Invited Lecture
Speaker: Myungho Kim (Kyung Hee University)
Date: TBA
Title: Cluster algebras and monoidal categories
Abstract:
Cluster algebras are special commutative rings introduced by Fomin and Zelevinsky in the early 2000s. Specifically, the cluster algebra refers to a subring generated by special elements called cluster variables in the field of rational functions, and the process of creating a new cluster variable from given cluster variables is called a mutation. Cluster algebra is being actively studied as it is observed that the mutation operation appears in various forms in various fields of mathematics.
A monoidal categorification of a given cluster algebra means that the Grothenieck ring is isomorphic to the cluster algebra and that special elements called cluster monomials correspond to simple objects. If there is such a monoidal categorification, then the given monoidal category and the cluster algebra are closely related and help understand each other's properties.
In this talk, I will explain that the category of finite-dimensional representations of quiver Hecke algebras and that of quantum affine algebras form monoidal categorifications of cluster algebras. This is based on several joint works with Seok-Jin Kang, Masaki Kashiwara, Se-jin Oh, and Euiyong Park.
Special Session
(SS-03) Representation Theory and Related Topics
Speaker: Jethro van Ekeren (Universidade Federal Fluminense)
Schedule: 2021.10. 21. AM 8:40~9:10
Title: Chiral Homology and Classical Series Identities
Abstract:
I will discuss results of an ongoing project on the chiral homology of elliptic curves with coefficients in conformal vertex algebras. We find interesting links between this structure and classical number theoretic identities of Rogers-Ramanujan type (joint work with George Andrews and Reimundo Heluani).
Speaker: Ryo Sato (Kyoto university)
Schedule: 2021.10.21. AM 9:10~9:40
Title: Feigin-Seikhatov duality in W-superalgebras
Abstract:
W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. It has been known that princial W-algebras satisfy a certain duality relation (Feigin-Frenkel duality) which can be regarded as a quantization of the geometric Langlands correspondence. Recently, D. Gaiotto and M. Rapčák found dualities between more general W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of thier conjecture is proved by T. Creutzig and A. Linshaw, and a more specific subclass (Feigin-Seikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will explain how to upgrade the latter case to the level of representation theory by using relative semi-infinite cohomology. This is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka.
Speaker: 유필상 (칭화대학교)
Schedule: 2021.10.21. AM 9:50~10:20
Title: Representation Theory via Quantum Field Theory
Abstract:
It is known that some subjects in mathematics may be enriched by finding their context in physics. In this talk, we argue that representation theory is no exception. After explaining the physical context for representation theory of a finite group as the most basic example, we discuss a research program on how to use ideas of quantum field theory to study certain objects of interest in geometric representation theory.
Speaker: George D. Nasr (University of Oregon)
Schedule: 2021.10.21. AM 10:20~10:50
Title: A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids and its connections to Representation Theory
Abstract:
In 2016, Elias, Proudfoot, and Wakefield introduced Kazhdan-Lusztig polynomials for a class of combinatorial objects called matroids. Later, they presented the equivariant (representation-theoretic) version of these polynomials. We will introduce both these topics and discuss results in the case of sparse paving matroids. For the ordinary Kazhdan-Lusztig polynomials, we present a combinatorial formula using skew Young tableaux for the coefficients of these polynomials for sparse paving matroids. In the case of uniform matroids (a special case of sparse paving matroids), this formula results in a nice combinatorial interpretation that arises in the equivariant version of these polynomials.
Speaker: 오재성 (고등과학원)
Schedule: 2021.10.21. AM 11:00~11:30
Title: A tugging symmetry conjecture for the modified Macdonald polynomials
Abstract:
In this talk, we propose a conjecture which is a symmetry relation for the modified Macdonald polynomials of stretched partitions, $\widetilde{H}_{k\mu}[X;q,q^k]=\widetilde{H}_{\mu^k}[X;q^k,q]$. Using the LLT-expansion of the modified Macdonald polynomials and linear relations of the LLT polynomials, we prove the conjecture for one column shape partition $\mu=(1^l)$. This is based on the joint work with Seung Jin Lee.
Speaker: 김영훈 (서울대학교, QSMS)
Schedule: 2021.10.21 AM 11:30~12:00
Title: Extensions of 0-Hecke modules for dual immaculate quasisymmetric functions by simple modules
Abstract:
For each composition $\alpha$, Berg {\it et al.} introduced an indecomposable $0$-Hecke module $\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the quasisymmetric characteristic image. In this talk, we study extensions of $\mathcal{V}_\alpha$ by simple modules. To do this, we construct a minimal projective presentation of $\mathcal{V}_\alpha$ and calculate $\mathrm{Ext}^1$-group between $\mathcal{V}_\alpha$ and simple modules. Then we describe all non-split extensions of $\mathcal{V}_\alpha$ by simple modules in a combinatorial manner. As a corollary, it is shown that $\mathcal{V}_\alpha$ is rigid. This is joint work with S.-I. Choi, S.-Y. Nam, and Y.-T. Oh.
(SS-01) Trends in Arithmetic Geometry
Speaker: Chang-Yeon Chough (서울대학교, QSMS)
Schedule: TBA
Title: Brauer groups in derived/spectral algebraic geometry
Abstract:
Toën gave an affirmative answer to Grothendieck's question of comparing the Brauer group and the cohomological Brauer group of a scheme for all quasi-compact and quasi-separated (derived) schemes by introducing the notion of derived Azumaya algebras. I'll give a glimpse of the extension of this result to algebraic stacks in the setting of derived/spectral algebraic geometry. If time permits, my latest work on twisted derived equivalences in the derived/spectral setting, which is based on the aforementioned extension, will be presented.
