Deformation theory and curvature of direct image sheaves

Deformation theory and curvature of direct image sheaves

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https://www.youtube.com/embed/jp6g-KtK07o  + 14
강연자 최영준
소속 부산대학교

An important problem in complex geometry is to classify complex manifolds up to biholomorphism. Deformation theory approaches this problem by studying how complex structures vary in holomorphic families and relating these variations to the geometry of their parameter spaces.

In this talk, we use compact Riemann surfaces as our main examples to explain this viewpoint. Kodaira–Spencer theory describes infinitesimal changes of complex structure, and the L² pairing of their harmonic representatives defines the Weil–Petersson metric. The curvature of this metric provides information about the geometry of the moduli space of Riemann surfaces.

These ideas extend to families of higher-dimensional compact Kähler manifolds through higher direct image sheaves, which organize the cohomology of the fibers. Their natural L² metrics provide a broader framework for understanding the relationship between deformation and curvature. We will present general results on the curvature of these metrics and discuss several applications.

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현재 Deformation theory and curvature of direct image sheaves

강연자 : 최영준 | 소속 : 부산대학교

An important problem in complex geometry is to classify complex manifolds up to biholomorphism. Deformation theory approaches this problem by studying how complex structures vary in holomorphic families and relating these variations to the geometry of their parameter spaces.

In this talk, we use compact Riemann surfaces as our main examples to explain this viewpoint. Kodaira–Spencer theory describes infinitesimal changes of complex structure, and the L² pairing of their harmonic representatives defines the Weil–Petersson metric. The curvature of this metric provides information about the geometry of the moduli space of Riemann surfaces.

These ideas extend to families of higher-dimensional compact Kähler manifolds through higher direct image sheaves, which organize the cohomology of the fibers. Their natural L² metrics provide a broader framework for understanding the relationship between deformation and curvature. We will present general results on the curvature of these metrics and discuss several applications.

Mutations of Fano simplices and Diophantine equations

강연자 : 조윤형 | 소속 : 성균관대학교

In this talk, I will introduce combinatorial mutations of convex lattice polytopes and discuss their connections with algebraic geometry, symplectic geometry, representation theory, and number theory. I will then focus on mutations of Fano simplices and the invariants they preserve. Finally, I will describe mutations of the Fano simplex associated with P³ and explain their relationship with Q-Gorenstein degenerations of P³. This talk is based in part on joint work with Yongnam Lee.

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.