Thursday, November 10, 2022

2022-11-15 / 11:00 ~ 12:00

IBS-KAIST 세미나 - 대수기하학: Coregularity of Fano varieties

by Joaquín Moraga(UCLA)

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2K_X|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

2022-11-11 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by ()

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this talk, we will present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we will also present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions.

2022-11-11 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 최영필(연세대)

In this talk, we discuss the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring interaction potentials generalizing various previously studied cases, including the Coulomb and Manev potentials. For the first time, we extend the local theory of classical solutions to interaction potentials which are more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known singularity formation result for attractive Vlasov-Poisson. Our local well-posedness and singularity formation results extend to cases with linear diffusion and damping in velocity.

2022-11-10 / 16:15 ~ 17:30

학과 세미나/콜로퀴엄 - 정수론:

by ()

A theorem of Khare-Wintenberger and Kisin (once Serre’s modularity conjecture) says that every two-dimensional odd absolutely irreducible representation \bar\rho of the Galois group of the rationals over a finite field comes from a modular form f, that is, \bar\rho ~ \bar\rho_f. The conjecture even provides a recipe for the weight, level and character of f, but does not give any information about the slope of f. In this talk, based on joint work with Kumar, we provide conditions on f - the main one being that the weight of f is close to 0 - which guarantee that the slope of a modular form g giving rise to the twist of \bar\rho_f by the cyclotomic character has slope one more than the slope of f. This provides a global explanation of some local patterns mentioned in our first talk. The proof uses the theta operator and Coleman-Hida families of overconvergent forms. (This is the second of the two KAIX Invited Lectures.)

2022-11-15 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Excluding single-crossing matching minors in bipartite graphs

by Sebastian Wiederrecht(IBS 이산수학그룹)

By a seminal result of Valiant, computing the permanent of (0, 1)-matrices is, in general, #P-hard. In 1913 Pólya asked for which (0, 1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding $K_{3,3}$ as a matching minor. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond K3,3. Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the permanent of the corresponding (0, 1)-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and $K_{3,3}$ which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains #P-hard on bipartite graphs which exclude $K_{5,5}$ as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes. As another application of our structure theorem, we obtain a strict generalisation of the algorithm for the k-vertex disjoint directed paths problem on digraphs of bounded directed treewidth. This is joint work with Archontia Giannopoulou and Dimitrios Thilikos.

2022-11-15 / 17:00 ~ 18:00

학과 세미나/콜로퀴엄 - Mathematics & Beyond Seminar:

by 최영상 상무(삼성전자 종합기술원)

2022-11-11 / 10:00 ~ 11:00

SAARC 세미나 - SAARC 세미나:

by 윤철희(KAIST 김재철 AI 대학원)

Stochastic finite-sum optimization problems are ubiquitous in many areas such as machine learning, and stochastic optimization algorithms to solve these finite-sum problems are actively studied in the literature. However, there is a major gap between practice and theory: practical algorithms shuffle and iterate through component indices, while most theoretical analyses of these algorithms assume uniformly sampling the indices. In this talk, we talk about recent research efforts to close this theory-practice gap. We will discuss recent developments in the theoretical convergence analysis of shuffling-based optimization methods. We will first consider minimization algorithms, mainly focusing on stochastic gradient descent (SGD) with shuffling; we will then briefly talk about some new progress on minimax optimization methods.

2022-11-17 / 11:50 ~ 12:30

대학원생 세미나 - 대학원생 세미나: Hyperbolicity in Groups

by 김준석(KAIST)

Geometric group theory concerns about how to see geometric properties in finitely generated groups. Defining Cayley graph of a finitely generated group with respect to finite generating set gives a perspective to describe geometric properties of finitely generated groups. Once we get a geometric perspective, we can classify finitely generated groups via quasi-isometry, since two Cayley graphs are quasi-isometric. In this talk, we will explain some basic notions appeared in geometric group theory (for example, quasi-isometry, hyperbolic groups, Švarc–Milnor lemma) and some theorems related to (relative) hyperbolicity of groups.

2022-11-10 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by ()

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that is equipped with data-adaptive methods of finding the number of the underlying communities. Central to our method is fused l-1 penalty applied on an induced graph from the given data. The proposed method shows promising results.

Events for the 취소된 행사 포함