiological functions. Understanding
the behavior of complex reaction networks is a challenging and
interesting task. In this talk, I would like to illustrate how the
methods of algebraic topology can shed light on the properties of
chemical reaction systems. In particular, we discuss the following two
problems: (1) response of reaction systems to external perturbations
and (2) simplification of complex reaction networks without altering
the behavior of the system.
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
This talk is concerned with the bifurcation and stability of the
compresible Taylor vortex. Consider the compressible Navier-Stokes
equations in a domain between two concentric infinite cylinders. If the
outer cylinder is at rest and the inner one rotates with sufficiently
small angular velocity, a laminar flow, called the Couette flow, is
stable. When the angular velocity of the inner cylinder increases,
beyond a certain value of the angular velocity, the Couette flow becomes
unstable and a vortex pattern, called the Taylor vortex, bifurcates and
is observed stably. This phenomena is mathematically formulated as a
bifurcation and stability problem. In this talk, the compressible Taylor
vortex is shown to bifurcate near the criticality for the incompressible
problem when the Mach number is sufficiently small. The localized
stability of the compressible Taylor vortex is considered under
sufficiently small axisymmetric perturbations; and it is shown that the
large time behavior of solutions around the Taylor vortex is described
by solutions of a system of diffusion equations.
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 review some of machine learning approaches and present a novel algorithm based on variational inference to solve (stochastic) systems. Numerical examples are provided to illustrate the proposed algorithm.
Helly-type theorems and problems form a nice area of discrete geometry. I will start with the notable theorems of Radon and Tverberg and mention the following conjectural extension.
For a set X of points x(1), x(2),...,x(n) in some real vector space V we denote by T(X,r) the set of points in X that belong to the convex hulls of r pairwise disjoint subsets of X.
We let
t(X,r) = 1 + dim(T(X,r)).
Radon's theorem asserts that
If t(X,1) < |X| then t(X, 2) > 0.
If t(X,1) + t(X,2) < | X | then t(X,3) >0.
In the lecture I will discuss connections with topology and with various problems in graph theory.
I will also mention questions regarding dimensions of intersection of convex sets.
1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos
2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804
3) A link to Barany's book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
Counting the number of points on a variety is a historical method for investigating the variety, for example, in the Weil conjecture. Nowadays, it is known that the point count helps us determine the E-polynomial. This E-polynomial, in turn, gives arithmetic-geometric information on the variety such as the dimension, the number of irreducible components and Euler characteristic.
In this talk, we will consider a specific type of variety, the character variety associated to the fundamental group of a surface. In short, we will discuss this variety for a punctured surface, with regular semisimple or regular unipotent monodromy at the punctures. This variety plays a crucial role in diverse areas of mathematics, including non-abelian Hodge theory, geometric Langlands program and mathematical physics. The complex representation theory of finite groups will be used to compute the number of points on such a variety.
9:30-10:30am
Title: Equations in Simple Groups
Abstract: Given a word w in a free group on variables x_1,...,x_n, a finite group G, and an element g in G, we consider the question of whether the equation w = g has solutions where the x_i take values in G, and if so, how many. I am particularly interested in what happens when the word is fixed and G is a large finite simple groups. I will say something about the ideas which have led to progress for certain families of words, with emphasis on open problems.
10:50-11:50
Title: Elliptic curves and field arithmetic
Abstract: Let E be an elliptic curve over a field K. When K is a number field, Mordell's theorem says that the points of E over K form a finitely generated group. We say a field is "anti-Mordellic" if the opposite is true for all E/K. I will discuss what is known about anti-Mordellic fields, with emphasis on a longterm joint project with Bo-Hae Im to understand the relation between the anti-Mordellic property and the absolute Galois group of K.
Ellipsoidal BGK model (ES-BGK) is a generalized version of the Boltzmann-BGK model.
In this model, the local Maxwellian in the relaxation operator is extended to an ellipsoidal Gaussian
with a Prandtl parameter ν, so that the correct Prandtl number can be computed in the Navier-Stokes
limit. In this talk, we review some of the recent results on ES-BGK model, such as the existence
(stationary or non-stationary) theory and the entropy-entropy production estimates. A dichotomy
is observed between −1/2 < v < 1 and ν=−1/2. In the former case, an equivalence relation between
the local temperature and the temperature tensor enables one to apply theories developed
for the original BGK model in a modified form. In the critical case (ν=−1/2), where the correct
Prandtl number is achieved, such equivalence breaks down, and the structure of the flow has
to be incorporated to estimate the temperature tensor from below. This is from joint works with
Stephane Brull, Doheon Kim, and Son Sung Jun.
We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981). As a consequence, the stability up to a translation is obtained by using a concentrated compactness method. As an application, we prove linear in time filamentation near Hill’s vortex: there exists an arbitrary small outward perturbation growing linearly for all times. These results rigorously confirm numerical simulations by Pozrikidis in 1986. The second part is joint work with In-Jee Jeong(SNU).
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
Ordinary differential equations are useful in modeling the periodic behavior of organisms, such as circadian rhythm, based on known biological knowledge and researchers' hypotheses. The theoretical mathematical models are calibrated to the experimental measurements by estimating a set of unknown model parameters. Traditional parameter estimation with mathematical models often focuses only on the point estimation relying on an optimization method such as simulated annealing, but it often neglects the uncertainty in point estimates and suffers from the local trap issue. This talk provides a gentle introduction to Bayesian analysis focusing on its usefulness in uncertainty quantification; introduces a Bayesian computing method with an advanced Markov chain Monte Carlo called the generalized multiset sampler; and illustrates the proposed Bayesian inference with circadian oscillations observed in a model filamentous fungus, Neurospora crassa.
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085
그래프 신경망은 그래프에서 높은 표현 능력과 함께 특징 정보를 추출하는 방법론으로 학계와 산업체에서 최근 폭발적인 관심을 받고 있다. 본 세미나에서는 그래프 신경망의 개요 및 주요 동작 원리를 다룬다. 구체적으로, message passing의 원리를 이해하고 state-of-the-art 알고리즘에서 사용한 다양한 message passing 함수를 소개한다. 그리고, 협업 필터링에 기반한 추천 시스템을 소개하고, 이러한 추천 시스템 설계에 그래프 신경망의 응용에 대해 학습한다. 경량화된 그래프 신경망을 사용한 state-of-the-art 추천 알고리즘을 소개하고, 해당 방법들이 가지는 challenge를 이해한다. 마지막으로, 발표자 연구실에서 제안한 그래프 신경망을 활용한 새로운 추천 시스템 방법을 간단히 소개한다.
Morihiko Saito's theory of mixed Hodge modules is a far generalisation of classical Hodge theory, which is based on the theory of perverse sheaves, D-modules, variations of Hodge structures. One can think of mixed Hodge modules as a certain class of D-modules with Hodge structures. Naturally they are accompanied by perverse sheaves via the Riemann–Hilbert correspondence. This guide consists of about 8 talks, which may cover: review of classical Hodge theory, D-modules and filtered D-modules, nearby and vanishing cycles, etc. The main goal is to understand the notion of mixed Hodge modules and to explain two important theorems: the structure theorem and the direct image theorem. If time permits, we discuss recent applications of the theory in algebraic geometry.
Zoom 회의 ID: 352 730 6970; 암호: 1778 ; 실명으로 들어오시면 대기실에서 개별 승인해 드립니다.
Zoom 회의 ID: 352 730 6970; 암호: 1778 ; 실명으로 들어오시면 대기실에서 개별 승인해 드립니다.