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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
No. Subject Author Date Views
Notice QSMS Mailing List Registration qsms 2021.09.09 4736
85 [QSMS-BK21 Seminar] Past seminars qsms 2022.08.26 1406
84 [SNU Number Theory Seminar 2022-04-15] A note on non-ordinary primes for some genus-zero arithmetic groups qsms 2022.04.12 1426
83 [QSMS Seminar 2022.04.19] Universal objects in vertex algebra theory qsms 2022.04.07 1460
82 [QSMS Monthly Seminar] The dimension of the kernel of an Eisenstein ideal qsms 2022.03.25 1461
81 [QSMS Seminar 2022-08-31] Localization of the category of modules over quiver Hecke algebras qsms 2022.08.25 1464
80 [Seminar 2020.12.17] Zeta함수와 L함수에 대한 역사적 고찰IV qsms 2020.12.08 1470
79 [QSMS Monthly Seminar] Quantization of integrable differential difference equations qsms 2022.04.22 1474
78 [QSMS Monthly Seminar] Yokonuma Hecke algebras and invariants of framed links qsms 2022.05.27 1475
77 [QSMS Seminar 2022-08-24] Vector bundles and representation theory on a curve qsms 2022.08.13 1475
76 [QSMS Mini-course] Pablo Portilla cuadrado (University of Lille) qsms 2022.07.04 1484
75 [SNU Number Theory Seminar 2022-06-21] Arithmetic local systems over the moduli space of curves qsms 2022.07.21 1487
74 [BK21-QSMS Toric Geometry Seminar] The Fredholm regularity of discs qsms 2022.03.14 1498
73 [SNU Number Theory Seminar 20220321, 0322, 0324] Three lectures on the Eisenstein ideal qsms 2022.03.19 1507
72 [SNU Number Theory Seminar 2022-08-08] Geometry of the $B_{dR}^+$-Grassmannian qsms 2022.08.04 1507
71 [SNU Number Theory Seminar 2022-05-13] A singular function containing all Lagrange numbers less than three qsms 2022.04.12 1512
70 [QSMS Geometry Seminar 2022-05-12] Tropical Lagrangian multi-sections and tropical locally free sheaves qsms 2022.05.07 1525
69 [SNU Number Theory Seminar 2022-08-11] The dual Lagrangian fibration of hyper-Kähler manifolds qsms 2022.08.04 1531
68 [QSMS Seminar 2022.04.05] Trialities of W-algebras qsms 2022.02.15 1535
67 [Seminar 2021.01.07] Zeta함수와 L함수에 대한 역사적 고찰VII qsms 2021.01.04 1544
66 [Seminar 2020.12.10] Zeta함수와 L함수에 대한 역사적 고찰III qsms 2020.12.08 1557
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