Timetable
*Time zone: KST, UTC/GMT +9, Seoul
Dec 06, Friday
09:00 ~ 09:25 Registration
09:25 ~ 09:30 Open Ceremony
*Chair: Hironori Oya (Institute of Science Tokyo)
09:30 ~ 10:30 Jaeseong Oh (KIAS)
10:50 ~ 11:50 Naoki Fujita (Kumamoto University)
11:50 ~ 12:00 Time for a group photo
12:00 ~ 14:00 Lunch
*Chair: Uhi Rinn Suh (Seoul National University)
14:00 ~ 15:00 Wen Chang (Shaanxi Normal University)
15:15 ~ 16:15 Ching Hung Lam (Academia Sinica)
16:30 ~ 17:30 Toshiro Kuwabara (University of Tsukuba)
Dec 07, Saturday
*Chair: Ching Hung Lam (Academia Sinica)
09:30 ~ 10:30 Hironori Oya (Institute of Science Tokyo)
10:50 ~ 11:50 Chi-Heng Lo (Purdue University)
11:50 ~ 14:00 Lunch
*Chair: Naoki Fujita (Kumamoto University)
14:00 ~ 15:00 Shiquan Ruan (Xiamen University)
15:15 ~ 16:15 Akito Uruno (Seoul National University)
16:30 ~ 17:30 Xiaomeng Xu (Peking University)
17:30 ~ Banquet
Dec 08, Sunday
*Chair: Myungho Kim (Kyung Hee University)
09:30 ~ 10:30 Linliang Song (Tongji University)
10:50 ~ 11:50 Haruto Murata (University of Tokyo)
11:50 ~ 14:00 Lunch
*Chair: Wen Chang (Shaanxi Normal University)
14:00 ~ 15:00 Valentin Buciumas (POSTECH)
15:15 ~ 16:15 Tzu-Jan Li (Academia Sinica)
16:15 ~ 16:30 Ending Ceremony
Titles and Abstracts
Dec 06, Friday
Speaker: Jaeseong Oh (KIAS)
Title: Science Fiction conjecture and q,t Catalan Numbers
Abstract: Macdonald polynomials form an important family of symmetric functions due to their connections to representation theory, geometry, probability theory, and combinatorics. To study these polynomials, Bergeron and Garsia introduced the Science Fiction conjecture. In this talk, I will introduce Macdonald intersection polynomials that arise in the context of the Science Fiction conjecture and explore their relation to q,t Catalan numbers. If time permits, I will also present new discoveries on extending the Science Fiction conjecture and a new proof of the Loehr-Warrington conjecture. Based on joint works with Seung Jin Lee and Donghyun Kim.
Speaker: Naoki Fujita (Kumamoto University)
Title: Semi-toric degenerations of Schubert varieties and symplectic Gelfand-Tsetlin polytopes
Abstract: One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin gave two realizations of a Schubert class as sums of reduced Kogan faces and reduced dual Kogan faces. In this talk, we generalize these two realizations to type C full flag variety and symplectic Gelfand-Tsetlin polytopes, using semi-toric degenerations of Schubert varieties. Our generalization of reduced Kogan faces (resp., reduced dual Kogan faces) inherits combinatorics of mitosis operators (resp., reduced subwords). This talk is partly based on a joint work with Yuta Nishiyama.
Speaker: Wen Chang (Shaanxi Normal University)
Title: Tilting-completion for gentle algebras
Abstract: It is proved that any almost tilting module over a gentle algebra is partial tilting, that is, it can be completed as a tilting module. Furthermore, it has at most $2n$ complements, which confirms a (deformed) conjecture of Happel for the case of gentle algebras. At the same time, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a connected gentle algebra with rank $n$ and a pre-tilting module over it with rank $m$ which is not partial tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is doing inductions by cutting the surface, which is expected to be useful elsewhere.
Speaker: Ching Hung Lam (Academia Sinica)
Title: Automorphism groups of parafermion vertex operator algebras
Abstract: We determine the full automorphism groups of all parafermion vertex operator algebras associated with simple Lie algebras and positive integral levels. In particular, we show that the automorphism group is isomorphic to the automorphism group of the root system of the simple Lie algebra if
(i) the level is at least 3 or
(ii) the level is 2 and the simple Lie algebra is non simply laced.
Our method is based on the representation theory of parafermion vertex operator algebras and the theory of simple current extensions. It is a joint work with Xingjun Lin and Hiroki Shimakura.
Speaker: Toshiro Kuwabara (University of Tsukuba)
Title: Vertex operator superalgebra arising from the Hilbert scheme of points on the complex plane
Abstract: The Hilbert scheme of points on the complex plane is one of the fundamental examples of conical symplectic resolutions, providing a resolution of the singularities of the quotient space C^{2N} / S_N under the action of the symmetric group S_N. According to Kashiwara and Rouquier, the rational Cherednik algebra of type A can be constructed from the noncommutative algebra of global sections of a C[[h]]-algebra sheaf over the Hilbert scheme. By considering a vertex superalgebra analog, we construct a sheaf of h-adic vertex superalgebras over the Hilbert scheme, from which a family of vertex superalgebras can be defined as its global sections.
Bonetti, Meneghelli, and Rastelli conjectured the existence of a supersymmetric vertex operator algebra W_G associated with a complex reflection group G, as part of the context of 4D/2D duality in four-dimensional conformal field theory. We prove that the vertex superalgebra constructed in our work satisfies the expected properties of W_G in the case of G = S_N. In particular, it naturally contains a vertex algebra called the N=4 superconformal vertex algebra and admits a free field realization in terms of an (N-1)-rank βγbc system. Furthermore, using the vanishing of the BRST cohomology of the construction, we determine the supercharacter of the vertex superalgebra. This talk is based on a joint work with Tomoyuki Arakawa and Sven Moeller.
Dec 07, Saturday
Speaker: Hironori Oya (Institute of Science Tokyo)
Title: Algebraic study of quantum configuration spaces of decorated flags
Abstract: Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$, and $U$ its maximal unipotent subgroup. An element of $G/U$ is called a decorated flag. In this talk, we study algebraic structures of a quantum analogue of the ring of regular functions on the configuration space of $n$ decorated flags.
I explain its relation with the quantum coordinate rings of $G$ and its Borel subgroup. It can be considered as a quantum analogue of Wilson lines on the moduli space of decorated twisted $G$-local systems on the polygons. We also discuss its quantum cluster algebra structure.
This talk is based on a joint work with Tsukasa Ishibashi.
Speaker: Chi-Heng Lo (Purdue University)
Title: On local Arthur packets and unitary dual of classical groups
Abstract: Recently, Tadić classified the unitary dual for representations of corank at most 3 of classical groups over p-adic fields. Based on the classification, he conjectured that a representation of critical type is unitary if and only if it is of Arthur type, and that any isolated representation in the unitary dual is of critical type. These conjectures indicate that representations of Arthur type form an important subset inside the whole unitary dual.
Jointly with A. Hazeltine, D. Jiang, B. Liu and Q. Zhang, recently we proposed a refinement of Tadić’s conjecture: A representation of good parity is unitary if and only if it is of Arthur type. Moreover, we gave a conjecture on a description of the whole unitary dual for classical groups. In this talk, I will introduce these two conjectures and our main result that they hold for representations of corank at most 3 of symplectic and split special odd orthogonal groups.
Speaker: Shiquan Ruan (Xiamen University)
Title: iHall algebras of weighted projective lines and iquantum loop algebras
Abstract: This is joint work with Ming Lu and Weiqiang Wang.
The iHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the iquantum loop algebra, which is a generalization of the iquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation.
Speaker: Akito Uruno (Seoul National University)
Title: Crystal bases of the negative half over the quantum orthosymplectic superalgebras
Abstract: The crystal base theory for the quantized enveloping algebra Uq(g) associated to a symmetrizable Kac-Moody algebra g has been one of the most important tools in the representation theory of Uq(g), reflecting its fundamental combinatorial structure.
In this talk, we show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q-analogue of a polynomial representation of a general linear Lie superalgebra. We also construct a crystal base of the negative half of the quantum orthosymplectic superalgebras and observe compatibility with a crystal base of a parabolic Verma module.
This is joint work with Jae-Hoon Kwon and Il-Seung Jang.
Speaker: Xiaomeng Xu (Peking University)
Title: Crystals arising from WKB analysis
Abstract: This talk gives an introduction to the Stokes phenomenon and the WKB approximation of meromorphic linear systems of ODEs at a k-th order pole. It then realizes $gl_n$-crystals, as well as the cactus group actions, from the Stokes phenomenon in the WKB approximation at a second order pole.
Dec 08, Sunday
Speaker: Linliang Song (Tongji University)
Title: Weakly triangular decomposition and categorification
Abstract: Ariki’s categorification theorem connects the various fundamental problems in representation theory of cyclotomic Hecke algebras to important invariants of the integrable highest weight module of (affine) type A. This talk establishes an analog of Ariki’s categorification theorem related to cyclotomic (oriented)Brauer categories and Kauffman categories. These categories are categorical versions of (cyclotomic walled ) Brauer algebras, cyclotomic BMW algebras which appear naturally in Lie theory via Schur-Weyl duality. In particular, the categorification theorem for cyclotomic oriented Brauer category was conjectured by Brundan-Comes-Nash-Reynolds in 2014. The proof relies on the notion of weakly triangular decomposition, which provides a sufficient condition to show that the corresponding categories are upper finite fully stratified category in the sense of Brundan-Stroppel. This talk is based on joint works with H.Rui, and joint works with M.Gao and H.Rui.
Speaker: Haruto Murata (University of Tokyo)
Title: Affine highest weight structures on module categories over quiver Hecke algebras
Abstract: Associated with a symmetrizable Kac-Moody algebra $\mathfrak{g}$, the category of graded modules over the quiver Hecke algebra provide a categorification of the quantum group $U_q(\mathfrak{n}_-)$. For any Weyl group element $w$, there is a full subcategory $\mathscr{C}_w^{\text{f.g.}}$ corresponding to the quantum unipontent subgroup $U_q(\mathfrak{n}_- \cap w \mathfrak{n}_+)$. In this talk, we show that $\mathscr{C}_w^{\text{f.g.}}$ is an affine highest weight category by concretely realizing the standard modules using determinantial modules.
Speaker: Valentin Buciumas (POSTECH)
Title: The fundamental local equivalence via special functions
Abstract: My talk will focus on an interesting class of special functions that are symmetric with respect to some twisted action of the symmetric group (called the Chinta-Gunnells action). These functions can be used to model a graded version of the representation theory of Lusztig's quantum group at a root of unity, as well as certain Whittaker spaces on p-adic groups (the fact that the quantum and p-adic settings are equivalent is what is known as the fundamental local equivalence in geometric Langlands). Moreover, certain structure coefficients for these functions have integral positivity properties; they can be seen to be equal to the transition coefficients from LLT polynomials to Schur polynomials.
In my talk, I will introduce these functions and at the end present some combinatorial results about them that can be attained using solvable lattice models.
Speaker: Tzu-Jan Li (Academia Sinica)
Title: On integral images of Curtis homomorphisms
Abstract: For a connected reductive group G defined over a finite field, Bonnafé and Kessar have characterized the integral images of the Curtis homomorphism associated with G under the assumption that the order of the Weyl group of G is invertible in the corresponding coefficient rings. In this talk, I would like to explain a partial refinement of Bonnafé and Kessar's characterization mentioned above, with emphasis on the role of the orders of Weyl groups in this topic of integral images of Curtis homomorphisms.
