Introduction to Symmetric Function Theory

Introduction to Symmetric Function Theory

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https://www.youtube.com/embed/dPCKdVSyq1k  + 47
강연자 최현재
소속 서울대학교

Algebraic combinatorics studies the connections between combinatorics and algebraic structures. Its central topics include symmetric functions, tableaux, crystals, and so on. In this talk, I will focus on symmetric function theory and positivity problems. Then we move on to q-analogues of symmetric functions, such as the Kostka-Foulkes polynomials. Finally, I will briefly introduce Lusztig’s q-weight multiplicities, a generalization of Kostka-Foulkes polynomials, and describe a type C combinatorial formula using semistandard oscillating tableaux. This talk is based on joint work with Donghyun Kim and Seung Jin Lee. 

프린트

Seeing Quadratic Lattices Through Their Substructures

강연자 : 김대준 | 소속 : 고려대학교

Representations by quadratic forms have long been studied through the geometric language of quadratic lattices. In this talk, we approach a quadratic lattice through two of its substructures: lattice cosets and full-rank sublattices. In the first part, we briefly introduce theta series of lattice cosets and then look at a graph structure on ternary cosets that reflects their algebraic relationships. In the second part, we turn to counting sublattices of a binary lattice up to isometry, and see how this leads to a Dirichlet series closely related to the arithmetic of imaginary quadratic fields.

Representation theory of 0-Hecke algebras and quasisymmetric functions

강연자 : 이소연 | 소속 : 서울대학교

The representation theory of the symmetric group plays a central role in mathematics, particularly in combinatorics through its connection with symmetric functions. In this setting, the Iwahori–Hecke algebra can be viewed as a q-deformation of the group algebra of the symmetric group. Its representation theory depends on the parameter q, leading to different behavior for different values of q. In this talk, I focus on the case q = 0, namely the 0-Hecke algebra, whose representation theory is closely related to quasisymmetric functions. I will briefly review the history of this subject and discuss recent developments involving quasisymmetric functions and related combinatorics.
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현재 Introduction to Symmetric Function Theory

강연자 : 최현재 | 소속 : 서울대학교

Algebraic combinatorics studies the connections between combinatorics and algebraic structures. Its central topics include symmetric functions, tableaux, crystals, and so on. In this talk, I will focus on symmetric function theory and positivity problems. Then we move on to q-analogues of symmetric functions, such as the Kostka-Foulkes polynomials. Finally, I will briefly introduce Lusztig’s q-weight multiplicities, a generalization of Kostka-Foulkes polynomials, and describe a type C combinatorial formula using semistandard oscillating tableaux. This talk is based on joint work with Donghyun Kim and Seung Jin Lee. 

Efficient and accurate structure preserving schemes for complex nonlinear systems

강연자 : Jie Shen | 소속 : Eastern Institute of Technology, Ningbo

Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme. I will present some recent advances on using the scalar auxiliary variable (SAV) approach and Lagrange multiplier approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, and can achieve higher-order accuracy. The computational cost is dominated by solving decoupled linear equations with constant coefficients at each time step.

학부생을 위한 ɛ 강연: Optimality of Gerver's Sofa

강연자 : 백진언 | 소속 : 고등과학원

We resolve the moving sofa problem, by showing that Gerver's construction with 18 curve sections attains the maximum area of 2.2195....

Formalizing mathematics: why it matters now

강연자 : 황병학 | 소속 : KIAS

Formalizing mathematics involves translating mathematical statements from natural language into a precise formal language that computers can interpret. As modern mathematics becomes deeper and more complex, the importance of formalization has grown significantly. In this talk, I will provide a brief introduction to the concept of formalization, discuss its significance, and explore how formalization and AI interact with each other.

Total domatic number of cubic graphs

강연자 : 권혜민 | 소속 : KIAS

The theory of dominating sets in graph theory has attracted considerable attention from numerous researchers, and various related notions and variations have been introduced. In this talk, we present the concepts of total dominating sets and the total domatic number. In particular, we discuss conditions under which a cubic graph has total domatic number at least 2.

학부생을 위한 ɛ 강연: A Zoo of Noncompact Surfaces

강연자 : 곽상훈 | 소속 : 서울대학교

학부 위상수학의 한 목표는 콤팩트한 곡면은 “구멍의 개수”로 완전히 분류됨을 소개하는 것입니다. 하지만 구멍이 무한히 많아지면 어떻게 될까요? 본 강연에서는 무한 곡면의 다양한 예를 살펴보고, Kerékjártó, Richards, Brown, Messer 등 다양한 수학자들이 제시한 무한 곡면에 대한 분류 정리들을 소개합니다.

Ancient solutions to the Yamabe flow

강연자 : 김승혁 | 소속 : 한양대학교

The Yamabe flow is a geometric evolution equation whose goal is to improve the geometry of a manifold by deforming its metric toward one with constant scalar curvature. In this talk, I will introduce the basic background and motivation for the Yamabe flow and then discuss recent developments concerning ancient solutions, namely, solutions that exist for all negative times. In the latter part of the talk, I will present joint work with Dr. Haixia Chen (Hanyang University) and Prof. Monica Musso (University of Bath, UK) on the existence of nonradial ancient solutions.

Strichartz estimates for the Schrödinger equation on the two-dimensional torus

강연자 : 곽범종 | 소속 : KAIST

The study of estimating the size of a solution to a linear dispersive partial differential equation, called the Strichartz estimate, has been of interest in both partial differential equations and harmonic analysis.

In this talk, I will focus on Strichartz estimates for the Schrödinger equation on a flat two-dimensional torus, which has rich connection to discrete mathematics. I will briefly introduce some of the connections, ranging from number theory to, more recently, incidence geometry and additive combinatorics.