Bi-orderability of knot groups

A group is biorderable if it admits a strict total ordering that is invariant under both left and right multiplication. A knot is said to be biorderable if the fundamental group of its complement is biorderable. In this talk, we review the theory of biorderable groups and explore the relationship between topological invariants of knots and their biorderability.

New approaches to physics informed deep neural operator networks 

Partial differential equations (PDEs) play a central role in modeling natural and social phenomena. Although classical numerical methods based on the finite difference method (FDM) and the finite element method (FEM) provide reliable solutions, they suffer from high computational costs in real-time analysis and large-scale experiments. In this talk, we introduce two deep neural operator–based approaches for efficiently solving second-order elliptic PDEs with mixed boundary conditions. The first approach is a convolutional operator learning model based on a U-Net architecture, which takes FDM/FEM-discretized representations as inputs. The second approach adopts a sparse neural operator framework, in which structural sparsity is imposed on the network architecture or learned representations to reduce model complexity while preserving predictive accuracy. Numerical experiments demonstrate that the proposed two approaches achieve both high accuracy and fast inference compared to conventional DeepONet-based methods, and remain stable and effective even for significantly larger computational scales. 

Characterization of Alternating Links 

Alternating links are traditionally defined by their diagrammatic representations. Therefore characterizing them through their intrinsic properties is one of the most fundamental questions in knot theory. This presentation explores a topological characterization of alternating links based on the intrinsic structures of the link exterior. 

Combinatorics of orthogonal polynomial sequences

Orthogonal polynomial sequences (OPS) form an orthogonal basis for the polynomial ring and appear widely in analysis, probability, and mathematical physics. Although orthogonality is defined through an inner product involving integrals, this inner product is often difficult to compute directly—this is where combinatorics enters the story. In this talk, we will explore how discrete models such as lattice paths and matchings naturally encode analytic properties of orthogonal polynomials.

Introduction to Quantum Groups and Crystal Bases

During the 1980s and 1990s, the theory of quantum groups was established by various mathematicians. Conceptually, a quantum group is a q-deformation of the universal enveloping algebra. The original algebra can be recovered in the classical limit where the parameter q → 1. In the 1990s, Kashiwara introduced and developed the theory of crystal bases. This theory serves as a powerful framework that encompasses the structure of existing Lie algebras while translating complex algebraic problems into manageable combinatorial ones. In this speaks, we will first examine how these structures are realized in the simplest case, sl2. Subsequently, we will discuss the case of type Al and demonstrate the correspondence between crystal bases and the Semistandard Young Tableaux (SSYT) via the case of sl3.

Why do we count with q? : From Rook Placements to Hessenberg Varieties

In this talk, we introduce the palindromic linked q-hit numbers, which arise from counting rook placements on specific board shapes. While these numbers are defined combinatorially, we address the fundamental question: "Why do we count with q?" We answer this by exploring the connection between these combinatorial objects and regular semisimple Hessenberg Varieties. We explain the geometric role of q as the grading of the cohomology ring of the variety. And then, we discuss how the cohomology character can be expressed as a co-palindromic linear combination of these palindromic linked q-hit numbers. This talk is based on joint work with Seung Jin Lee. 

원 위의 군 작용과 그 응용 

원 (circle)은 가장 쉬운 다양체인 동시에 저차원 위상수학의 여러 문제에서 자연스럽게 등장하는 근본적인 공간이기도 하다. 본 강연에서는 우선 원 위의 group action이 유도되는 두 가지 대표적인 상황인 곡면의 hyperbolic geometry와 3차원 다양체의 pseudo-Anosov 흐름에 대해서 설명할 것이다. 곡면의 hyperbolic geometry나 pseudo-Anosov flow를 통해 얻어진 원 위의 작용들은 특별한 성질들을 가지게 되는데 이에 대해서도 알아볼 것이다.  최종적으로 이들의 inverse problem, 즉, 원에 특별한 방식으로 작용하는 군에 대해서 우리가 어떤 이야기를 할 수 있는지에 대해서 다뤄 볼 것이다. 

박창제 - Point Derivation and Directional Derivative

이민혁 - Braid Form of Knot

최푸른하늘 - What is Complex Geometry? 

국예성 - A Homology Sphere

On the Convergence of the Metropolis-Hastings Kernel 

The Metropolis-Hastings algorithm is a well-known Markov Chain Monte Carlo method for estimating posterior summaries in Bayesian statistics. In this talk, we prove the convergence and explore the behaviour of the Metropolis kernel via its ergodicity. The main reference is Tierney, L. (1994). Markov Chains for Exploring Posterior Distributions.   

The Many Faces of n! : An Invitation to Algebraic Combinatorics

We all know that there are n! ways to order n distinct objects. In Algebraic Combinatorics, however, this is not the end of the story—it is only the beginning. In this talk, we will explore the surprising appearances of n! across the mathematical landscape. We will see how n! arises naturally in Algebra (finite fields), Geometry (flag varieties), Probability (random matrices), and Representation Theory (coinvariant rings), revealing deep structural connections between these seemingly disparate areas. 

영지식 증명(zero-knowledge proof)과 적용

영지식 증명(zero-knowledge proof)은 증명자가 자신의 비밀 정보를 전혀 드러내지 않으면서도, 그 정보가 참임을 검증자에게 수학적으로 확신시키는 기술입니다. ZKP의 고전적 모델인 QR 프로토콜을 통해 ZKP의 3대 속성(completeness, soundness, zero knowledge)과 고전 모델의 한계를 살펴보고, 한계를 극복한 zk-SNARK 모델을 확인합니다. 끝으로 ZKP의 적용 방향을 제시합니다.

Relativistic Quantum Theory: From Birth to Death, and Rebirth 

In this lecture, we will examine the development of relativistic quantum theory (RQT), formulated in the 1920s by pioneers such as P. Dirac, J. Schwinger, and others. RQT originated by imposing the Lorentz symmetry on (original) quantum mechanics. We will begin by studying the Klein-Gordon equation and the Dirac equation, and then confront the limitations of them. This will lead us to the "death" of RQT in its original form.

The story continues, however, with the introduction of a completely new perspective: the field-theoretic approach. We will see how RQT undergoes a "rebirth" through this framework, guided by the concept of second quantization. Along the way, we will quantize the Klein-Gordon field and the Dirac field, and introduce the powerful idea of operator-valued fields, which lies at the foundation of modern quantum field theory (QFT).

Convex Optimization and Minimax Problems

Convex optimization is a cornerstone of modern machine learning and optimization theory. In this talk, I will introduce several methods for solving convex-concave minimax problems (such as GDA and Extragradient) and extend the discussion to stochastic settings.

오민규 - Intuition for differential Galois Theory

조수연 - Bradley-Terry Model for Pairwise Comparisons 

한채영 - From Stanley–Stembridge Conjecture to Positivity in Unit Interval Graphs 

황다현 - Hairy Ball Theorem