2025 Algebraic and Analytic Structures on Manifolds and their Applications

• 고승찬 

제목: Mathematical Theory of Neural Network and Its Application to Scientific Machine Learning 

초록: 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. 

• 김세정 

제목: 새로운 관점에서 바라본 기하평균과 행렬로의 확장 (New perspective on the geometric mean and its extension to matrices)

초록: 기하평균(geometric mean)은 산술평균과 함께 가장 널리 알려진 평균 개념 중 하나로, 주어진 값들의 곱을 기반으로 한다. 특히 비율이나 성장률과 같은 곱셈적 데이터의 대표값으로 적합하다. 본 강연에서는 실수에 대한 기하평균을 새로운 관점에서 바라보며 그 특성을 살펴보고, 행렬에 대한 기하평균을 정의하는 방법에 대해 소개한다.

• 이미라 

제목: Recent trends in machine learning and the role of advanced mathematics 

초록: As artificial intelligence (AI) has evolved into a crucial tool influencing both scientific research and everyday life, there has been a growing interest in the development of machine learning methodologies. At the same time, many recent studies have tried to apply advanced mathematical theories, reflecting the increasing significance of mathematical approaches in this area. In this talk, we will explore the historical development and recent trends of machine learning and introduce how mathematical theories, including stochastic analysis, have been applied in this field.

• 이재용 

제목: Two approaches using deep learning to solve partial differential equations

초록: Many differential equations and partial differential equations (PDEs) are being studied to model physical phenomena in nature with mathematical expressions. Recently, new numerical approaches using machine learning and deep learning have been actively studied. There are two mainstream deep learning approaches to approximate solutions to the PDEs, i.e., using neural networks directly to parametrize the solution to the PDE and learning operators from the parameters of the PDEs to their solutions. As the first direction, Physics-Informed Neural Network was introduced in (Raissi, Perdikaris, and Karniadakis 2019), which learns the neural network parameters to minimize the PDE residuals in the least-squares sense. On the other side, operator learning using neural networks has been studied to approximate a PDE solution operator, which is nonlinear and complex in general. In this talk, I will introduce these two ways to approximate the solution of PDE and my research related to them.

• 정미란 

제목: From Distance to Quantum Divergence

초록: A divergence function is originated from differences between probability distributions in statistics and the degree of difference between data in the fields of information theory and data science. In mathematics one can see that the divergence is a generalization of a squared distance. In general, it satisfies non-negativity and positive definiteness but doesn't satisfy symmetry and triangle inequality. In this talk, we introduce divergence on the real vector space and quantum divergence on the open convex cone of positive definite matrices. Moreover, we introduce some examples of divergence and quantum one.

• 김정수 

제목: 데이터의 모양을 이해하는 방법: 위상적 데이터 분석(TDA) 소개 

초록: 이번 발표에서는 데이터의 모양을 이해하기 위한 현대적 접근법인 위상적 데이터 분석(Topological Data Analysis, TDA) 의 기본 개념을 소개합니다. 핵심 개념인 simplicial complex, homology, persistent homology 등을 간단히 살펴본 뒤, 이러한 기법들이 데이터 과학 및 공학의 다양한 문제들에 어떻게 적용될 수 있는지 소개하고자 합니다. 

• 양정윤 

제목: The strange binomial identities of professor Moriarty 

초록: 이번 발표에서는 생성함수론을 공부하는 과정에서 알게 된 흥미로운 항등식, Moriarty identity를 소개합니다. 이 항등식은 셜록 홈스 시리즈에 등장하는 모리아티 교수가 수학자라는 설정에서 유래한 이름으로, 실제 수학자 H. T. Davis가 책 속에서 붙인 별칭입니다. 이 항등식을 증명하는 다양한 방법이 알려져 있는데, 특히 생성함수론에서 자주 쓰이는 snake oil method를 이용하면 이 항등식을 손쉽게 증명할 수 있습니다. 발표에서는 어떻게 이 정체불명의 이름을 가진 항등식이 등장하게 되었는지에 대한 배경과 snake oil method를 이용한 증명이 얼마나 깔끔하게 이루어지는지를 흥미롭게 풀어보고자 합니다.

• 지민환 

제목: An Energy-Stable and Efficient Linear Convex Splitting for the Parabolic Sine-Gordon Equation and Its Applications 

초록: The parabolic sine-Gordon equation, regarded as a phase-field model describing phase transitions in microstructures, exhibits multi-phase characteristics that enable a various applications. In this talk, we first provide the basic background for the numerical treatment of the parabolic sine-Gordon equation. We then propose a linear convex splitting scheme to parabolic sine-Gordon equation and analyze its properties, including energy stability and the discrete maximum principle. We also discuss how the method can be extended to achieve higher-order accuracy in time. Finally, we present several numerical results to show the performance of the scheme and to demonstrate its possible applications.