Diophantine equations and moduli spaces with nonlinear symmetry
A fundamental result in number theory is that, under certain linear actions of arithmetic groups on homogeneous varieties, the integral points of the varieties decompose into finitely many orbits. For a classical example, the set of integra...
Symplectic Geometry, Mirror symmetry and Holomorphic Curves
Symplectic geometry arose from the study of classical mechanics, and later many interesting symplectic invariants has been found since Gromov introduced techniques of J-holomorphic curves. Miraculously, such invariants are closely related wi...
Toward bridging a connection between machine learning and applied mathematics
This lecture explores the topics and areas that have guided my research in computational mathematics and deep learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and dee...
1. 금본위제, 달러, 비트코인 등 돈의 흐름으로 보는 세계사 2. 사람은 어떻게 생각하고 행동하는가 ? (행동경제학, 비선형성) 3. 돈에 대한 생각, 행동, 습관을 바꾸어보자. (부자들은 무엇이 다른가 ? 지금부터 준비해보자.) 4. 주식, 부동산 등 자산관리 [...
<학부생을 위한 ɛ 강연> Self-Supervised Learning in Computer Vision
In recent years, artificial intelligence has made remarkable progress in developing algorithms that can learn from vast amounts of carefully labeled data. This paradigm of supervised learning has made great success in training specialist mo...
Limit computations in algebraic geometry and their complexity
Given a one-parameter family of algebraic varieties, its point-wise limit is usually too small whereas its algebraic limit is usually too big. I will introduce a notion of meaningful geometric limit and explain how it can be effectively comp...
Geometry, algebra and computation in moduli theory
I will explain the basic concepts of moduli and how moduli spaces can be constructed in algebraic geometry. Exploring the moduli spaces and issues arising from their construction lead to interesting interplay of geometry, algebra and computa...
<학부생을 위한 ɛ 강연> Geometry and algebra of computational complexity
학부생을 위한 이 강연에서는 고전적 튜링 기계의 기본적 정의로부터 시작하여 • 튜링기계를 비롯한 다양한 컴퓨터 모델의 복잡도 개념; • 계산(불)가능성 – 특히 디오판틴 방정식의 알고리즘적 해결법 (힐버트의 10번째 문제); • Non-deterministic 튜링 기계...
Mechanization of proof: from 4-Color theorem to compiler verification
I will give a broad introduction to how to mechanize mathematics (or proof), which will be mainly about the proof assistant Coq. Mechanizing mathematics consists of (i) defining a set theory, (2) developing a tool that allows writing definit...
젊은과학자상 수상기념강연: From particle to kinetic and hydrodynamic descriptions to flocking and synchronization
In this talk, I will report a recent progress for the modeling of collective behaviors of complex systems, in particular ocking and synchronization. Flocking and synchro-nization are ubiquitous in our daily life, for example, ocking of birds...
Coulomb Gases are point processes consisting of particles whose pair interaction is governed by the Coulomb potential. There is also an external potential which confines the particles to a region. Wigner introduced this toy model for the Gi...
The spaces admitting a rational parameterization are called rational. In particular plane conics, including circles, are rational. We will explain a few interesting applications of the rational parameterization of a circle. Also several exam...
학부생을 위한 ε 강연회: Constructions by ruler and compass together with a conic
Trisection of an angle and duplication of a cube are among the famous problems of Greeks. Although they were proven later to be impossible in general, Greeks already knew that one can trisect an angle and duplicate a cube by supplimenting se...
Birational Geometry of varieties with effective anti-canonical divisors
Fano varieties are fundamental objects in algebraic geometry. These can be considered as the unique output of the -K -minimal model program on the varieties with effective anticanonical divisors. Thus the initial models should encode the in...
A modified separation method to solve a heat-transfer boundary value problem
We derive a general solution of the heat equation through two modied separation methods. The obtained solution is expressed as linearly combined kernel solutions in terms of Hermite polynomials, which appears to provide an explanation of non...