Moduli space of vector bundles on curves

Moduli space of vector bundles on curves

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https://www.youtube.com/embed/Pt2cpsl0tXs  + 4
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.

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현재 Moduli space of vector bundles on curves

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

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

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

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

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

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.

젊은과학자상 수상 기념강연: 최소주기궤도를 찾아서

강연자 : 강정수 | 소속 : 서울대학교
해밀턴 역학은 고전역학을 수학적으로 기술하는 방법 중 하나로, 곡면 또는 다양체 위의 벡터장으로 표현할 수 있다. 푸앵카레는 해밀턴 역학의 중요한 예인 3체 문제를 연구하면서 주기 궤도의 중요성을 강조하였고, 이러한 통찰은 현대 사교 기하학 이론에서도 엿볼 수 있다. 이번 강연에서는 해밀턴 역학과 사교 기하학의 관계를 소개하고, 주기 궤도와 관련된 문제 및 결과를 살펴보고자 한다.

What I talk about when I talk about solitons

강연자 : 권순식 | 소속 : 카이스트
The study of solitons—localized, non-dispersive wave structures—began with a single observation by Scott Russell in 1834 and has since evolved into a rich and far-reaching mathematical theory. In this talk, I will give a brief history of soliton theory, from the discovery of solitary water waves and the integrability of the KdV equation, to the development of inverse scattering and the broader class of nonlinear dispersive equations. A central modern question in the field is the soliton resolution conjecture, which asserts that generic solutions to dispersive nonlinear equations asymptotically resolve into a finite number of solitons plus radiation. I will introduce the conjecture and discuss its current status, including rigorous results in completely integrable models and recent breakthroughs in non-integrable settings such as radial energy-critical wave equations and other models.

학부생을 위한 ɛ 강연: 우리는 무엇을 어떻게 '잘' 셀 수 있을까?

강연자 : 이승재 | 소속 : 인천대학교
수학에서 '무엇인가를 센다'는 것은 겉으로 보기엔 쉬운 문제처럼 보이지만, 그 안에는 깊고 흥미로운 세계가 숨어 있습니다. 이번 강연에서는 우리가 일상에서 흔히 접하는 간단한 counting 문제들로 시작하여, 점차 더 복잡하고 흥미로운 수학적 문제들이 어떻게 발전해 나가는지 살펴봅니다. 이를 통해 수학에서 counting이 단순히 숫자를 세는 것을 넘어, 문제 속 숨겨진 구조와 성질을 탐구하고 규칙성을 발견하는 데 얼마나 필수적인 역할을 하는지 알아봅니다. 특히 대수학, 해석학, 기하학 등 수학의 다양한 분야들이 counting 문제와 어떻게 연결되어 있는지, 우리가 무엇을 어떻게 '잘' 셀 수 있게 도와주는지 알아봅니다.

Symplectic geometry of $A_n$ Milnor fibers

강연자 : 김종명 | 소속 : 서울대학교

 $A_n$ Milnor fibers are one of the most extensively studied symplectic manifolds. In this talk, we will first review their definitions and basic properties, then explore how they can be studied using Lefschetz fibrations. In particular, we will see how the Lagrangian Floer theory of $A_n$ Milnor fibers can be understood through the topology of the base of a Lefschetz fibration. If time permits, I will also discuss a joint work in progress with Hanwool Bae and Wonbo Jeong related to this topic.