## Activities

1. YMSC Courses Mini Courses Seminars Lectures Conferences

## 几何表示论讨论班

Geometric Representation Seminar

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

Upcoming talks

Title: Category O for a hybrid quantum group

Speaker: Quan Situ

Time: Fri., 3:30-4:30 pm, Dec.30, 2022

Venue: Zoom Meeting ID: 271 534 5558 Passcode: YMSC

Abstract:

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 permitted, we will discuss a deformed version of these results.

Past talks

Title: Quasimaps to quivers with potentials

Speaker: Gufang Zhao (University of Melbourne)

Time: Fri., 10:00-11:00am, Dec.16, 2022

Venue: Zoom Meeting ID: 687 513 9542  Passcode: YMSC

Abstract:

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.

Title: Cluster Nature of Quantum Groups

Speaker: Linhui Shen (Michigan State University)

Time: Fri., 10:00-11:00am, Dec.9, 2022

Venue: Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract:

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.

Title: Microlocal sheaves and affine Springer fibers

Speaker: Pablo Boixeda Alvarez (Yale University)

Time: Fri., 9:00-10:30am, Dec.2, 2022

Venue: Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract:

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.

Title: Vertex operator algebras and conformal blocks

Speaker: Bin Gui (YMSC)

Time: Fri.,10:00-11:00am,Nov.25,2022

Venue: Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract:

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.

Title: The chromatic Lagrangian

Time: Fri.,10:00-11:00am, Nov.18,2022

Venue:(Online)Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract: 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.

Title：Enumerative Geometry and Geometric Representation Theory of some Elliptic Surfaces

Speaker：Sam DeHority (Columbia University)

Time：11/04 10am-11am

On site venue: Ning Zhai W11；Zoom Meeting ID: 276 366 7254 Passcode: YMSC

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.

Title：Stable envelopes and Bott-Samelson resolution

Speaker：Jakub Koncki (University of Warsaw)

Time：10/28 (4 pm - 5 pm)

Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract：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.

Title: Mirror symmetry for quiver stacks and machine learning

Speaker: Siu-Cheong Lau (Boston University)

Time: 10/20 (9 am -10 am)

Venue: online only (updated)

Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract: 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.

Title:Local types of equivariant G-bundles and parahoric group schemes

Speaker: Jiuzu Hong (University of North Carolina at Chapel Hill)

Time: 10/14， 9:30-10:30 am

Onsite venue: Ning Zhai, W11

Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract: 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.

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

Speaker: Gustavo Jasso (Lunds universitet)

Time： Fri., 4:00 pm - 5:00 pm, Oct.7,2022

Onsite Venue: Ning Zhai, W11.

Zoom Meeting ID: 276 366 7254 Passcode: YMSC

Abstract: 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.

Title: Revisiting Jacobi-Trudi identities via the BGG category O

Speaker：Tao Gui (AMSS)

Date and time: Sep. 30, 10：00-11：00

Venue: Ning Zhai, W11

Abstract: 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.

Title: Virtual Coulomb branch and quantum K-theory

Speaker：Zhou, Zijun (IPMU)

Date and time: Sep. 23, 9:50-11:30

Venue: Zoom Meeting ID: 271 534 5558 Passcode: YMSC

Abstract: 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.

Title: Character sheaves and Hecke algebras

Speaker: Ting Xue (University of Melbourne)
Time: Apr. 13th, 4:00-5:00 pm (Beijing time)
Venue：Ning Zhai W11 + Zoom Meeting ID: 4552601552 Passcode: YMSC

Abstract:
We discuss character sheaves in the setting of graded Lie algebras. Via a nearby cycle construction irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. Recent work of Lusztig and Yun relates Fourier transforms of character sheaves to irreducible representations of trigonometric double affine Hecke algebras.  We will explain the connection between the work of Lusztig-Yun and our work, and discuss some conjectures arising from this connection. If time permits, we will discuss applications to cohomology of Hessenberg varieties and affine Springer fibres. This is based on joint work with Kari Vilonen and partly with Tsao-Hsien Chen and Misha Grinberg.

Date: Dec. 1st, Wed, 5:00-6:00 pm

Location: online

Zoom Meeting ID: 849 963 1368

Speaker: Navid Nabijou (Cambridge)

Title: Enumerative invariants of 3-fold flops: hyperplane arrangements and wall-crossing

Abstract: 3-fold flopping contractions form a fundamental building block of the higher-dimensional Minimal Model Program. They exhibit extremely rich geometry, which has been investigated by many people over the past half-century. I will present an elegant and visually-pleasing relationship between enumerative invariants of flopping contractions and certain hyperplane arrangements constructed combinatorially from root system data. I will discuss both Gopakumar-Vafa (GV) and Gromov-Witten (GW) invariants, explaining how these are related to one another and how they are encoded in finite and infinite arrangements, respectively. Finally, I will discuss wall-crossing: our combinatorial approach allows us to explicitly construct flops from root system data, leading to a new “direct” proof of the Crepant Transformation Conjecture, with a very explicit formulation. This is joint work with Michael Wemyss.

Date: Nov. 16th, Tuesday, 10:00-11:00 am

Location: online

Zoom Meeting ID: 849 963 1368

Speaker: Justin Campbell (Chicago)

Title: Affine Harish-Chandra bimodules and Steinberg-Whittaker localization

Abstract: This talk will be about my paper of the same title with Gurbir Dhillon. It is well-known that the center of the enveloping algebra of an affine Kac-Moody algebra at noncritical level is trivial. Nonetheless, its representation theory shares many features with that of a finite-dimensional semisimple Lie algebra, including a block decomposition of category O. We propose an analogue, for any affine Weyl group orbit, of the category of Kac-Moody representations with the corresponding "generalized central character." Namely, we consider the subcategory generated by the relevant Verma modules under the categorical loop group action. We also construct equivalences relating various categories of affine Harish-Chandra bimodules, Whittaker modules, and Whittaker D-modules on the loop group, generalizing known equivalences in the finite-dimensional case proved by Bernstein-Gelfand, Beilinson-Bernstein, Milicic-Soergel, and others.

Date: Nov. 12th, Friday, 2:40-3:40 pm

Location: online

Zoom Meeting ID: 388 528 9728

Passcode: BIMSA

Speaker: Michael McBreen (CUHK)

Title: Deletion and contraction for Hausel-Proudfoot spaces

Abstract: Dolbeault hypertoric manifolds are hyperkahler integrable systems generalizing the Ooguri-Vafa space. They approximate the Hitchin fibration near a totally degenerate nodal spectral curve. On the other hand, Betti hypertoric varieties are smooth affine varieties parametrizing microlocal sheaves on the same nodal spectral curve. I will review joint work with Zsuzsanna Dansco and Vivek Shende (arXiv:1910.00979) which constructs a diffeomorphism between the Dolbeault and Betti hypertorics, and proves that it intertwines the perverse and weight filtrations on their cohomologies. I will describe our main tool : deletion-contraction sequences arising from either smoothing a node of the spectral curve or separating its branches. I will discuss some more recent developments and open questions.

Date: Nov. 5th, Friday, 4:00-5:00 pm

Location: online, also displayed in Lecture Hall, 3rd floor, Jin Chun Yuan West Building, Yau MSC

Zoom Meeting ID: 8499631368

Speaker: Jens Niklas Eberhardt (Bonn)

Title: Motivic Springer Theory

Abstract: Algebras and their representations can often be constructed geometrically in terms of convolution of cycles.

For example, the Springer correspondence describes how irreducible representations of a Weyl group can be realised in terms of a convolution action on the vector spaces of irreducible components of Springer fibers. Similar situations yield the affine Hecke algebra, quiver Hecke algebra (KLR algebra), quiver Schur algebra or Soergel bimodules.

In this spirit, we show that these algebras and their representations can be realised in terms of certain equivariant motivic sheaves called Springer motives.

On our way, we will discuss weight structures and their applications to motives as well as Koszul and Ringel duality.

This is joint work with Catharina Stroppel.

Date: Oct. 29th, Friday, 3:20-4:20 pm

Location: Ning Zhai W11, Yau MSC

Speaker: 迟敬人 Jingren Chi (Morningside Center)

Title: Local and global approach to geometry of affine Springer fibers

Abstract: In this talk, I will first review the classical works of Kazhdan-Lusztig and Bezrukavnikov on basic geometric properties of affine Springer fibers (mainly the dimension formula and equi-dimensionality properties) and explain certain generalizations to the setting of mixed-characteristic local fields. Then I will explain an alternative approach to some of these results using global geometry of the Hitchin fibration.

Date: Oct. 26th, Tuesday, 9:00-10:00 am

Zoom Meeting ID: 8499631368

Speaker: Joel Kamnitzer (Toronto)

Title: Symplectic duality and (generalized) affine Grassmannian slices

Abstract: Under the geometric Satake equivalence, slices in the affine Grassmannian give a geometric incarnation of dominant weight spaces in representations of reductive groups.  These affine Grassmannian slices are quantized by algebras known as truncated shifted Yangians.  From this perspective, we expect to categorify these weight spaces using category O for these truncated shifted Yangians.

The slices in the affine Grassmannian and truncated shifted Yangians can also be defined as special cases of the Coulomb branch construction of Braverman-Finkelberg-Nakajima.  From this perspective, we find many insights.  First, we can generalize affine Grassmannian slices to the case of non-dominant weights and arbitrary symmetric Kac-Moody Lie algebras.  Second, we establish a link with modules for KLRW algebras.  Finally, we defined a categorical g-action on the categories O, using Hamiltonian reduction.

Date: Oct. 15th, Friday,3:20-4:20 pm

Speaker: Dongyu Wu 吴冬羽 (BIMSA)

Location:  Ning Zhai W11, Yau MSC

Title: The Stable Limit DAHA and the Double Dyck Path Algebra

Abstract:The double Dyck path algebra (DDPA) is the key algebraic structure that governs the phenomena behind the shuffle and rational shuffle conjectures. The structure emerged from their considerations and computational experiments while attacking the conjecture. Nevertheless, the DDPA bears some resemblance to the structure of a type A double affine Hecke algebra (DAHA). I will explain how the DDPA emerges naturally and canonically (as a stable limit) from the family of GLn DAHAs.

Date: Oct. 8th, Friday, 10:00-11:00 am

Speaker:  李鹏程 Pengcheng Li (PKU)

Location:  Lecture Hall, 3rd floor, Jin Chun Yuan West Building, Yau MSC

Title:Categorical actions on representations of finite orthogonal groups of odd dimension

Abstract:Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of isolated representations of finite orthogonal groups of odd dimension in non-defining characteristic. We also prove that isolated Rouquier blocks at linear primes are derived equivalent to their Brauer correspondents. Using the reduction theorem of Bonnafé-Dat-Rouquier, we prove that Broué's abelian defect group conjecture is true for the groups SO_{2n+1}(q) with odd q at odd linear primes. This is a joint work with Yanjun Liu.

Date: Sep. 29th,Wednesday, 4:00-5:00 pm

Speaker: Merlin Christ (Hamburg)

Zoom Meeting ID：849 963 1368
Passcode：YMSC

Title:An introduction to perverse schobers on surfaces

Abstract:Perverse schobers are a conjectured categorification of perverse sheaves. They are expected to describe Fukaya categories "with coefficients" and may further allow the study of their categories of global sections via local-to-global principles. In the case of perverse schobers on surfaces, their theory has made much progress. The main goal of the talk is to discuss a new framework for the description of perverse schobers on surfaces, based on ribbon graphs.

Date: Sep. 24th, Friday, 2:40-3:40 pm

Location: Conference Room 3, Jin Chun Yuan West Building, Yau MSC

Speaker: 苏桃 Tao Su (Yau MSC)

Title: Dual boundary complexes of Betti moduli spaces

Abstract: The homotopy type conjecture is part of the geometric P=W conjecture in non-abelian Hodge theory. It states that the smooth Betti moduli space of complex dimension d over a general punctured Riemann surface, has dual boundary complex homotopy equivalent to a sphere of dimension d-1. In this talk, via a microlocal/contact geometric perspective, I will explain a proof of the conjecture for a class of rank n wild character varieties over the two sphere with one puncture, associated to any “Stokes Legendrian knot” defined by a n-strand positive braid.

Date: June 4, 2021, 3:20 to 4:20 PM

Zoom Meeting ID：3610386975
Speaker: Quoc Ho (IST Austria)
Title: Factorization homology and the arithmetic and topology of configuration spaces
Abstract: The last decade has witnessed many interesting interplays between homological/representation stability phenomena and questions in arithmetic statistics. In this talk, I will show how the algebro-geometric version of factorization homology provides a unifying framework for studying these phenomena in the case of configuration spaces. In particular, I will explain the relationship between various zeta values coming out of point-counts on configuration spaces and homological stability phenomena exhibited by these spaces, answering questions of Farb--Wolfson--Wood. Time permitting, I will explain how these ideas can be further developed to study representation stability for ordered configuration spaces.

Date: May 14, 2021, 3:20 PM to 4:20 PM
Speaker: Lin Xun
Title:Noncommutative Hodge conjecture
Abstract: I will propose a rational Hodge conjecture for small smooth proper dg categories. The Hodge conjecture of Per_{dg}(X) is equivalent to the rational Hodge conjecture of projective smooth variety X. The Hodge conjecture is additive to the semi-orthogonal decompositions. Using semi-orthogonal decompositions, especially from HPD,  we obtain some interesting examples. Motivated from these examples, we expect that the dual statement of Hodge conjecture for linear sections of projective dual varieties can be proved by geometric methods. In this talk, there will be more questions than theorems.

Date: May 7, 2021, 3:20 PM to 4:20 PM
Zoom Meeting ID：3610386975
Speaker: Syu Kato (Kyoto University)
Title: Categorification of DAHA and Macdonald polynomials
Abstract: We exhibit a categorification of the double affine Hecke algebra (DAHA) associated with an untwisted affine root system (except for type $G$) and its polynomial representation by using the (derived) module category of some Lie superalgebras associated to the root system. This particularly yields a categorification of symmetric Macdonald polynomials. This is a joint work with Anton Khoroshkin and Ievgen Makedonskyi.

Date: Apr 23, 2021, 3:20 PM to 4:20 PM

Place: Ning Zhai W11
Speaker:  李鹏辉 Penghui Li

Title：Eisenstein series via factorization homology of Hecke categories.

Abstract: Motivated by the spectral gluing patterns in the Betti Langlands program. We define the E_2 Hecke category as the category of coherent sheaves on moduli stacks of G-bundles on a disk with parabolic reduction on the boundary circle. We prove that its factorization homology is the (enhanced) Eisenstein series category. Our results naturally extend previous known computations of Ben-Zvi--Francis--Nadler and Beraldo. This is a joint work with Quoc Ho.

Date: Apr 16, 2021, 3:20 PM to 4:20 PM

Zoom Meeting ID：3610386975
Speaker: Tatsuki Kuwagaki

Title：Sheaf quantization and irregular singularity
Abstract：Constructible sheaves have played an important role in the development of representation theory. The topic of this talk is sheaf quantization, which is a geometric refinement of the notion of constructible sheaf (“constructible sheaf (or local system) of 21st century”). I will give an introduction to sheaf quantization and discuss how it is difficult (at present) to construct it in general and its relation to irregular singularities.

Date: Apr 1, 2021, 8:00 AM to 9:00 AM
Place: Ning Zhai W11

Zoom Meeting ID：3610386975

Title：A categorical sl_2 action on some moduli spaces of sheaves
Abstract：We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman, Yoshioka, and Nakajima. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. I'll spend most of my time on a nice example. This is joint with my student Ryan Takahashi.

Date: Mar. 24th, Wednesday, 8:00-9:00 am
Zoom Meeting ID：3610386975
Title: Hecke bubbling in Betti Geometric Langlands
Abstract: I'll discuss joint work with Zhiwei Yun that reformulates Hecke actions in terms of bubbling projective lines at marked points of curves. Our motivation is to relate automorphic categories for smooth curves and their nodal degenerations.

Date: Mar. 12th, 2021
Place: Ning Zhai W11
Speaker: 陈伟彦 CHEN Weiyan
Title： Topology and Arithmetic Statistics
Abstract： Topology studies the shape of spaces. Arithmetic statistics studies the behavior of random algebraic objects such as integers and polynomials. I will talk about a circle of ideas connecting these two seemingly unrelated areas. To illuminate the connection, I will focus on two concrete examples: (1) the Burau representation and superelliptic curves, and (2) cohomology of the space of multivariate irreducible polynomials.

Date: Mar. 5th
Speaker: Lin Chen 陈麟 (Harvard University)
Title: nearby cycles and long intertwining functor
Abstract: Let G be a reductive group and (N,N^-) be the unipotent radicals of a pair of opposite Borel subgroups (B,B^-). The well-known long intertwining functor is an equivalence from the category Shv(Fl)^N of N-equivariant sheaves on the flag variety Fl to the similar category Shv(Fl)^{N^-} for N^-. We will interpret this equivalence as a duality between the above two categories, and explain that the unit object for this duality can be obtained as nearby cycles along a Vinberg-degeneration of Fl. We also describe an affine analogue of this result and explain its relation with Bernstein’s second adjointness.

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