# 학과 세미나 및 콜로퀴엄

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One of the important work in graph theory is the graph minor theory developed by Robertson and Seymour in 1980-2010. This provides a complete description of the class of graphs that do not contain a fixed graph H as a minor. Later on, several generalizations of H-minor free graphs, which are sparse, have been defined and studied. Also, similar topics on dense graph classes have been deeply studied. In this talk, I will survey topics in graph minor theory, and discuss related topics in structural graph theory.

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

In this talk, we consider the Ising and Potts model defined on large lattices of dimension two or three at very low temperature regime. Under this regime, each monochromatic spin configuration is metastable in that exit from the energetic valley around that configuration is exponentially difficult. It is well-known that, under the presence of external magnetic fields, the metastable transition from a monochromatic configuration to another one is characterized solely by the appearance of a critical droplet. On the other hand, for the model without external field, the saddle structure is no longer characterized by a sharp droplet but has a huge and complex plateau structure. In this talk, we explain our recent research on the analysis of this energy landscape and its application to the demonstration of Eyring-Kramers formula for models on fixed two or three dimensional lattice (cf. https://arxiv.org/abs/2102.05565) or models on growing two-dimensional lattice (cf. https://arxiv.org/abs/2109.13583).

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.

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Online(Zoom)
콜로퀴엄
Gil Kalai (Hebrew University)
The Cascade Conjecture and other Helly-type problems

Online(Zoom)

콜로퀴엄

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.