# Department Seminars & Colloquia

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산업경영학동(E2-1) 세미나실 (2216)
ACM Seminars
Dohyun Kwon (Dept. of Mathematics, University of Seoul)
Applications of De Giorgi\'s Minimizing Movements and Optimal Transport

산업경영학동(E2-1) 세미나실 (2216)

ACM Seminars

The study of gradient flows has been extensive in the fields of partial differential equations, optimization, and machine learning. In this talk, we aim to explore the relationship between gradient flows and their discretized formulations, known as De Giorgi's minimizing movements, in various spaces. Our discussion begins with examining the backward Euler method in Euclidean space, and mean curvature flow in the space of sets. Then, we investigate gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. Subsequently, we provide a theoretical understanding of score-based generative models, demonstrating their convergence in the Wasserstein distance.

Abstract: In 1993, Demeyer and Ford computed the Brauer group of a smooth toric variety over an algebraically closed field of
characteristic zero. One may pose the same question to the toric varieties over any field of positive characteristic. Another
interesting question is what will happen if we replace the base field by a discrete valuation ring, thereby replacing smooth toric varieties by smooth toric schemes over a discrete valuation ring in the sense of Kempf-Knudsen-Mumford-Saint-Donat. In this talk. I am going to discuss the answers to these questions. This is joint work with Roy Joshua.

Zoom information will be provided a few days before the zoom talk.

Zoom information will be provided a few days before the zoom talk.

In the analysis of singularities, uniqueness of limits often arises as an important question: that is, whether the geometry depends on the scales one takes to approach the singularity. In his seminal work, Simon demonstrated that Lojasiewicz inequalities, originally known in real algebraic geometry in finite dimensions, can be applied to show uniqueness of limits in geometric analysis in infinite dimensional settings. We will discuss some instances of this very successful technique and its applications.

Finite path integral is a finite version of Feynman’s path integral, which is a mathematical methodology to construct TQFT’s (topological quantum field theories) from finite gauge theory. It was introudced by Dijkgraaf and Witten in 1990. We study finite path integral model by replacing finite gauge theory with homological algebra based on bicommutative Hopf algebras. It turns out that Mayer-Vietoris functors such as homology theories extend to TQFT which preserves compositions up to a scalar. This talk concerns the second cohomology class of cobordism (more generally, cospan) categories induced by such scalars. In particular, we will explain that the obstruction class is described purely by homological algebra, not via finite path integral.

Zeta functions and zeta values play a central role in Modern Number Theory and are connected to practical applications in codes and cryptography. The significance of these objects is demonstrated by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems are related to these objects, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. We first recall results and well-known conjectures concerning these objects over number fields. If time permits, we will present recent developments in the setting of function fields. This is a joint work with Im Bo-Hae and Kim Hojin among others.

There will be a tea time at 15:30 before the lecture.

Contact: Professor Bo-Hae Im ()

https://mathsci.kaist.ac.kr/bk21four/index.php/boards/view/board_seminar/3/

There will be a tea time at 15:30 before the lecture.

Contact: Professor Bo-Hae Im ()

https://mathsci.kaist.ac.kr/bk21four/index.php/boards/view/board_seminar/3/

The mapping class group Map(S) of a surface S is the group of isotopy classes of diffeomorphisms of S. When S is a finite-type surface, the classical mapping class group Map(S) has been well understood. On the other hand, there are recent developments on mapping class groups of infinite-type surfaces. In this talk, we discuss mapping class groups of finite-type and infinite-type surfaces and elements of these groups. Also, we define surface Houghton groups, which are subgroups of mapping class groups of certain infinite-type surfaces. Then we discuss finiteness properties of surface Houghton groups, which is a joint work with Aramayona, Bux, and Leininger.

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산업경영학동(E2-1) 세미나실 (2216)
ACM Seminars
Hayoung Choi (Dept. of Mathematics, Kyungpook National Univ.)
Solving group-sparse problems via deep neural networks with theoretical guarantee

산업경영학동(E2-1) 세미나실 (2216)

ACM Seminars

In this talk, we consider a group-sparse matrix estimation problem. This problem can be solved by applying the existing compressed sensing techniques, which either suffer from high computational complexities or lack of algorithm robustness. To overcome the situation, we propose a novel algorithm unrolling framework based on the deep neural network to simultaneously achieve low computational complexity and high robustness. Specifically, we map the original iterative shrinkage thresholding algorithm (ISTA) into an unrolled recurrent neural network (RNN), thereby improving the convergence rate and computational efficiency through end-to-end training. Moreover, the proposed algorithm unrolling approach inherits the structure and domain knowledge of the ISTA, thereby maintaining the algorithm robustness, which can handle non-Gaussian preamble sequence matrix in massive access. We further simplify the unrolled network structure with rigorous theoretical analysis by reducing the redundant training parameters. Furthermore, we prove that the simplified unrolled deep neural network structures enjoy a linear convergence rate. Extensive simulations based on various preamble signatures show that the proposed unrolled networks outperform the existing methods regarding convergence rate, robustness, and estimation accuracy.

In this talk, I will introduce the use of deep neural networks (DNNs) to solve high-dimensional evolution equations. Unlike some existing methods (e.g., least squares method/physics-informed neural networks) that simultaneously deal with time and space variables, we propose a deep adaptive basis approximation structure. On the one hand, orthogonal polynomials are employed to form the temporal basis to achieve high accuracy in time. On the other hand, DNNs are employed to create the adaptive spatial basis for high dimensions in space. Numerical examples, including high-dimensional linear parabolic and hyperbolic equations and a nonlinear Allen–Cahn equation, are presented to demonstrate that the performance of the proposed DABG method is better than that of existing DNNs.
zoom link:
https://kaist.zoom.us/j/3844475577
zoom ID: 384 447 5577

https://kaist.zoom.us/j/3844475577 회의 ID: 384 447 5577

https://kaist.zoom.us/j/3844475577 회의 ID: 384 447 5577

In this talk, we address a question whether a mean-field approach for a large particle system is always a good approximation for a large particle system or not. For definiteness, we consider an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University).

(KAI-X Distinguished Lecture Series)
We have multiple approaches to vanishing theorems for the cohomology of Shimura varieties, via either algebraic geometry or automorphic forms. Such theorems have been of interest with either complex or torsion coefficients. Recently, results have been obtained under various genericity hypotheses by Caraiani-Scholze, Koshikawa, Hamann-Lee et al. I will survey different approaches. If time permits, I may discuss an ongoing project with Koshikawa to understand the non-generic case.