Tuesday, April 9, 2024

2024-04-09 / 16:00 ~ 17:00

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I will discuss some recent progress on the freeness problem for groups of 2x2 rational matrices generated by two parabolic matrices. In particular, I will discuss recent progress on determining the structural properties of such groups (beyond freeness) and when they have finite index in the finitely presented group SL(2,Z[1/m]), for appropriately chosen m.

2024-04-16 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Algorithmic aspects of linear delta-matroids

by Magnus Wahlström(Royal Holloway, University of London)

Delta-matroids are a generalization of matroids with connections to many parts of graph theory and combinatorics (such as matching theory and the structure of topological graph embeddings). Formally, a delta-matroid is a pair $D=(V,\mathcal F)$ where $\mathcal F$ is a collection of subsets of V known as "feasible sets." (They can be thought of as generalizing the set of bases of a matroid, while relaxing the condition that all bases must have the same cardinality.) Like with matroids, an important class of delta-matroids are linear delta-matroids, where the feasible sets are represented via a skew-symmetric matrix. Prominent examples of linear delta-matroids include linear matroids and matching delta-matroids (where the latter are represented via the famous Tutte matrix). However, the study of algorithms over delta-matroids seems to have been much less developed than over matroids. In this talk, we review recent results on representations of and algorithms over linear delta-matroids. We first focus on classical polynomial-time aspects. We present a new (equivalent) representation of linear delta-matroids that is more suitable for algorithmic purposes, and we show that so-called delta-sums and unions of linear delta-matroids are linear. As a result, we get faster (randomized) algorithms for Linear Delta-matroid Parity and Linear Delta-matroid Intersection, improving results from Geelen et al. (2004). We then move on to parameterized complexity aspects of linear delta-matroids. We find that many results regarding linear matroids which have had applications in FPT algorithms and kernelization directly generalize to linear delta-matroids of bounded rank. On the other hand, unlike with matroids, there is a significant difference between the "rank" and "cardinality" parameters - the structure of bounded-cardinality feasible sets in a delta-matroid of unbounded rank is significantly harder to deal with than feasible sets in a bounded-rank delta-matroid.

2024-04-12 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by 최준호(카이스트)

In the past decade, machine learning methods (MLMs) for solving partial differential equations (PDEs) have gained significant attention as a novel numerical approach. Indeed, a tremendous number of research projects have surged that apply MLMs to various applications, ranging from geophysics to biophysics. This surge in interest stems from the ability of MLMs to rapidly predict solutions for complex physical systems, even those involving multi-physics phenomena, uncertainty, and real-world data assimilation. This trend has led many to hopeful thinking MLMs as a potential game-changer in PDE solving. However, despite the hopeful thinking on MLMs, there are still significant challenges to overcome. These include limits compared to conventional numerical approaches, a lack of thorough analytical understanding of its accuracy, and the potentially long training times involved. In this talk, I will first assess the current state of MLMs for solving PDEs. Following this, we will explore what roles MLMs should play to become a conventional numerical scheme.

2024-04-12 / 14:00 ~ 16:00

학과 세미나/콜로퀴엄 - 기타: Introduction to étale cohomology 3

by 이제학(KAIST)

This is an introductory reading seminar presented by a senior undergraduate student, Jaehak Lee, who is studying the subject.

2024-04-09 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Positive discrepancy, MaxCut and eigenvalues of graphs

by Eero Räty(Umeå University)

The positive discrepancy of a graph $G$ of edge density $p$ is defined as the maximum of $e(U) - p|U|(|U|-1)/2$, where the maximum is taken over subsets of vertices in G. In 1993 Alon proved that if G is a $d$-regular graph on $n$ vertices and $d = O(n^{1/9})$, then the positive discrepancy of $G$ is at least $c d^{1/2}n$ for some constant $c$. We extend this result by proving lower bounds for the positive discrepancy with average degree d when $d < (1/2 - \epsilon)n$. We prove that the same lower bound remains true when $d < n^(2/3)$, while in the ranges $n^{2/3} < d < n^{4/5}$ and $n^{4/5} < d < (1/2 - \epsilon)n$ we prove that the positive discrepancy is at least $n^2/d$ and $d^{1/4}n/log(n)$ respectively. Our proofs are based on semidefinite programming and linear algebraic techniques. Our results are tight when $d < n^{3/4}$, thus demonstrating a change in the behaviour around $d = n^{2/3}$ when a random graph no longer minimises the positive discrepancy. As a by-product, we also present lower bounds for the second largest eigenvalue of a $d$-regular graph when $d < (1/2 - \epsilon)n$, thus extending the celebrated Alon-Boppana theorem. This is joint work with Benjamin Sudakov and István Tomon.

2024-04-09 / 17:00 ~ 18:00

편미분방정식 통합연구실 세미나 - 편미분방정식: Evolution of Crystalline Mean Curvature Flow with a Volume Constraint: Part 2

by 권도현(서울시립대학교)

In this talk, we focus on the global existence of volume-preserving mean curvature flows. In the isotropic case, leveraging the gradient flow framework, we demonstrate the convergence of solutions to a ball for star-shaped initial data. On the other hand, for anisotropic and crystalline flows, we establish the global-in-time existence for a class of initial data with the reflection property, utilizing explicit discrete-in-time approximation methods.

2024-04-09 / 16:00 ~ 17:00

SAARC 세미나 - SAARC 세미나:

by 권도현(서울시립대학교)

Atoms and molecules aim to minimize surface energy, while crystals exhibit directional preferences. Starting from the classical isoperimetric problem, we investigate the evolution of volume-preserving crystalline mean curvature flow. Defining a notion of viscosity solutions, we demonstrate the preservation of geometric properties associated with the Wulff shape. We establish global-in-time existence and regularity for a class of initial data. Furthermore, we discuss recent findings on the long-time behavior of the flow towards the critical point of the anisotropic perimeter functional in a planar setting.

2024-04-12 / 11:00 ~ 12:00

IBS-KAIST 세미나 - 수리생물학:

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