Graphs, Surfaces, and their Symmetries

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.

Structural stability of meandering-hyperbolic group actions

강연자 : 김성운 | 소속 : 제주대학교
Sullivan sketched a proof of his structural stability theorem for differentiabl group actions satisfying certain expansion-hyperbolicity axioms. We relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions. This is a joint work with Michael Kapovich and Jaejeong Lee.

Regularity for non-uniformly elliptic problems

강연자 : 오제한 | 소속 : 경북대학교
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 characterized by the fact that its ellipticity rate and growth radically change with the position. We show gradient Hölder continuity and Calderón-Zygmund type estimates for distributional solutions to double phase problems in divergence form. We next introduce a general class of degenerate/singular fully nonlinear elliptic equations which covers the problems of double phase type. We provide C^1 regularity under minimal assumptions on associated operators whose ellipticity may degenerate or blow up along a region where the gradient of a viscosity solution vanishes.

Signed Graph Theory : Extensions & Applications

강연자 : 허철원 | 소속 : KIAS



What a colorful world : a biased personal introduction to discrete Geometry

강연자 : 이승훈 | 소속 : KAIST


Now

현재 Graphs, Surfaces, and their Symmetries

강연자 : 곽상훈 | 소속 : KIAS




Introduction to Poincare-Einstein Manifolds

강연자 : 이상훈 | 소속 : KIAS




Collective Dynamicsand Spectral Analysis

강연자 : Wang Xinyu | 소속 : 서울대학교



Harmonic Analysis

강연자 : 오정태 | 소속 : 서울대학교