Math Seminar 

Title: Evolutionary coding based AI agents for mathematics 

Abstract: The rectilinear crossing number is the minimum number of edge crossings in a straight-line drawing of a graph in the plane. Despite its elementary definition, even the case of complete graphs remains poorly understood, and progress over the past several decades has relied on a blend of geometric intuition, combinatorial reasoning, and increasingly sophisticated computational methods. In this talk, I will present a progress report on an attempt to use artificial intelligence as a new exploratory tool for this classical problem. After briefly surveying the history of the rectilinear crossing number and its known constructions, I will introduce OpenEvolve, an open-source framework inspired by AlphaEvolve, which has recently been applied to the study of mathematical conjectures. I will describe the framework at a high level and explain how it can be adapted to the rectilinear crossing number. I will then discuss results from this ongoing work. OpenEvolve was able to not only rediscover known optimal constructions for small complete graphs, but also improve upon a best known construction for a larger instance. I conclude by reflecting on the potential role of AI-assisted exploration in mathematics. 

Venue: Room 103, Building 10-1 (Map), Seoul National University
Host: Boram Park
참가신청: https://forms.gle/xSzjqytGdTj4zyJX6 

Title: Fundamental PDEs

Abstract: Partial differential equations (PDEs) form a fundamental mathematical language for describing nature, from electrostatics and heat diffusion to wave propagation and fluid motion. In this talk, we revisit three basic examples---the Poisson, heat, and wave equations---and discuss why they are central to both mathematics and physics. We begin with the physical principles from which these equations arise, including conservation laws and Maxwell's equations, and then turn to some of the main analytical ideas used to study them. These include fundamental solutions, Fourier analysis, and energy methods, all of which provide insight into the qualitative behavior of solutions. Along the way, we highlight characteristic features of each equation: equilibrium and harmonicity for the Poisson equation, diffusion and smoothing for the heat equation, and propagation for the wave equation. More broadly, these examples illustrate several recurring themes in PDE: averaging principles, maximum principles, and rigidity phenomena such as Liouville-type theorems. They also reveal natural connections with other parts of mathematics, including complex analysis. The goal of the talk is to present an accessible introduction to some of the ideas that make these equations truly fundamental.

Venue: Room 207,  Building 10, Seoul National University

Host: Jinmyoung Seok

Speaker: Cheolwon Heo (SUNY Korea), Kang-ju Lee (SNU), Minho Cho (KIAS)

Website: Click here
Venue:  Room 207, Building 10 (Map), Seoul National University
Host: Boram Park

Website: Click here
Venue: Room 103, Building 10-1 (Map), Seoul National University 
Host: Boram Park

Title: How Many Sequences Are Needed to Test Continuity?

Abstract: 

As explained in Calculus I by Hong Jong Kim, continuity of a function at a point is equivalent to the following sequential condition: for every sequence converging to that point, the corresponding sequence of function values converges to the value of the function at the point. However, for functions defined on the real line, continuity at a point can also be determined by checking whether the left hand and right hand limits both agree with the function value. In other words, continuity can sometimes be tested without directly examining every possible type of convergent sequence, such as sequences that approach the point by alternating between its two sides. This leads to a natural question: do we really need to test every convergent sequence in order to determine continuity? If not, how far can we reduce the collection of sequences without losing the ability to detect discontinuity?

In this talk, we define a test set to be a family of convergent sequences that can detect every discontinuity, and ask whether there exists a minimal test set that cannot be reduced any further. Under suitable topological assumptions, we show that such a minimal test set exists and introduce a simple application of the axiom of choice that appears in its construction. Finally, we present a concrete example in which continuity can be tested using far fewer sequences than the full collection of convergent sequences.

(Korean) 『미적분학 1』(김홍종)에서 설명하듯이, 함수가 한 점에서 연속이라는 것은 그 점으로 수렴하는 모든 수열에 대하여 함숫값의 수열이 그 점에서의 함숫값으로 수렴한다는 것과 동치이다. 그러나 정의역이 실수인 함수의 경우, 한 점에서의 좌극한과 우극한이 모두 그 점에서의 함숫값과 일치하면 함수가 연속임을 알 수 있다. 즉, 그 점의 좌우를 번갈아 오가며 접근하는 수열과 같은 모든 형태의 수렴수열을 직접 검사하지 않고도 연속성을 판정할 수 있다. 이는 자연스러운 질문을 떠올리게 한다. 함수의 연속성을 판정하기 위해 정말 모든 수렴수열을 검사해야 할까? 그렇지 않다면, 연속성을 판정하는 능력을 잃지 않으면서 수열들의 모음을 어디까지 줄일 수 있을까?

  본 발표에서는 모든 불연속을 검출할 수 있는 수렴수열의 모음을 test set이라 정의하고, 그중 더 이상 줄일 수 없는 최소 test set이 존재하는지를 살펴본다. 적절한 위상적 조건 아래에서 이러한 최소 test set이 존재함을 보이고, 그 구성 과정에서 나타나는 선택공리의 간단한 응용을 소개한다. 마지막으로 전체 수렴수열의 모음보다 훨씬 작은 수열의 모음만으로도 연속성을 판정할 수 있는 구체적인 예를 살펴본다.  

Venue: Room 207, Building 10, Seoul National University 

Host: Sanghoon Kwak

Title: Group Properties and Profinite Completion

Abstract:  In geometric topology and geometric group theory, a central question is which groups and properties can be detected through their finite quotients. This perspective leads to the study of profinite rigidity. In this talk, we provide background on this line of research and present some recent developments in this area. We then present two recent results, one joint with Junseok Kim (Technion) and the other with Hyungryul Baik (KAIST). If time permits, we will also outline the proof.

Venue: Room 207, Building 10, Seoul National University 

Host: Sanghoon Kwak

Title: TBA

Abstract:  TBA

Venue: Room 207, Building 10, Seoul National University 

Host: Sanghoon Kwak