Homogeneous dynamics and its application to number theory

Homogeneous dynamics and its application to number theory

6578
https://www.youtube.com/embed/II08PztOz04  + 1265
강연자 임선희
소속 서울대학교
Homogeneous dynamics, the theory of flows on homogeneous spaces, has been proved useful for certain problems in Number theory.
In this talk, we will explain what kind of geometry and dynamics we need to solve certain number theoretic questions such as counting matrices of integer entries, or some problems in Diophantine approximation. The appropriate manifold can often be seen as a space of lattices, and its asymptotic geometry is governed by the smallest length of a non-zero vector in a given lattice, which is also the backbone of post-quantum cryptography.
We will then explain how (partial) solutions of Oppenheim conjecture and Littlewood conjecture were obtained using homogeneous dynamics. We will also survey some recent results and remaining challenges.
프린트

Prekosmic Grothendieck/Galois categories

강연자 : 이재혁 | 소속 : 서울대학교

In this talk, I will present a duality between mathematical objects equipped with group structures and the categories of their internal representations. We establish the duality by providing a complete abstract characterization of such categories. The primary motivation is to simultaneously generalize and unify the two categorical concepts, Galois categories and Tannakian categories. I will explain the motivation and a concrete example in detail. Then I will explain that the duality exists in a general context, namely over a prekosmos. This is a joint work with Jae-Suk Park.

Introduction to crystal base theory

강연자 : 우루노아끼또 | 소속 : 서울대학교
The representation theory of quantum groups has become an important area of research. For an integrable highest weight module of quantum groups, there exists a remarkable basis at q = 0, known as the crystal base, which was introduced by Kashiwara. It has a nice structure with respect to tensor products, and can be viewed as a colored oriented graph, called a crystal.
In this talk, we will briefly review crystal base theory, focusing on the case of gl_n. We will also explain how the crystal of an irreducible highest weight module can be identified with the set of semistandard Young tableaux.

SDEs and PDEs: an approach via Semigroups and Invariant Measures

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.

Machine learning meets scientific computing

강연자 : 홍영준 | 소속 : 수리과학부

In recent years, advances in computational power and data availability have propelled machine learning (ML) to the forefront of scientific computing, complementing and enhancing traditional methods. This lecture explores the integration of ML with numerical methods for multi-scale problems, highlighting new perspectives in scientific computing. Key topics include the synergy between ML and numerical analysis, convergence analysis from both fields, and error estimation techniques. Additionally, we will discuss how neural networks contribute to solving complex partial differential equations (PDEs) efficiently. By leveraging these approaches, we can tackle challenging scientific problems with greater accuracy and computational efficiency. If time permits, we will also touch on emerging topics such as foundation models and ML-driven material discovery.


Survey on a geography of model theory

강연자 : 이정욱 | 소속 : 창원대학교

In this survey talk, we will explore a geography of Model Theory under the guidance of Classification Theory. First, we review basic first-order logic and several dividing lines developed in Classification Theory. Second, we will focus on Classification Theory for fields and examine some interactions between Field Arithmetic and dividing lines.

The lace expansion in the past, present and future

강연자 : Akira Sakai | 소속 : Hokkaido University
The lace expansion is one of the few methods to rigorously prove critical behavior for various models in high dimensions. It was initiated by David Brydges and Thomas Spencer in 1985 to show degeneracy of the critical behavior for weakly self-avoiding walk in d > 4 dimensions to that for random walk. This is one of the reasons Brydges was awarded the Poincaré Prize at ICMP this summer. Self-avoiding walk is a standard model for linear polymers in a good solvent. Other models to which the lace expansion has been successfully applied are percolation, lattice trees/animals, the contact process, the Ising and phi^4 models. In the colloquium talk, I will explain the lace expansion for self-avoiding walk (past) and how it has been extended to other models listed above (present). If time permits, I will also explain the current status of my ongoing work on the lace expansion for self-avoiding walk on random conductors (future).

<학부생을 위한 ɛ 강연> 색과 그래프: 그래프 색칠 문제의 매력과 도전

강연자 : 박보람 | 소속 : 아주대학교
그래프 색칠 문제는 그래프 이론에서 중요한 주제로, 인접한 두 정점이 같은 색을 가지지 않도록 그래프를 색칠하는 방법을 연구합니다. 이는 단순한 퍼즐처럼 보일 수 있지만, 수학적 깊이와 다양한 응용 가능성을 지니고 있습니다. 이번 강연에서는 그래프 색칠 문제의 역사적 배경과 기본 개념을 소개하고, 4색 정리와 같은 유명한 정리와 다양한 연구 방향, 매력적인 추측들을 소개합니다.

Regularity theory for nonlocal equations

강연자 : 김민현 | 소속 : 한양대학교
Nonlocal equations, often modeled using the fractional Laplacian, have received significant attention in recent years. In this talk, we will briefly overview how the classical regularity theory for second-order (elliptic) PDEs (in divergence form) has been extended to fractional-order nonlocal equations. We will explore the Schauder, De Giorgi–Nash–Moser, Morrey–Campanato, and Calderón–Zygmund theories, and present some open problems in these fields.
Now

현재 Homogeneous dynamics and its application to number theory

강연자 : 임선희 | 소속 : 서울대학교
Homogeneous dynamics, the theory of flows on homogeneous spaces, has been proved useful for certain problems in Number theory.
In this talk, we will explain what kind of geometry and dynamics we need to solve certain number theoretic questions such as counting matrices of integer entries, or some problems in Diophantine approximation. The appropriate manifold can often be seen as a space of lattices, and its asymptotic geometry is governed by the smallest length of a non-zero vector in a given lattice, which is also the backbone of post-quantum cryptography.
We will then explain how (partial) solutions of Oppenheim conjecture and Littlewood conjecture were obtained using homogeneous dynamics. We will also survey some recent results and remaining challenges.

On classification of long-term dynamics for some critical PDEs

강연자 : 김기현 | 소속 : 서울대학교
This talk concerns the problem of classifying long-term dynamics for critical evolutionary PDEs. I will first discuss what the critical PDEs are and soliton resolution for these equations. Building upon soliton resolution, I will further introduce the classification problem. Finally, I will also touch on a potential instability mechanism of finite-time singularities for some critical PDEs, suggesting the global existence of generic solutions.