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

Dirichlet problem and regular boundary points for elliptic equations i…

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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.

프린트

New and old results around the strong convergence of random matrices

강연자 : Benoît Collins | 소속 : Kyoto University

A family of random matrices converges strongly if, in addition to the convergence of their joint moments, the operator norm of any noncommutative polynomial in these matrices converges to that of its limiting free probability model. We will survey this notion, from its original motivation in operator algebras and the first breakthrough of Haagerup and Thorbjørnsen for Gaussian random matrices, to the remarkable acceleration of the last decade, with consequences ranging from operator algebras to spectral gaps of random graphs. Along the way, we will discuss contributions by Bordenave and the speaker, Parraud, Bandeira, Boedihardjo and van Handel, and Chen, Garza-Vargas, Tropp and van Handel, and conclude with recent joint work with Yamagishi. The talk is intended for a broad mathematical audience.

<학부생을 위한 ɛ 강연> Exploring Variational Problems over Graphons with a Tailored Neural Architecture and LLMs

강연자 : 양홍석 | 소속 : 고등과학원

In this talk, I will describe our neural framework for exploring variational problems over graphons—symmetric measurable functions from [0,1]^2 to [0,1] that serve as limit objects for dense graph sequences. These problems arise in two areas of mathematics: extremal graph theory, which studies the optimisation of graph parameters under constraints, often in the limit as the number of vertices grows; and large-deviation theory for dense random graphs, which uses constrained graphon optimisation to study the probabilities and typical structures of rare events. We represent graphons with implicit neural networks and optimise graphon objectives by gradient descent. Our approach combines three ingredients: a multiscale sinusoidal residual architecture designed to capture sharp, step-like graphons; an embedded solver that enforces a single empirical density constraint, with gradients computed by implicit differentiation; and symmetry-aware Monte Carlo estimators. For generalised Turán problems whose known solutions required substantial human effort, our framework recovers known optimal graphons without human intervention. Applied to open instances, it produces candidate optima, including a previously unreported family of candidate extremal structures. This led us to a Goodman-type conjecture for odd-cycle densities, which we subsequently proved and formalised in Lean with the help of LLMs. For variational problems arising in large-deviation theory for Erdős–Rényi random graphs, the framework produces new candidate optimal graphons in both upper- and lower-tail regimes for several pattern graphs. In the upper-tail regime, these candidates achieve better objective values than the best-known benchmark construction due to Lubetzky and Zhao.

Knots and 4-Manifolds

강연자 : 박정환 | 소속 : 서울대학교

Knots live in three-dimensional space, but many fundamental questions about them are four-dimensional: does a knot bound a smoothly embedded disk in the 4-ball, and do distinct spanning surfaces become equivalent there? In this talk, I will discuss two results. First, I will explain why the (2,1)-cable of the figure-eight knot is not smoothly slice. Second, I will describe two minimal-genus Seifert surfaces for the same knot that remain non-isotopic after their interiors are pushed into the 4-ball.

Homogeneous dynamics and uniform distribution

강연자 : Michael Bersudsky | 소속 : 서울대학교

Homogeneous dynamics studies the asymptotic properties of group actions on homogeneous spaces. It has played an important role in number theory by providing conceptual and dynamical tools that have led to the resolution of several long-standing problems. In this talk, I will give an overview of the field, focusing on the theory of qualitative uniform distribution, and present some of my joint work in this area.

Diophantine Approximation on Kleinian Circle Packings

강연자 : 박강래 | 소속 : 서울대학교

Kleinian circle packings are fractal limit sets that can be visualized as infinite patterns of mutually tangent circles. In this talk, we will discuss a Diophantine approximation problem on such limit sets: how well a point in the limit set can be approximated by tangent points coming from the packing. We will focus on a Good approximation theorem, a Lagrange-type theorem relating approximation quality to geometric lengths, and epsilon-badly approximable points, whose size is measured by Hausdorff dimension. This talk is based on joint work with Seonhee Lim and Yongquan Zhang.

Geometry, Analysis, and Probability of Conformal Fields

강연자 : 강남규 | 소속 : 고등과학원

Over the past twenty-five years, several striking predictions from statistical physics have been rigorously proven using Schramm–Loewner evolution (SLE). Under fairly general conditions, establishing the convergence of interface curves in critical 2D lattice models to their scaling limits requires only a single martingale observable -- a stochastic analogue of an integral of motion -- which uniquely determines the law. In this talk, I will describe how to construct families of such martingale-observables in various conformal settings, deriving them from Ward’s equations in conformal field theory and variational formulas in geometric analysis. This presentation is based on a series of joint works with Tom Alberts, Sung-Soo Byun, and Nikolai Makarov.

Coherent Lagrangian classes

강연자 : 좌동욱 | 소속 : 충북대학교

Moduli problems often manifest a remarkable symmetry now known as a shifted symplectic/Lagrangian structure. They are crucial because it provides the right framework for defining and counting objects in these spaces.

In the first part of the talk. I will explain why such symmetries arise in a natural way. Then I will present a construction of a homomorphism from the K-group of matrix factorisations of the critical locus to the K-group of the Lagrangian. The key step is the construction of a specialisation functor for categories of matrix factorisations along the deformation to the normal cone. This is joint work with Jeongseok Oh.

Seeing Quadratic Lattices Through Their Substructures

강연자 : 김대준 | 소속 : 고려대학교

Representations by quadratic forms have long been studied through the geometric language of quadratic lattices. In this talk, we approach a quadratic lattice through two of its substructures: lattice cosets and full-rank sublattices. In the first part, we briefly introduce theta series of lattice cosets and then look at a graph structure on ternary cosets that reflects their algebraic relationships. In the second part, we turn to counting sublattices of a binary lattice up to isometry, and see how this leads to a Dirichlet series closely related to the arithmetic of imaginary quadratic fields.

Representation theory of 0-Hecke algebras and quasisymmetric functions

강연자 : 이소연 | 소속 : 서울대학교

The representation theory of the symmetric group plays a central role in mathematics, particularly in combinatorics through its connection with symmetric functions. In this setting, the Iwahori–Hecke algebra can be viewed as a q-deformation of the group algebra of the symmetric group. Its representation theory depends on the parameter q, leading to different behavior for different values of q. In this talk, I focus on the case q = 0, namely the 0-Hecke algebra, whose representation theory is closely related to quasisymmetric functions. I will briefly review the history of this subject and discuss recent developments involving quasisymmetric functions and related combinatorics.

Introduction to Symmetric Function Theory

강연자 : 최현재 | 소속 : 서울대학교

Algebraic combinatorics studies the connections between combinatorics and algebraic structures. Its central topics include symmetric functions, tableaux, crystals, and so on. In this talk, I will focus on symmetric function theory and positivity problems. Then we move on to q-analogues of symmetric functions, such as the Kostka-Foulkes polynomials. Finally, I will briefly introduce Lusztig’s q-weight multiplicities, a generalization of Kostka-Foulkes polynomials, and describe a type C combinatorial formula using semistandard oscillating tableaux. This talk is based on joint work with Donghyun Kim and Seung Jin Lee.