Bochner-Riesz means and spectral projection for Hermite expansions

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현재 Bochner-Riesz means and spectral projection for Hermite expansions

수학강연회 | 강연자 : 유재현 | 소속 : 이화여자대학교

Hermite functions play an important role in many areas, including quantum mechanics, partial differential equations, and probability theory. In this talk, I will introduce two problems about Hermite functions that have recently attracted much attention in harmonic analysis. To help the audience follow along, I will begin with some background.

Moduli space of vector bundles on curves

수학강연회 | 강연자 : 임우남 | 소속 : 연세대학교

The moduli spaces of vector bundles on curves lie at the crossroads of geometry, topology, and representation theory. In this talk, I will introduce these moduli spaces with a focus on their topological aspects. In particular, I will present formulas for their Betti numbers and describe their intersection theory, along with some applications. If time permits, I will also discuss some recent developments about these classical moduli spaces.

No anomalous dissipation in two dimensional fluids

수학강연회 | 강연자 : 박재민 | 소속 : 연세대학교

In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for rough initial data. This is a joint work with Mikael Latocca (Univ. Evry) and Luigi De Rosa (GSSI).

Mathematical theory of neural network and its application to scientific machine learning

수학강연회 | 강연자 : 고승찬 | 소속 : 인하대학교

In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success in various fields. From a mathematical point of view, deep learning essentially involves approximating a target function, relying on the approximation power of deep neural networks. Therefore, it is important to understand the approximation and generalization properties of neural networks in high dimensions. The primary objective of this talk is to mathematically analyze the approximation of neural networks within the classical numerical analysis framework. We will explore the proper regularity of target functions which is suitable for neural network approximation, and investigate how these properties are reflected in the approximation and learning complexity of neural networks. Next, I will apply these theories to my recent work on the operator learning method for solving parametric PDEs. I will analyze the intrinsic structure of the proposed method through the theory described above, deriving some useful results both theoretically and practically. Furthermore, I will demonstrate some relevant numerical experiments, confirming that these theory-guided strategies can be utilized to significantly improve the performance of the method.

Contextual Bandits and Reinforcement Learning with Function Approximation

수학강연회 | 강연자 : 이다빈 | 소속 : 서울대학교

In this talk, we discuss contextual bandits and reinforcement learning problems based on function approximation frameworks. For the first part, we consider neural logistic bandits, where the main task is to learn an unknown reward function within a logistic link function using a neural network. For the second part, we explain algorithms for learning Markov decision processes whose transition is governed by a multinomial logit model.

On the p-adic Group Cohomology of Finite Group Schemes

BK21 FOUR Rookies Pitch | 강연자 : 권혁준 | 소속 : 서울대학교

We introduce a cohomology theory for finite group schemes with commutative formal groups as coefficients. Using Fontaine's Witt covectors, this theory provides a p-adic cohomology theory for finite group schemes and is motivated by the failure of étale cohomology to detect inseparable extensions. We define a G-module structure on commutative formal group schemes and prove that their category forms a Grothendieck category, so it has enough injectives. We show that, with Witt covectors as coefficients, the derived functors of the invariants functor coincide with the cohomology computed via the bar resolution. As an application, we identify the first cohomology of a finite commutative p-group scheme G with its Dieudonné module.

The conductor density of abelian extensions with local behaviors

BK21 FOUR Rookies Pitch | 강연자 : 오경원 | 소속 : 전남대학교

Arithmetic statistics is the study of the statistical properties of arithmetic invariants associated with number-theoretic objects. An essential aspect of arithmetic statistics is the enumeration of number-theoretic objects of interest, such as primes, number fields, and elliptic curves. In this talk, we will explain how such number-theoretic objects can be enumerated using L-functions, a fundamental tool in number theory. Additionally, we will demonstrate how to compute the number of abelian extensions over an arbitrary number field K that satisfying given local conditions at a finite set of primes with a power saving error term.