SDEs and PDEs: an approach via Semigroups and Invariant Measures

SDEs and PDEs: an approach via Semigroups and Invariant Measures

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강연자 이해성
소속 국립금오공과대학교

In this talk, we explore Stochastic Differential Equations (SDEs) and Partial Differential Equations (PDEs) with rough coefficients, focusing on the interplay between analysis and stochastic analysis. Starting from the Ornstein-Uhlenbeck process as a key example, we investigate the role of semigroups and invariant measures in general time-homogeneous Itô-SDEs. Furthermore, based on general second-order partial differential operators, we develop an analytic framework for constructing (infinitesimally) invariant measures and regularized semigroups. This enables us to construct solutions to the corresponding SDEs with a potential candidate for an invariant measure, or even with prescribed invariant measure. The presented approach naturally extends to more challenging situations. For instance, it applies to the study of degenerate SDEs, which illustrates the fundamental role of semigroups and invariant measures in both SDEs and PDEs.

프린트

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.

Orbit Structures in Hamiltonian Dynamics

강연자 : 손범준 | 소속 : 서울대학교

Hamiltonian systems arise naturally in classical mechanics as models of physical motion, describing how particles evolve over time. Their motions correspond to critical points of the Hamiltonian action functional, which solves the Euler–Lagrange equation. Through the study of this action functional, Hamiltonian systems have been shown to differ from general dynamical systems in several fundamental ways. In this talk, we highlight these differences by comparing the orbit structures of Hamiltonian systems with those of more general dynamical systems, tracing developments from Poincaré to Floer. In particular, we focus on how these results can be understood using modern symplectic geometry, especially through the framework of Hamiltonian Floer homology. Building on this framework, I will also present my recent result on a rigidity phenomenon in the topological entropy of Hamiltonian systems, which I call effective robustness.

학부생을 위한 ɛ 강연: 위상, 기하, 대수의 만남 - 토릭 다양체로의 초대

강연자 : 이은정 | 소속 : 충북대학교

토릭 다양체는 토러스가 잘 작용하는, 좋은 구조를 지닌 다양체입니다. 토릭 다양체의 위상과 기하에 관한 연구는 조합론과 표현론을 비롯한 여러 수학 분야와 깊이 연결되어 있습니다. 이 강연에서는 이러한 연관성의 구체적인 예들을 살펴보고, 특히 토릭 다양체의 위상과 조합론 사이의 관계를 중심으로 논의할 예정입니다.

Physics-informed Neural Networks: A Neural Network-Based PDE Solver

강연자 : 이명수 | 소속 : 서울대학교

Physics-informed Neural Networks (PINNs) are deep learning-based methods for predicting solutions of partial differential equations (PDEs). Unlike conventional machine learning approaches, PINNs do not rely solely on data; instead, they directly incorporate physical laws, expressed as PDEs, into the loss functions used for training neural networks. This feature allows PINNs to offer a unified framework suitable for solving a wide range of PDE problems, including both forward and inverse problems, making them broadly applicable in science and engineering. In this talk, I will briefly introduce the basic concepts of PINNs, followed by several illustrative applications of PINNs to simple PDE problems.

Combining AI and Traditional Numerical Methods for Efficient and Accurate PDE Solutions

강연자 : 이진실 | 소속 : 서울대학교

This presentation introduces a method that combines traditional numerical techniques with artificial intelligence (AI) approaches to enhance the efficiency and accuracy of solving partial differential equations (PDEs). By utilizing AI models, particularly Neural Operators, we approximate the solutions of PDEs using data, which reduces computation time and improves predictive capabilities for more complex systems. The presentation will explore the potential applications of both traditional numerical methods and AI in various engineering and scientific domains.