강연영상 1 페이지 > 서울대학교 수리과학부

We introduce a cohomology theory for finite group schemes with commutative formal groups as coefficients. Using Fontaine's Witt covectors, this theory provides a p-adic cohomology theory for finite group schemes and is motivated by the failure of étale cohomology to detect inseparable extensions. We define a G-module structure on commutative formal group schemes and prove that their category forms a Grothendieck category, so it has enough injectives. We show that, with Witt covectors as coefficients, the derived functors of the invariants functor coincide with the cohomology computed via the bar resolution. As an application, we identify the first cohomology of a finite commutative p-group scheme G with its Dieudonné module.

Arithmetic statistics is the study of the statistical properties of arithmetic invariants associated with number-theoretic objects. An essential aspect of arithmetic statistics is the enumeration of number-theoretic objects of interest, such as primes, number fields, and elliptic curves. In this talk, we will explain how such number-theoretic objects can be enumerated using L-functions, a fundamental tool in number theory. Additionally, we will demonstrate how to compute the number of abelian extensions over an arbitrary number field K that satisfying given local conditions at a finite set of primes with a power saving error term.

 $A_n$ Milnor fibers are one of the most extensively studied symplectic manifolds. In this talk, we will first review their definitions and basic properties, then explore how they can be studied using Lefschetz fibrations. In particular, we will see how the Lagrangian Floer theory of $A_n$ Milnor fibers can be understood through the topology of the base of a Lefschetz fibration. If time permits, I will also discuss a joint work in progress with Hanwool Bae and Wonbo Jeong related to this topic.

Hamiltonian systems arise naturally in classical mechanics as models of physical motion, describing how particles evolve over time. Their motions correspond to critical points of the Hamiltonian action functional, which solves the Euler–Lagrange equation. Through the study of this action functional, Hamiltonian systems have been shown to differ from general dynamical systems in several fundamental ways. In this talk, we highlight these differences by comparing the orbit structures of Hamiltonian systems with those of more general dynamical systems, tracing developments from Poincaré to Floer. In particular, we focus on how these results can be understood using modern symplectic geometry, especially through the framework of Hamiltonian Floer homology. Building on this framework, I will also present my recent result on a rigidity phenomenon in the topological entropy of Hamiltonian systems, which I call effective robustness.

Physics-informed Neural Networks (PINNs) are deep learning-based methods for predicting solutions of partial differential equations (PDEs). Unlike conventional machine learning approaches, PINNs do not rely solely on data; instead, they directly incorporate physical laws, expressed as PDEs, into the loss functions used for training neural networks. This feature allows PINNs to offer a unified framework suitable for solving a wide range of PDE problems, including both forward and inverse problems, making them broadly applicable in science and engineering. In this talk, I will briefly introduce the basic concepts of PINNs, followed by several illustrative applications of PINNs to simple PDE problems.

This presentation introduces a method that combines traditional numerical techniques with artificial intelligence (AI) approaches to enhance the efficiency and accuracy of solving partial differential equations (PDEs). By utilizing AI models, particularly Neural Operators, we approximate the solutions of PDEs using data, which reduces computation time and improves predictive capabilities for more complex systems. The presentation will explore the potential applications of both traditional numerical methods and AI in various engineering and scientific domains.

In this talk, I will present a duality between mathematical objects equipped with group structures and the categories of their internal representations. We establish the duality by providing a complete abstract characterization of such categories. The primary motivation is to simultaneously generalize and unify the two categorical concepts, Galois categories and Tannakian categories. I will explain the motivation and a concrete example in detail. Then I will explain that the duality exists in a general context, namely over a prekosmos. This is a joint work with Jae-Suk Park.