SDEs and PDEs: an approach via Semigroups and Invariant Measures

SDEs and PDEs: an approach via Semigroups and Invariant Measures

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강연자 이해성
소속 국립금오공과대학교

In this talk, we explore Stochastic Differential Equations (SDEs) and Partial Differential Equations (PDEs) with rough coefficients, focusing on the interplay between analysis and stochastic analysis. Starting from the Ornstein-Uhlenbeck process as a key example, we investigate the role of semigroups and invariant measures in general time-homogeneous Itô-SDEs. Furthermore, based on general second-order partial differential operators, we develop an analytic framework for constructing (infinitesimally) invariant measures and regularized semigroups. This enables us to construct solutions to the corresponding SDEs with a potential candidate for an invariant measure, or even with prescribed invariant measure. The presented approach naturally extends to more challenging situations. For instance, it applies to the study of degenerate SDEs, which illustrates the fundamental role of semigroups and invariant measures in both SDEs and PDEs.

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새글 Rigidity and random walks in homogeneous dynamics

강연자 : Timothée Bénard | 소속 : Université Sorbonne Paris Nord

Homogeneous dynamics studies group actions on spaces with a rich algebraic structure, such as tori and quotients of Lie groups.. It is the setting of remarkable and often unexpected rigidity phenomena, as illustrated by the pioneering works of Mostow, Margulis, and Ratner. I will recall some of these classical results and explain how the questions have evolved since then, particularly with the introduction of a new tool: random walks.

The Mathematics of Insider Trading: An Introduction to the Kyle Model

강연자 : 김성진 | 소속 : 성균관대학교

How is private information reflected in market prices through trading? An informed trader seeks to profit from private information, but trading reveals that information to the market and reduces future profit opportunities. This talk introduces the Kyle model and explains how strategic trading and the market’s learning from order flow jointly determine equilibrium prices in discrete and continuous time. We then introduce several extensions, including models with information arriving over time, public disclosure of insider trades, and imperfect competition among market makers.

What Can Valuation Tell Us About Uncertainty?

강연자 : 박종진 | 소속 : 서울대학교

How should we value a financial contract when its underlying probability model is uncertain? Starting from simple examples, this talk introduces model uncertainty and nonlinear valuation, then explores an inverse question: what can a valuation rule reveal about the uncertainty it reflects? We discuss how this question extends to valuations that evolve consistently over time and outline the mathematical challenges in recovering compatible probabilistic models. 

Profinite Groups in Geometry

강연자 : 장원용 | 소속 : 서울대학교

In geometric group theory, a classical aim is to study finitely generated groups by exploring the connections between their algebraic properties and the topological and geometric properties of spaces on which these groups act. In this talk, we introduce a different viewpoint for studying finitely generated groups, namely, profinite rigidity. A central question in this area is which group structures or properties can be detected through their finite quotients. We provide some background on this line of research and present recent developments in the area. We then present two recent results, one joint with Junseok Kim (Technion) and the other with Hyungryul Baik (KAIST).

An Introduction to the Regularity Theory of Multi-Valued Harmonic Map

강연자 : 이상훈 | 소속 : 서울대학교, 쎈펠로우

 We will discuss the regularity theory of multi-valued maps that are stationary with respect to the Dirichlet energy, highlighting its connection to the regularity theory of minimal surfaces.

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.