Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, these complicated spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the vacuum Einstein equation. Looking thes...

In this lecture, I will convey subtle interplay between dynamics of Hamiltonian flows and La-grangian intersection theory via the analytic theory of Floer homology in symplectic geometry. I will explain how Floer homology theory (`closed str...

Gromov introduced the analytic method of pseudoholomorphic curves into the study of symplectic topology in the mid 80's and then Floer broke the conformal symmetry of the equation by twisting the equation by Hamiltonian vector fields. We sur...

We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Ha...

Counting circles in Apollonian circle packings and beyond

수학의 기하학, 위상학 그리고 알고리즘의 건축디자인의 적용 사례 및 이론적 배경

※ 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...

We introduce the notion of congruences (modulo a prime number) between modular forms of different levels. One of the main questions is to show the existence of a certain newform of an expected level which is congruent to a given modular form...

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...

There are three Bieberbach theorems on flat Riemannian manifolds; characterization, rigidity and finiteness. These extend to almost flat manifolds. We discuss characterization, rigidity and finiteness of infra-nilmanifolds (almost flat manif...

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...

In this talk, we will consider the hyperbolic Kac-Moody algebra associated to a certain rank 3 Cartan matrix and generalized Kac-Moody algebras that contain the hyperbolic Kac-Moody algebra. The denominator funtions of the generalized Kac-Mo...

우리의 직관에 의하면 우주는 4차원으로 이루어져있다. 달리 말하면 우리가 어떤 사건을 기술 할 때 네 개의 숫자(공간3 + 시간1) 이면 충분하다는 것 이다. 세상에 다른 여분의 차원이 더 있다고 가정 여러 물리 법칙을 통일적으로 기술할 수 있음이 알려졌다...

Dynamics of many particle system can be described by PDE of probability density function. The Boltzmann equation in kinetic theory is one of the most famous equation which describes rarefied gas dynamics. One of main property of the Boltzman...

※ 강연 앞 부분이 잘렸습니다. (강연자료 다운: Mirror symmetry of pairings.pdf ) 초록: Mirror symmetry has served as a rich source of striking coincidences of various kinds. In this talk we will first review two kinds of mirror symmetry statem...

Convergence of Fourier series and integrals is the most fundamental question in classical harmonic analysis from its beginning. In one dimension convergence in Lebesgue spaces is fairly well understood. However in higher dimensions the probl...

A hyperplane arrangement is an arrangement of a finite set of hyperplanes in some vector space. Hyperplane arrangements generalize other famous combinatorial objects such as graphs and matroids. In this talk, we introduce a characteristic po...

Modern mathematics with axiomatic systems has been developed to create a complete reasoning system. This was one of the most exciting mathematical experiments. However, even after the failure of the experiment, mathematical research is still...