Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, h...

인터넷 혁명을 거치면서 인간이 생산 및 소비하는 정보의 양이 비약적으로 증가하고 있다. 다루는 정보와 신호의 차원이 커짐에 따라 정보를 획득, 압축, 저장 그리고 복원하는 일련의 과정에 대한 새로운 패러다임이 요구되어 왔으며, 특히 큰 차원의 신호에...

최근, 노트북, 비디오 카메라, 휴대용 전화기 등과 같은 휴대용 전자 제품의 수요가 급격하게 증대되고, 전기 자동차, 에너지 저장용 축전지, 로봇, 위성 등의 개발이 본격화됨에 따라, 반복적인 충방전이 가능한 고성능 배터리에 대한 관심이 늘어나고 있다. ...

오랜 세월동안 기득권을 누려오던 통신 업계는 이제 성장의 한계에 직면해 있다. 통신의 고유 영역이던 음성통화와 데이터 통신은 스마트폰의 활성화, 사업구조의 변화 등으로 인해 여타 산업군들에게 그 자리를 내어주며 더이상 캐시카우의 역할을 하지 못하...

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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...

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...

In this talk, we investigate some regularity results for non-uniformly elliptic problems. We first present uniformly elliptic problems and the definition of non-uniform ellipticity. We then introduce a double phase problem which is characte...

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...