WYMK 2023

• 최범준 교수님 (POSTECH)

Title : 그대들, 어떻게 수학과 살 것인가

• 유재원 (POSTECH)

Title : Hodge Decomposition of the Cohomology of Complex Projective Smooth Variety

Abstract : 복소대수다양체의 코호몰로지를 계산하는 것은 어려운 작업이다. 이번 발표에서는 복소대수다양체의 Dolbeault  코호몰로지 $H^{p,q}(X)$와 K\"ahler 타입의 복소다양체에 대해서 가장 기본적으로 알려진 Hodge Decomposition을 소개하고자 한다. 더 자세하게는, 복소대수다양체의 코호몰로지가 기본적으로 $\mathfrak{sl}(2,C)$의 표현을 준다는 것을 간단하게 보일 것이다. 시간이 된다면 예제 몇 개를 소개하고자 한다.

• 김영종 (KAIST)

Title : Introduction to Online Convex Optimization

Abstract : In this talk, I will explain the setting of online convex optimization and the definition of regret and constraint violation. I then will introduce various algorithms and their theoretical guarantees under various assumptions. The connection with some topics in machine learning such as stochastic gradient descent, multi-armed bandit, and reinforcement learning will also be briefly discussed.

• 정다솔 (POSTECH)

Title : Basic theory on algebraic surfaces

Abstract : In this talk, I will introduce a basic algebraic surface theory. Before treating surfaces, I will review basic notions of algebraic geometry roughly: varieties, divisors, and cohomolgies. I then will introduce the intersection theory on smooth surfaces. As an application, I also introduce Castelnuovo's contraction theorem, which is the start of the minimal model program(MMP). Finally, further generalized topics will be discussed: ruled/rational surfaces, schematic picture of MMP, etc.

• 최도영 (KAIST)

Title : Geometry of secant varieties

Abstract : Secant varieties have long been a classical research area in algebraic geometry.  This talk aims to provide an introduction to the theory of secant varieties, highlighting their relevance and applications in various fields. We begin by presenting several equivalent definitions and basic geometric properties of secant varieties. Additionally, we introduce notable applications, including matrix multiplications and polynomial Waring rank problems from a geometric perspective. Lastly, recent results of secant varieties are discussed.

• 윤재영 (서울대학교)

Title : Winfree model with higher-order couplings and influences

Abstract : We introduce Winfree model, which was proposed by Arthur Winfree to describe the collective behavior of pulsatile oscillators. In this talk, we focus on Winfree model with higher-order influences, which is the first attempt to make mathematical analysis for the approximated pulse-coupled model. We study the sufficient conditions for coupling strengths, in terms of order in influence, for death, locking and incoherence. Next, we add randomness on the order in influence. In this case, we mainly consider the complete oscillator death. We prove exponential-relaxation toward equilibrium and provide a local sensitivity in probability space.

• 권재성 (UNIST)

Title : On the μ-invariant in Iwasawa theory

Abstract : Iwasawa theory is a powerful framework, which provides deep insights into the arithmetic object, such as elliptic curves and modular forms, etc.

Iwasawa studied the class groups of an infinite tower of field extensions, and he proved that there is an invariant which explains the aformentioned class groups. This invariant is called mu-invariant. In 1979, Ferrero and Washington proved the vanishing of mu-invariant.

The lecture gives an analytic method to prove the celebrated theorem of Ferrero-Washington, which is based on the celebrated conjecture, called Iwasawa main conjecture, proved by Mazur and Wiles in 1984.

If time permits, I also would like to introduce various generalized versions of this problem, for example, Selmer groups of elliptic curves and Mazur-Tate-Teitelbaum p-adic L-functions.

• 윤태호 (서울대학교)

Title : Data Sampling via Probability Distribution Flow along Stochastic Processes

Abstract : Methods for generating samples following a particular distribution is a crucial topic in statistics and machine learning. In this talk, we discuss sampling algorithms utilizing stochastic processes. We briefly review the mathematical characterization of how density functions evolve along stochastic differential equations, and how it can be used for designing practical sampling algorithms. Through a unified perspective, we look into both the Langevin algorithm and its variants (from a traditional context of sampling from a known density) and diffusion probabilistic models (from a modern generative modeling context in machine learning).