Organizers: Will Donovan, Penghui Li, Peng Shan, Changjian Su

Time: Friday 10:00 am-11:00 am, Beijing time

Place: Ning Zhai 宁斋 W11, Yau Mathematical Sciences Center, Tsinghua University

official seminar webpage, which contains the zoom information for online talks

some related minicourses in YMSC this semester:

Prof. Liang Xiao's minicourse, Cycles on special fiber of Shimura varieties and arithmetic applications

Prof. Cédric Bonnafé's minicourse, Introduction to Deligne-Lusztig theory

• 9/23 Zijun Zhou (Kavli IPMU)

Virtual Coulomb branch and quantum K-theory

In this talk, I will introduce a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of quasimaps into the holomorphic symplectic quotient T^*N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under variation of GIT.

• 9/30 Tao Gui (AMSS, CAS)

Revisiting Jacobi-Trudi identities via the BGG category O

The talk aims to introduce two problems I am thinking about. I will first give a new proof (joint with Arthur L. B. Yang) of the (generalized) Jacobi-Trudi identity via the BGG category O of sl_n(C). Then the talk will be devoted to the Stanley-Stembridge conjecture about the chromatic symmetric function, which can be reformulated using the immanants of the Jacobi-Krudi matrices. Finally, I will talk about Haiman's conjecture on the evaluation of virtual characters of Hecke algebra of the symmetric group on the Kazhdan-Lusztig basis, which implies the Stanley-Stembridge conjecture.

• 10/07 (4 pm - 5 pm) Gustavo Jasso (Lunds universitet)

The Donovan-Wemyss Conjecture via the Derived Auslander-Iyama Correspondence

The Donovan-Wemyss Conjecture predicts that the isomorphism type of an isolated compound Du Val singularity R that admits a crepant resolution is completely determined by the derived-equivalence class of any of its contraction algebras. Crucial results of August and Hua-Keller reduced the conjecture to the question of whether the singularity category of R admits a unique DG enhancement. I will explain, based on an observation by Bernhard Keller, how the conjecture follows from a recent theorem of Fernando Muro and myself that we call the Derived Auslander-Iyama Correspondence.

• 10/14 (9:30 am - 10:30 am)Jiuzu Hong (University of North Carolina at Chapel Hill)

Local types of equivariant G-bundles and parahoric group schemes

Parahoric Bruhat-Tits group schemes over an algebraic curve X is a smooth group scheme over X, which is generically reductive, and parahoric at ramified points. This notion was introduced by Pappas-Rapoport and formalized by Heinloth. There is increasing research going on related to the moduli stack of parahoric bundles, conformal blocks and global Schubert varieties of parahoric group schemes.

When the parahoric BT group scheme is generically split, the structure theory is established by Balaji-Seshadri. For general case, this is a recent result of Damiolini and myself, also Pappas-Rapoport (in a different approach). In our approach, this global result replies on a local counterpart. The basic idea is that any parahoric group scheme over X arises from a equivariant G-bundle on a cover of X. This requires a study of local types of equivariant G-bundles. When G is of adjoint type, it turns out that the local types is closely related to Kac’s classification of finite order automorphisms on simple Lie algebras.

• 10/20 (9 am -10 am) Siu-Cheong Lau (Boston University)

Mirror symmetry for quiver stacks and machine learning

Quiver representation emerges from Lie theory and mathematical physics. Its simplicity and beautiful theory have attracted a lot of mathematicians and physicists. In this talk, I will explain localizations of a quiver algebra, and the relations with SYZ and noncommutative mirror symmetry. I will also explore the applications of quivers to computational models in machine learning.

• 10/28 (4 pm - 5 pm) Jakub Koncki (University of Warsaw)

Stable envelopes and Bott-Samelson resolution

Schubert varieties have a well-studied resolution of singularities called Bott-Samelson resolution. We study certain characteristic classes of Schubert varieties in the equivariant K-theory of a flag variety. In particular we show that the stable envelope can be constructed using Bott-Samelson resolution. As a consequence we generalize inductive formulas computing stable envelopes.

• 11/04 Sam DeHority (Columbia University)

Enumerative Geometry and Geometric Representation Theory of some Elliptic Surfaces

I will discuss new algebraic structures associated to moduli of sheaves on elliptic surfaces, and describe their relation with other parts of mathematical physics. These algebraic structures control the enumerative geometry of these moduli spaces analogous to how quantum groups control enumerative invariants of quiver varieties. The main results discussed will include a description of the quantum differential equation in these geometries and work in progress describing the relevant algebras as Hopf algebras with generalized versions of R matrices.

• 11/11 (9:30 am-10:30 am) Ben Webster (University of Waterloo)

colloquium talk

• 11/18 Gus Schrader (Northwestern University)

The chromatic Lagrangian

The chromatic Lagrangian is a Lagrangian subvariety inside a symplectic leaf of the cluster Poisson moduli space of Borel-decorated PGL(2) local systems on a punctured surface. I will describe the cluster quantization of the chromatic Lagrangian, and explain how it canonically determines wavefunctions associated to a certain class of Lagrangian 3-manifolds L in Kahler \mathbb{C}^3, equipped with some additional framing data. These wavefunctions are formal power series, which we conjecture encode the all-genus open Gromov-Witten invariants of L. Based on joint work with Linhui Shen and Eric Zaslow.

• 11/25 Bin Gui (YMSC)

Vertex operator algebras and conformal blocks

Conformal blocks are central objects in the study of 2d conformal field theory and vertex operator algebras (VOAs). Indeed, many important problems in VOAs are related to conformal blocks, including modular invariance of VOA characters (the earliest such type of problem is the famous monstrous moonshine conjecture), the construction of tensor categories for VOA representations, the study of the relationship between VOAs and low dimensional topology, and so on. I will give a brief review of the development of VOA conformal block theory and some recent progress.

• 12/02 （9am-10:30 am) Pablo Boixeda Alvarez (Yale University)

Microlocal sheaves and affine Springer fibers

The resolutions of Slodowy slices $\widetilde{\mathcal{S}}_e$ are symplectic varieties that contain the Springer fiber $(G/B)_e$ as a Lagrangian subvariety.In joint work with R. Bezrukavnikov, M. McBreen and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element $ts$ for $s$ a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

• 12/09 Linhui Shen (Michigan State University)

Cluster Nature of Quantum Groups

We present a rigid cluster model to realize the quantum group $U_q(g)$ for $g$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group to a quotient algebra of the Weyl group invariants of a Fock-Goncharov quantum cluster algebra. By applying the quantum duality of cluster algebras, we show that the quantum group admits a cluster canonical basis $\Theta$ whose structural coefficients are in $\mathbb{N}[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$. The basis $\Theta$ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution. Based on a recent preprint arXiv: 2209.06258.

• 12/16 Gufang Zhao (University of Melbourne)

Quasimaps to quivers with potentials

This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.

• 12/30 (3:30 pm-4:30 pm) Quan Situ (YMSC, Tsinghua University)

Category O for a hybrid quantum group

In this talk, we introduce a hybrid quantum group at root of unity. We consider its category O and discuss some basic properties including linkage principle and BGG reciprocity. Then we show that there is an isomorphism between the center of a block (of arbitrary singular type) of the category O with the cohomology ring of the partial affine flag variety (of the corresponding parahoric type). A key ingredient is an abelian equivalence between the Steinberg block of O and the category of equivariant coherent sheaves on the Springer resolution. If time permits, we will discuss a deformed version of these results.