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.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

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Lecture to be recorded by KAI-X

Distributed optimization is a concept that multi-agent systems find a minimal point of a global cost functions which is a sum of local cost functions known to the agents. It appears in diverse fields of applications such as federated learning for machine learning problems and the multi-robotics systems. In this talk, I will introduce motivations for distributed optimization and related algorithms with their theoretical issues for developing efficient and robust algorithms.

We prove that the zero function is the only solution to a certain degenerate PDE defined in the upper half-plane under some geometric assumptions. This result implies that the Euclidean metric is the only adapted compactification of the standard half-plane model of hyperbolic space when the scalar curvature of the compactified metric has a certain sign. These Liouville-type theorems are expected to handle the boundary curvature blow-up to prove compactness results of CCE(conformally compact Einstein) manifolds with positive scalar curvature on the conformal infinity.

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심사위원장: 김동수, 심사위원:안드레아스 홈슨, 엄상일, 이주영(전산학부), 서승현(강원대학교)

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.

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

In this talk, we will discuss nonlocal elliptic and parabolic equations on C^{1,τ} open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Levy processes, whose Levy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. This talk is based on a joint work with Hongjie Dong (Brown University).

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ID: 853 0775 9189, PW: 342420

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심사위원장: 박철우, 심사위원: 정연승, 전현호, 안정연(산업및시스템공학과), 전용호(연세대학교)