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일정시작 2021-10-21
일정종료 2021-10-21
배경색상 #33FFF6

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.


List of Articles
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Notice QSMS Mailing List Registration qsms 2021.09.09 13291
» [2021 KMS Annual Meeting] Related talks qsms 2021.10.05 3951
149 [2021 KMS Spring Meeting] Related talks qsms 2021.04.30 7770
148 [2022 Global KMS International Conference 10/19 ~ 10/21] Related talks qsms 2023.01.05 3565
147 [2022 KMS Spring Meeting] Related talks qsms 2022.04.28 3583
146 [BK21-QSMS Toric Geometry Seminar] Moment Polytope qsms 2022.01.03 3893
145 [BK21-QSMS Toric Geometry Seminar] The Fredholm regularity of discs qsms 2022.03.14 3604
144 [KMS Specical Conference with 2022 Fields Medalists] Related talks qsms 2023.10.11 3772
143 [QSMS Geometry Seminar 2022-05-12] Tropical Lagrangian multi-sections and tropical locally free sheaves qsms 2022.05.07 3439
142 [QSMS Geometry Seminar 2022-05-17, 19] Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications qsms 2022.05.09 3480
141 [QSMS Geometry Seminar 20221230] Morphism spaces between coisotropic A-branes qsms 2022.12.27 3413
140 [QSMS Geometry seminar 2023-05-25] Symplectic field theory and codimension-2 stable Hamiltonian submanifolds qsms 2023.05.25 3382
139 [QSMS Mini-course] Pablo Portilla cuadrado (University of Lille) qsms 2022.07.04 3465
138 [QSMS Monthly Seminar 2022-09-30] Monotone Lagrangians via toric degenerations: algebraic approach qsms 2022.09.30 3437
137 [QSMS Monthly Seminar 2022-10-28] Oscillator representations of quantum affine algebras   qsms 2022.10.30 3395
136 [QSMS Monthly Seminar 2022-12-02] Extended crystal structures of Hernandez-Leclerc categories qsms 2022.12.02 3435
135 [QSMS Monthly Seminar 2023-03-24] Rational points on abelian varieties qsms 2023.03.24 3356
134 [QSMS Monthly Seminar 2023-04-21] Lagrangian submanifolds and their Legendrian lifts qsms 2023.04.21 3399
133 [QSMS Monthly Seminar 2023-05-26] Lagrangian submanifolds and their Legendrian lifts qsms 2023.05.26 3517
132 [QSMS Monthly Seminar 2023-10-13] Quantum Grothendieck rings of Hernandez-Leclerc categories / Regular Schur labeled skew shape posets and their 0-Hecke modules qsms 2023.10.13 3499
131 [QSMS Monthly Seminar 2023-11-11] Cluster structures on polygon spaces / From classical to quantum integrability qsms 2023.11.10 3757
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