Ill-posedness for incompressible Euler equations at critical regularit
We obtain a quantitative and robust proof that incompressible fluid models are strongly ill-posed in critical Sobolev spaces, in the sense that norm inflation and even nonexistence occur for critical initial data. We then show how to use th...
WGAN with an Infinitely wide generator has no spurious stationary points
Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer...
<학부생을 위한 ɛ 강연> Symplectic geometry and the three-body problem
We describe some of the history of the three-body problem and how it lead to symplectic geometry. We start by sketching Poincare’s prize-winning work, and discuss how it lead to the birth of the fields of dynamical systems and symplec...
<정년퇴임 기념강연> Hardy, Beurling, and invariant subspaces
The invariant subspace problem is one of the longstanding open problem in the field of functional analysis and operator theory. It is due to J. von Neumann (in 1932) and is stated as: Does every operator have a nontrivial invariant subspace...
Diophantine equations and moduli spaces with nonlinear symmetry
A fundamental result in number theory is that, under certain linear actions of arithmetic groups on homogeneous varieties, the integral points of the varieties decompose into finitely many orbits. For a classical example, the set of integra...
Sufficient conditions for the Jensen polynomials of the derivatives of a real entire function to be hyperbolic are obtained. The conditions are given in terms of the growth rate and zero distribution of the function. As a consequence some r...
초록: Let X be a homogeneous space for a Lie group G. A (G,X)-structure on a manifold M is an atlas of coordinate charts valued in X, such that the changes of coordinates locally lie in G. It is a fundamental question to ask how many ways o...
Geometric Langlands theory: A bridge between number theory and physics
※ 강연 앞 부분이 잘렸습니다. (강연자료 다운: Geometric Langlands Theory [A Bridge between Number Theory and Physics] (2022.04.28).pdf ) 초록: The Langlands program consists of a tantalizing collection of surprising results and conjectures w...
<2020년도 젊은 과학자상 수상 기념강연> Metastability of stochastic systems
Metastability란 random process가 여러 개의 안정된 상태를 가질 때 반드시 나타나는 현상으로, 수리물리학이나 화학의 여러 모형들은 물론 딥러닝의 알고리즘 등 다양한 곳에서 공통적으로 나타나는 현상이다. 본 강연에서는 이 Metastability를 수학적으로...
작용수대수에서 순서구조가 중요한 역할을 한다. C*-대수의 시작이라 할 수 있는 Gelfand-Naimark-Segal 표현정리는 양선형범함수로부터 *-준동형을 만들어내는데, 그 표현정리 이후 여러 가지 종류의 양사상에 대한 연구가 이루어졌다. 최근 활발하게 연구되...
There have been at least two surprising events to geometers in 80-90s that they had to admit physics really helps to solve classical problems in geometry. Donaldson proved the existence of exotic 4-dimensional Euclidean space using gauge th...
동형암호(Homomorphic Encryption)는 암호화된 상태에서 복호화없이 계산을 수행하는 암호로서 1978년 제안된 이후 오랜 연구를 거쳐 최근 실용화를 앞두고 있다. 본 강연에서는 우선 동형암호의 개념과 최근 연구결과 그리고 이의 기계학습에의 응용을 소개한...
Free probability is a young mathematical theory that started in the theory of operator algebras. One of the main features of free probability theory is its connection with random matrices. Indeed, free probability provides operator algebrai...
Riemann-Hilbert correspondence for irregular holonomic D-modules
The original Riemann-Hilbert problem is to construct a liner ordinary differential equation with regular singularities whose solutions have a given monodromy. Nowadays, it is formulated as a categorical equivalence of the category of regular...
Toward bridging a connection between machine learning and applied mathematics
This lecture explores the topics and areas that have guided my research in computational mathematics and deep learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and dee...
For a given compact Lie group G, classifying all manifolds equipped with G-actions is one of the most fundamental and important problems in differential geometry. In this talk, We will discuss the problem in the symplectic category and expl...
<학부생을 위한 ɛ 강연> Secure computation: Promise and challenges
This talk discusses modern cryptographic techniques, such as zero-knowledge proof, multi-party computation and homomorphic encryption, which provide advanced functionality and security guarantees beyond data privacy and authenticity. I will...
Category수학강연회소속송용수강연자<학부생을 위한 ɛ 강연> Secure computation: Promise and challenges
