강연영상 3 페이지 > 서울대학교 수리과학부

In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success in various fields. From a mathematical point of view, deep learning essentially involves approximating a target function, relying on the approximation power of deep neural networks. Therefore, it is important to understand the approximation and generalization properties of neural networks in high dimensions. The primary objective of this talk is to mathematically analyze the approximation of neural networks within the classical numerical analysis framework. We will explore the proper regularity of target functions which is suitable for neural network approximation, and investigate how these properties are reflected in the approximation and learning complexity of neural networks. Next, I will apply these theories to my recent work on the operator learning method for solving parametric PDEs. I will analyze the intrinsic structure of the proposed method through the theory described above, deriving some useful results both theoretically and practically. Furthermore, I will demonstrate some relevant numerical experiments, confirming that these theory-guided strategies can be utilized to significantly improve the performance of the method.

In this talk, we discuss contextual bandits and reinforcement learning problems based on function approximation frameworks. For the first part, we consider neural logistic bandits, where the main task is to learn an unknown reward function within a logistic link function using a neural network. For the second part, we explain algorithms for learning Markov decision processes whose transition is governed by a multinomial logit model. 

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.