2025.2.9- 2.13, Yangpyeong
Date: February 9-13, 2025
Venue: Yangpyeong Bloomvista [Website] [How to get there]
Address: 경기도 양평군 강하면 강남로 316 [Naver Maps]
Time | 2/9 (일) | 2/10 (월) | 2/11 (화) | 2/12 (수) | 2/13 (목) |
---|---|---|---|---|---|
9:20 | 개회사 | ||||
9:30 | 오정석 | 오정석 | 오정석 | Free Discussion | |
10:30 | Coffee Break | ||||
11:00 | 권재훈 | 권재훈 | 권재훈 | ||
12:00 | Group Photo | Lunch | Lunch | ||
Lunch | |||||
14:00 | 최동준 | Sylvain Carpentier | 천지원 | ||
15:00 | Free Discussion | Coffee Break | |||
15:30 | 이수홍 | 이소연 | Dmitriy Voloshyn | ||
16:30 | Coffee Break | ||||
17:00 | Free Discussion | Free Discussion | |||
17:40 | Dinner | Dinner | Dinner | ||
18:00 | Banquet |
Time | 2/9 (일) | 2/10 (월) | 2/11 (화) | 2/12 (수) | 2/13 (목) |
---|---|---|---|---|---|
9:20 | 개회사 | ||||
9:30 | 오정석 | 오정석 | 오정석 | Free Discussion | |
10:30 | Coffee Break | ||||
11:00 | 권재훈 | 권재훈 | 권재훈 | ||
12:00 | Group Photo | Lunch | Lunch | ||
Lunch | |||||
14:00 | 유화종 | 김찬호 | 권재성 | ||
15:00 | Free Discussion | Coffee Break | |||
15:30 | 김찬호 | 김영균 | 김인서 | ||
16:10 | Coffee Break | ||||
16:30 | Coffee Break | Problem Session | 최진우 | ||
17:00 | 조석준 | ||||
17:10 | |||||
17:40 | Dinner | Dinner | Dinner | ||
18:00 | Banquet |
Speaker: 오정석
Abstract: This is a short lecture on Grassmannians.
Speaker: 권재훈
Abstract: This series of lectures provides a basic introduction to the representation theory of quantum groups and crystal bases.
Lecture 1: Quantum groups and their representations
Lecture 2: Crystal bases for integrable highest-weight modules
Lecture 3 Combinatorial realization of crystals.
Slides: Day 1
Speaker: 유화종
Abstract: This will be a preliminary lecture for Chan-Ho Kim's lectures on Euler systems.
Slides: Day 1
Speaker: 김찬호
Abstract: This will be an introductory lecture on Euler systems. The theory of Euler systems has played the fundamental role in the progress towards various conjectures on special values of \(L\)-functions including the famous Birch and Swinnerton-Dyer conjecture. The goal of this lecture is to make the participants feel how it works.
Main Reference: Chapter 3 of https://arxiv.org/abs/2404.05186
Slides: Day 1
Speaker: 조석준
Abstract: Let \(N\) be a positive integer, and \(J_0(N)\) the Jacobian of the modular curve \(X_0(N)\). By the theorem of Manin and Drinfeld, the rational torsion subgroup of \(J_0(N)\) contains the rational cuspidal subgroup \(C_N(\mathbb{Q})\), and the generalized Ogg’s conjecture predicts that these groups are equal. As both groups are hard to understand, we focus on a more computable subgroup of \(C_N(\mathbb{Q})\), namely the rational cuspidal divisor class group \(C(N)\). But it is still hard to determine the group completely, so instead we explain how to compute the orders of its generators using eta quotients.
Slides: Link
Speaker: 최동준
Abstract: The Zhu algebra, introduced by Yongchang Zhu in 1990, is an associative algebra constructed from a given vertex algebra. For instance, the Zhu algebras of affine \(\mathcal{W}\)-algebras correspond to finite \(\mathcal{W}\)-algebras. In 2022, Alexander Molev introduced \(\mathcal{W}\)-algebras associated with centralizers in type A, which possess a natural vertex algebra structure. In this talk, I would like to introduce \(\mathcal{W}\)-algebras associated with centralizers in type A for a general nilpotent element, which we call generalized affine \(\mathcal{W}\)-algebras, and their Zhu algebras, which we call generalized finite \(\mathcal{W}\)-algebras. This is joint work with Alexander Molev and Uhi Rinn Suh.
Slides: Link
Speaker: 이수홍
Abstract: We introduce a non-semisimple Fock space \(\mathcal{F}^{\infty}\otimes\mathcal{M}\) of infinite rank, which has commuting actions of a parabolic \(q\)-boson algebra and \(U_p(\mathfrak{gl}_{>0})\) with \(p=-q^{-1}\). It contains the classical fermionic Fock space \(\mathcal{F}^n\) of level \(n\), whose semisimple decomposition is explained by Howe duality. Although crystal bases may not be unique when a module in consideration is not semisimple, sometimes (and actually quite often) we can find a largest crystal lattice, which we call a saturated crystal. Its existence is highly non-trivial and has been observed for quantum affine algebra by Kashiwara. We explain how we can (non-constructively) construct a saturated crystal on \(\mathcal{F}^{\infty}\otimes\mathcal{M}\), and how it can be used to describe the socle filtration of \(\mathcal{F}^{\infty}\otimes\mathcal{M}\) and that of tensor products of extremal weight modules of type \(A_{+\infty}\). This is a joint work with Jae-Hoon Kwon.
Slides: Link
Speaker: 김찬호
Abstract: This will be an introductory lecture on Euler systems. The theory of Euler systems has played the fundamental role in the progress towards various conjectures on special values of \(L\)-functions including the famous Birch and Swinnerton-Dyer conjecture. The goal of this lecture is to make the participants feel how it works.
Main Reference: Chapter 3 of https://arxiv.org/abs/2404.05186
Speaker: 김영균
Abstract: We review results on upper bounds for the size of periodic and preperiodic orbits in arithmetic dynamical systems. We then show that if an integral algebraic dynamical system admits an unramified reduction, the size of its preperiodic orbits are controlled by the corresponding dynamical system modulo that reduction. Using this, we establish an upper bound for preperiodic orbits in integral dynamical systems with morphisms of unramified reduction.
Speaker: Sylvain Carpentier
Abstract: Classical \(\mathcal{W}\) algebras are conformal differential algebras attached to a simple Lie algebra \(\mathfrak{g}\) and a nilpotent element \(f \in \mathfrak{g}\). A major discovery of Drinfeld and Sokolov is that these algebras can be realized as algebras of functions on linear differential matrix operators. More recently, De Sole Kac and Valeri have shown that the Poisson brackets on the \(\mathcal{W}\) algebra \(\mathcal{W}(\mathfrak{g}, f)\) identifies with the famous Gelfand Dickey brackets on differential operators. This identification enabled them to construct integrable systems of PDEs on each of the classical \(\mathcal{W}\) algebra. For example, the Korteweg de Vries equation corresponds to the pair \((\mathfrak{sl}_2, f)\). In this talk, we discuss the generalization of this picture to the supersymmetric case, which is a joint work with professor U. Suh.
Speaker: 이소연
Abstract: Due to its connection with quasisymmetric functions, the representation theory of \(0\)-Hecke algebras has drawn a lot of interest. Recently, poset modules have been intensively studied as a unified framework for studying various \(0\)-Hecke modules. In this talk, we review a study of a family of poset modules whose image under quasisymmetric characteristic is a symmetric function, namely, poset modules associated with regular Schur labeled skew shape posets. This provides a hint for a possible relationship between generic Hecke modules and \(0\)-Hecke modules. We will also discuss some approaches for studying this connection.
Speaker: 권재성
Abstract: We investigate the derived Hecke action on the cohomology of an arithmetic manifold associated to the multiplicative group over a number field. The degree one part of the action is proved to be non-vanishing modulo \(p\) under mild assumptions. The main ingredient is the Grunwald--Wang theorem. This work is joint with Dohyeong Kim.
Speaker: 김인서
Abstract: I will extend the classical Legendre symbol to the Rédei symbol using Milnor numbers obtained from the Magnus expansion. Koch’s theorem provides the relation between Milnor numbers and the Legendre symbol. I will explain how the Legendre symbol is generalized and demonstrate that the Rédei symbol is a specific case of this generalization.
Speaker: 최진우
Abstract:Unlike the case of real quadratic fields, it is in general hard to determine a complete set of fundamental units for a given family of non-Galois totally real cubic number fields. Ennola’s conjecture is one of unsolved problems on a pair of fundamental units of such a family. Stéphane Louboutin suggested a weak form of Ennola’s conjecture and provided a conditional proof of it. In this talk, I introduce both conjectures and make the conditional proof a complete proof. This can be done by showing the two main assumptions Louboutin made in his previous work hold true. This is a joint work with Professor Dohyeong Kim.
Speaker: 천지원
Abstract: Khovanov-Lauda-Rouquier algebras, which is also called quiver hecke algebra categorifies the negative half of quantized enveloping algebras. More precisely, the Grothendieck ring of a category of the graded modules of KLR algebra is isomorphic to the negative half. For example, there are some modules categorifying (dual) canonical bases and (dual) PBW basis in \(U_q^{-}(\mathfrak{g})\). In this talk, I will explain concrete description of such modules in type \(A\).
Speaker: Dmitriy Voloshyn
Abstract: I will start the talk with the story of the origins of cluster algebras (in the sense of Fomin and Zelevinsky) via total positivity. The story is a very instructive example of how a solution to a mathematical problem can indicate that the problem itself should have been posed differently. In a sense, the story is also a part of the general mathematical concept of completion: cluster algebras emerged as a way of completing solutions in total positivity. I will then move to the connection between cluster algebras and Poisson geometry, and explain how to use Poisson geometry to construct a cluster algebra. Surprisingly, in all examples in Lie theory that we know of, if there's a cluster algebra, there is a Poisson geometry in the background. Time permitting, I will explain some unexplored directions of research related to quantization. Mainly, there are examples of quantum cluster algebras on quantum space (for instance, quantum groups), as defined by Fock and Goncharov; if one considers a classical object in Lie theory endowed with a cluster algebra in the sense of Fomin and Zelevinsky, it is not clear how to quantize both the object and the cluster algebra and receive a quantum space with a quantum cluster algebra.
Tel: 02-880-4253
서울특별시 관악구 관악로1 서울대학교 25동 102호 QSMS 행정실 우편번호 08826
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