Dirichlet problem and regular boundary points for elliptic equations in non-divergence and double divergence form

학부생을 위한 ɛ 강연: A Zoo of Noncompact Surfaces

수학강연회 | 강연자 : 곽상훈 | 소속 : 서울대학교

학부 위상수학의 한 목표는 콤팩트한 곡면은 “구멍의 개수”로 완전히 분류됨을 소개하는 것입니다. 하지만 구멍이 무한히 많아지면 어떻게 될까요? 본 강연에서는 무한 곡면의 다양한 예를 살펴보고, Kerékjártó, Richards, Brown, Messer 등 다양한 수학자들이 제시한 무한 곡면에 대한 분류 정리들을 소개합니다.

Ancient solutions to the Yamabe flow

수학강연회 | 강연자 : 김승혁 | 소속 : 한양대학교

The Yamabe flow is a geometric evolution equation whose goal is to improve the geometry of a manifold by deforming its metric toward one with constant scalar curvature. In this talk, I will introduce the basic background and motivation for the Yamabe flow and then discuss recent developments concerning ancient solutions, namely, solutions that exist for all negative times. In the latter part of the talk, I will present joint work with Dr. Haixia Chen (Hanyang University) and Prof. Monica Musso (University of Bath, UK) on the existence of nonradial ancient solutions.

※ 동영상 5월 10일 공개 예정

Strichartz estimates for the Schrödinger equation on the two-dimensional torus

BK21 FOUR Rookies Pitch | 강연자 : 곽범종 | 소속 : KAIST

The study of estimating the size of a solution to a linear dispersive partial differential equation, called the Strichartz estimate, has been of interest in both partial differential equations and harmonic analysis.

In this talk, I will focus on Strichartz estimates for the Schrödinger equation on a flat two-dimensional torus, which has rich connection to discrete mathematics. I will briefly introduce some of the connections, ranging from number theory to, more recently, incidence geometry and additive combinatorics.

Now

현재 Dirichlet problem and regular boundary points for elliptic equations in non-divergence and double divergence form

BK21 FOUR Rookies Pitch | 강연자 : 김동하 | 소속 : 서울대학교 쎈 펠로우

In the theory of elliptic partial differential equations, the Dirichlet problem has long been a central topic. A fundamental question is to identify domains for which the Dirichlet problem is always solvable.Equivalently, this question leads to the characterization of regular boundary points, a classical problem in the theory of elliptic equations.. For the Laplace equation, this problem was completely solved by Wiener in 1924. He established the celebrated Wiener criterion, which characterizes regular boundary points in terms of a series of capacities. In this talk, we consider the Dirichlet problem for second-order elliptic equations in non-divergence form and double divergence form. We introduce a potential theory framework for these operators, including Perron’s method, capacity theory, and Wiener’s criterion. Assuming that the leading coefficients satisfy the Dini mean oscillation condition, we also establish the equivalence between regular boundary points for these operators and those for the Laplace operator.

Real spectra of large asymmetric Gaussian random matrices

BK21 FOUR Rookies Pitch | 강연자 : 이용우 | 소속 : 서울대학교

Gaussian random matrices play a central role in random matrix theory, serving as canonical models whose spectral properties often persist across much broader classes of ensembles. Many observables in non-Gaussian settings—including the limiting spectrum, local correlation functions, and counting statistics—exhibit the same asymptotic behavior as in the Gaussian case. In this talk, we briefly review the importance of Gaussian random matrices. We then focus on the real eigenvalues of asymmetric Gaussian random matrices, and discuss recent progress on their statistical properties.

Regularity theory of very weak solutions

BK21 FOUR Rookies Pitch | 강연자 : 임민규 | 소속 : KAIST

A very weak solution refers to a solution of a partial differential equation that has the lowest regularity and is defined in terms of distribution. Although the solutions best suited for establishing regularity theory are weak solutions, there are also regularity results for very weak solutions under certain conditions. This presentation will introduce the historical background as well as recent developments in the regularity theory of very weak solutions.

A brief introduction of noncommutative harmonic analysis

BK21 FOUR Rookies Pitch | 강연자 : Zhang Hao | 소속 : 서울대학교

Noncommutative harmonic analysis has been developing rapidly in recent years and has attracted considerable attention. It is deeply connected with classical Banach space theory, operator spaces, operator theory, geometric group theory, noncommutative geometry, Boolean analysis, quantum probability, and quantum information, among others, and has found wide-ranging applications across these fields. In this talk, I will present a brief overview of several important applications of noncommutative harmonic analysis. If time permits, I will also introduce my recent work on Schatten class membership of commutators.

Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups

BK21 FOUR Rookies Pitch | 강연자 : 이헌 | 소속 : 하얼빈공업대학교

In this talk, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a Beurling-Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the underlying Lie group. For this purpose, we need to introduce a particular dense subalgebra of the Beurling-Fourier algebra, which we call an analytic subalgebra. We first introduce an analytic subalgebra allowing a ``local" solution for general connected Lie groups. We will demonstrate that a ``global" solution is also possible for simply-connected nilpotent Lie groups.

Finding Spacecraft Orbits around Moons and Planets

BK21 FOUR Rookies Pitch | 강연자 : 정찬규 | 소속 : 서울대학교

The three-body problem asks a simple question: how do three objects move under mutual gravity? Unlike the two-body problem, which has general closed-form solutions, there is no single unified method for finding periodic solutions, even under the restricted assumption that the mass of the third body is negligible. In this talk, I will give a tour of some of the mathematical ideas behind finding periodic solutions, from classical ideas of Poincar\’e, to modern computer-assisted proofs. I will highlight how these ideas connect to developments in symplectic dynamics and also to applications in space missions.

Hamiltonian Dynamics and Three-body Problem

BK21 FOUR Rookies Pitch | 강연자 : 이동호 | 소속 : QSMS

Hamiltonian dynamics, originated in classical mechanics, lies at the root of symplectic geometry and provides a wealth of examples. One of the central topics is the study of periodic orbits, and over the years many tools have been developed to investigate them. Among the most interesting and historically important examples is the three-body problem. Despite centuries of research, however, a complete picture of its periodic orbits is still lacking, with only partial results available. In this talk, I will present the basic ideas and techniques of Hamiltonian dynamics and symplectic geometry including Floer theory, and introduce the three-body problem and related systems. I will also briefly describe some of my recent results related with the rotating Kepler problem and three-body problem.