Upcoming talks
12/18 Dylan G Allegretti (YMSC)
Character varieties, Coulomb branches, and duality
I will describe a project with Peng Shan which aims to relate character varieties of surfaces to K-theoretic Coulomb branches in the sense of Braverman, Finkelberg, and Nakajima. Focusing on the case where the surface is a once-punctured torus, I will show how our results provide a rigorous framework for studying the dualities of certain quantum field theories.
Past talks
12/11 Wenbin Yan (YMSC)
4d mirror symmetry for class-S theories
I will explain a (conjectural) correspondence between the representation theory of a vertex operator algebra (VOA) associated to a genus g Riemann surface with n-punctures and the geometry of Hitchin moduli space / character variety of the same Riemann surface. They are coming from Higgs / Coulomb branches 0f 4d class-S theories respectively. We propose that there exists a bijection between modular invariant modules of the VOA and fixed manifolds of Hitchin moduli spaces. The modularity is also related to dimensions of fixed manifolds. Moreover, all these data can be read from the mixed Hodge polynomials of the character variety. I will explain this correspondence in detail and provide examples.
12/04 Penghui Li (YMSC)
Geometric representation of Hecke categories
Hecke categories are the geometrization/categorification of Hecke algebras and play a key role in geometric representation theory. We shall survey some recent progress regarding Hecke categories and their applications in number theory, combinatorics, Lie theory, and topology. Part of this talk is based on joint work with Quoc P. Ho, David Nadler, and Zhiwei Yun.
11/27 苏长剑 (YMSC)
Symmetric functions via motivic Chern classes
The Macdonald polynomials are very important symmetric functions, generalizing the Schur functions, Hall-Littlewood polynomials, etc. I will report a joint work in progress with B. Ion and L. Mihalcea, in which we define a symmetric function via the motivic Chern classes of the Schubert cells in the affine Grassmannian. These polynomials depends on two variables q and t, and enjoy very similar specializations as the Macdonald polynomials.
11/20 Qizheng Yin (BICMR)
D-equivalence conjecture for K3^[n]
I will explain how to use Markman’s hyperholomorphic bundles to show that birational hyper-Kähler varieties of K3^[n] type are derived equivalent. This is joint work with Davesh Maulik, Junliang Shen, and Ruxuan Zhang.
11/13 Shang Li
Wonderful compactification over an arbitrary base scheme
Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this talk, we will construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes, which parameterize classical wonderful compactifications of De Concini and Procesi as geometric fibers. Our construction is based on a variant of the Artin–Weil method of birational group laws. In particular, our construction gives a new intrinsic construction of wonderful compactifications. If time permits, we will also discuss several applications of our compactification in the study of torsors under reductive group schemes.
11/06 Semeon Arthamonov (BIMSA)
Genus two Double Affine Hecke Algebra and its Classical Limit.
Double Affine Hecke Algebras were originally introduced by I.Cherednik and used in his 1995 proof of Macdonald conjecture from algebraic combinatorics. These algebras come equipped with a large automorphism group SL(2,Z) which has geometric origin, namely it is the modular group of a torus. It was subsequently shown that spherical Double Affine Hecke Algebras realize universal flat deformations of the quantum chracter variety of a torus and their existence is closely related to the fact that classical SL(n,C)-character varieties admit symplectic resolution of singularities via the Hilbert Scheme Hilb_n(\mathbb C*\times\mathbb C*).
In 2019 G. Belamy and T. Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare-Birkhoff-Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". (joint with Sh. Shakirov)
• 10/30 David Hernandez (Université Paris Cité)
Monoidal Jantzen Filtrations and quantization of Grothendieck rings
We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal categories with generic braidings.
It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring. This is a joint work with Ryo Fujita.
• 10/23 Cédric Bonnafé (CNRS, Université Montpellier)
Calogero-Moser spaces vs unipotent representations
Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group W (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from W . Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with W (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this talk is to gather these observations, state precise conjectures and provide general facts and examples supporting these conjectures.
2024/09/25 易灵飞 Lingfei Yi (上海数学中心)
Title: Slices in the loop spaces of symmetric varieties
Abstract: Let X be a symmetric variety. J. Mars and T. Springer constructed conical transversal slices to the closure of Borel orbits on X and used them to show that the IC-complexes for the orbit closures are pointwise pure. This is an important geometric ingredient in their work providing a more geometric approach to the results of Lusztig-Vogan. In the talk, I will discuss a generalization of Mars-Springer's construction of transversal slices to the setting of the loop space LX of X where we consider closures of spherical orbits on LX. I will also explain its applications to the formality conjecture in the relative Langlands duality. If time permits, I will discuss similar constructions for Iwahori orbits. This is a joint work with Tsao-Hsien Chen.
Title: Genus 2 Macdonald functions
Speaker: Shamil Shakirov (BIMSA)
Time: 2024/09/18, 3:30 pm - 4:30 pm
Venue: Shuangqing B627
Abstract:
We present a system of 3 integrable difference equations in 3 variables. We prove that automorphisms associated to this system form a genus 2 mapping class group. We study eigenfunctions of this system and call them genus 2 Macdonald functions. Finally, we work on genus 3 and higher genus generalization of this system.
Title: Opers--what they are and what they are good for
Speaker: Peter Koroteev (UC Berkeley)
Time: 2024/09/11, 3:30 pm - 4:30 pm
Venue: Shuangqing B627
Abstract:
I will introduce the space of (q-)opers on a projective line for a simple simply-conncted Lie group G. I will explain how this space is related to previously known results in geometry, physics, and integrable systems.
Title: An introduction to Cohomological Hall algebras II
Speaker: Yaping Yang
Time: 2024/08/29, 10:00 am - 11:30 am
Venue: Shuangqing B627
Abstract:
I plan to discuss a class of representations of COHA constructed from the cohomology of the moduli spaces of perverse coherent sheaves on a toric Calabi-Yau 3-fold X. I will explain the action of COHA of Kontsevich and Soibelman on the cohomology via “raising operators”. I will also discuss the “double” of the COHA that acts on the cohomology by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system and the shift is given by an intersection pairing.
Title: An introduction to Cohomological Hall algebras I
Speaker: Gufang Zhao
Time: 2024/08/27, 10:00 am -11:30 am
Venue: Shuangqing B627
Abstract:
Cohomological Hall algebra (COHA) is introduced by Kontsevich and Soibelman as a categorification of Donaldson-Thomas-type invariants of 3- Calabi-Yau categories. In this lecture we review this construction and basic properties. We also discuss various examples of COHA.
Title: Line bundles on moduli stack of parahoric bundles
Speaker: Jiuzu Hong (University of North Carolina at Chapel Hill)
Time: 2024/07/29 11 am -12 pm
Venue: B627, Shuangqing Complex Building A
Abstract:
Line bundles on moduli spaces/stacks of G-bundles were studied intensively in 90’s by many mathematicians including Beauville, Laszlo, Sorger, Faltings, Kumar, Narasimhan, Teleman, etc. The main problems are the Verlinde formula for the dimension of global sections of these line bundles, and the determination of the Picard groups of the moduli spaces/stack of G-bundles.
In this talk, I will discuss some results on the same problems for the line bundles on the moduli stack of bundles over parahoric Bruhat-Tits group schemes over curves. These questions for parahoric bundles were first proposed by Pappas-Rapoport, and they generalize the classical story of parabolic bundles. This talk will be based on my previous work with Shrawan Kumar, and an ongoing joint work with Chiara Damiolini.
Venue: C548, Shuangqing Complex Building A
07/25 9:30 am --10:30 am, Zheng Hua (University of Hong Kong)
Title: A modular construction of Positroid varieties
Abstract: We construct a family of Poisson structures on Grassmannian $G(k, n)$ parametrised by a Calabi-Yau curve, a simple vector bundle of degree $n$ on it. When the curve is a Kodaira cycle of $n$ irreducible components and for a particular choice of the vector bundle, we recover the standard Poisson structure of Drinfeld and Jimbo. In this case, the positroid varieties are isomorphic to certain moduli spaces of coherent systems on the Kodaira cycle. This leads to several new results and new proof of known results about the symplectic geometry of the positroid varieties. When we pick different vector bundles and curves, we speculate that one might get new cluster algebra structures on Grassmannian.
07/25, 10:45 am -- 11:45 am, Che Shen (Columbia University)
Title: Affine Laumon spaces and the dual Verma module of quantum affine algebra
Abstract: Laumon spaces parametrize flags of locally-free sheaves on the projective line satisfying certain conditions. Braverman-Finkelberg showed that the localized equivariant K-theory of Laumon spaces has a natural action of the quantum group U_q(sl_n) and can be identified with the universal Verma module. I will explain a refinement of this result that identifies the (non-localized) equivariant K-theory with the dual Verma module. The above result also has an affine analog where we consider affine Laumon spaces and the action of quantum affine algebra U_q(\hat{gl_n}), refining an earlier result of Negut.
07/25, 2 pm- 3pm, Laurentiu Maxim (University of Wisconsin-Madison)
Title: A geometric perspective on generalized weighted Ehrhart theory
Abstract:Classical Ehrhart theory for a lattice polytope encodes the relation between the volume of the polytope and the number of lattice points the polytope contains. In this talk, I will discuss a geometric interpretation, via the (equivariant) Hirzebruch-Riemann-Roch formalism, of a generalized weighted Ehrhart theory depending on a homogeneous function on the polytope and with Laurent polynomial weights attached to each of its faces. In the special case when the weights correspond to Stanley's g-function of the polar polytope, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. (Based on archive preprints arXiv:2403.17747 and arXiv:2405.02900, joint work with Jörg Schürmann.)
Title: Maulik-Okounkov Lie algebras and BPS Lie algebras II
Speaker: Tommaso Maria Botta (ETH Zurich)
Time: 2024/06/14 4:10 pm - 5:30 pm
Venue: ONLINE, Zoom Meeting ID: 271 534 5558 Passcode: YMSC
Abstract:
The Maulik-Okounkov (MO) Lie algebra associated to a quiver Q controls the R-matrix formalism developed by Maulik and Okounkov in the context of (quantum) cohomology of Nakajima quiver varieties. On the other hand, the BPS Lie algebra originates from cohomological DT theory, and in particular from the theory of cohomological Hall algebras associated to 3 Calabi-Yau categories. In this talk, I will explain how to identify the MO Lie algebra of Q with the BPS Lie algebra of the tripled quiver Q̃ with its canonical cubic potential. To link these seemingly diverse words, I will review the theory of non-abelian stable envelopes and use them to relate representations of the MO Lie algebra to representations of the BPS Lie algebra. As a byproduct, I will present a proof of Okounkov's conjecture, equating the graded dimensions of the MO Lie algebra with the coefficients of Kac polynomials. This is joint work with Ben Davison.
Title: Maulik-Okounkov Lie algebras and BPS Lie algebras
Speaker: Tommaso Maria Botta (ETH Zurich)
Time: 2024/06/07 4:00 pm - 5:30 pm
Venue: ONLINE, Zoom Meeting ID: 4552601552 Passcode: YMSC
Abstract:
The Maulik-Okounkov (MO) Lie algebra associated to a quiver Q controls the R-matrix formalism developed by Maulik and Okounkov in the context of (quantum) cohomology of Nakajima quiver varieties. On the other hand, the BPS Lie algebra originates from cohomological DT theory, and in particular from the theory of cohomological Hall algebras associated to 3 Calabi-Yau categories. In this talk, I will explain how to identify the MO Lie algebra of Q with the BPS Lie algebra of the tripled quiver Q̃ with its canonical cubic potential. To link these seemingly diverse words, I will review the theory of non-abelian stable envelopes and use them to relate representations of the MO Lie algebra to representations of the BPS Lie algebra. As a byproduct, I will present a proof of Okounkov's conjecture, equating the graded dimensions of the MO Lie algebra with the coefficients of Kac polynomials. This is joint work with Ben Davison.
Title: A geometrization of Zelevinsky's derivatives
Speaker: Taiwang Deng (BIMSA)
Time: 2024/05/31 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
In the 1970s, Bernstein and Zelevinsky introduced a set of operators that act on the Grothendieck group of the category of admissible representations for $GL_n(Q_p)$. These operators play a crucial role in their classification of irreducible representations of $GL_n$. Later, Zelevinsky's derivatives, also known as Bernstein-Zelevinsky operators, found several important applications in automorphic theory. However, determining the Zelevinsky derivative of an irreducible representation is generally challenging. In this talk, we provide an interpretation of Zelevinsky's derivatives as dual to Lusztig's geometric inductions. As a byproduct, we derive a multiplicity formula for computing Zelevinsky's derivatives.
Title: Higher rank $N=1$ triplet vertex superalgebras
Speaker: Hao Li (YMSC)
Time: 2024/05/24 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
Sheaf cohomology of specific bundles on flag varieties provides an important source of representations of certain algebra objects. Borel-Weil-Bott theorem is the classical example of this idea. It is one of the starting points of geometric representation theory. In the context of vertex algebras, Feigin and Tipunin proposed a geometric method to construct a class of logarithmic vertex algebras associated with simply laced root systems of simple Lie algebras and their irreducible representations. Later, Sugimoto rigorously proved the existence of such logarithmic vertex algebras and realized their representations as global sections of some bundles on flag variety. These examples are higher-rank generalizations of triplet $W$-algebras.
Adamovic and Milas introduced $N=1$ triplet vertex operator superalgebras. We use Feigin-Tipunin's idea to construct the higher-rank $N=1$ triplet vertex operator superalgebras. These are logarithmic vertex superalgebras associated with the type $B$ root system. We will also discuss the structures of their representations. This is the joint work with Myungbo Shim and Shoma Sugimoto.
Title: Cells in modified iquantum groups of type AIII and related Schur algebras
Speaker: Weideng Cui (Shandong University)
Time: 2024/05/17 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
For an associative algebra with a given basis, Lusztig introduced the notion of left, right and two-sided cells. In this talk, we shall provide a combinatorial description of two-sided cells in modified iquantum groups of type AIII and some related Schur-type algebras with respect to the canonical bases on them.
Title: Elliptic classes via the periodic module
Speaker: Changlong Zhong (State University of New York at Albany)
Time: 2024/05/10 10:00 am-11:30 am
Venue: Shuangqing Complex Building B627
Abstract:
Equivariant elliptic cohomology of symplectic resolutions was recently studied by Okounkov and his collaborators. For example, the elliptic stable envelope is defined and it is closely related to geometric representation theory, mathematical physics and 3d mirror symmetry. For the cotangent bundle T^*G/B, it is proved that the restriction to torus fixed points of elliptic stable envelopes are related with that for the Langlands dual. In this talk, I will focus on the elliptic Demazure-Lusztig operators that generate the elliptic classes corresponding to the elliptic stable envelope. The (sheaf of) modules spanned by these classes are called the periodic module, which is obtained from a certain twist of the Poincare line bundle. Our main result shows that the elliptic Demazure-Lusztig operators can be assembled naturally to obtain a canonical isomorphism between the periodic module and that for the Langlands dual system. This is joint work with C. Lenart and G. Zhao.
Title: Quantum cluster algebras associated to weighted projective lines
Speaker: Fang Yang (Tsinghua University)
Time: 2024/04/26 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
In the first part of this talk, I will briefly introduce categorification of acyclic quantum cluster algebras by cluster categories of acyclic quivers, based on the work of Fan Qin. In the second part, I will explain how to categorify certain quantum cluster algebras using cluster categories of coherent sheaves on weighted projective lines. Concretely, we firstly define specialized quantum cluster characters of objects in the cluster category over finite fields and then show a cluster multiplication formula, which gives rise to mutation relations of quantum cluster algebras. Moreover, we can show quantum cluster characters of indecomposable rigid objects are generic and then coincide with quantum cluster variables. If time permitted, I will also introduce some applications of this categorification, such as finding good bases.
Title: A Plucker coordinate mirror for flag varieties and quantum Schubert calculus
Speaker: Mingzhi Yang (Sun Yat-sen University)
Time: 2024/04/19 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
I will give an elementary talk on a new interpretation of Rietsch's mirror for flag varieties of type A, which enables us to prove the mirror symmetry prediction that the first Chern class is mapped to the superpotential under the isomorphism between the quantum cohomology ring and the Jacobi ring. Ideas of the proof will be shown in explicit examples, using techniques mainly from Linear Algebra. This talk is based on a recent work joint with Changzheng Li, Konstanze Rietsch, and Chi Zhang, and our paper is available on arxiv: 2401.15640.
Title: Categorical action for finite classical groups and its applications
Speaker: Pengcheng Li (YMSC)
Time: 2024/03/29 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
In this talk, we will discuss the categorical action on the representation category of finite classical groups and its applications in representation theory. We construct a categorical double quantum Heisenberg action on the representation category of finite classical groups. Over a field of characteristic zero or positive characteristic, we deduce a categorical action of a Kac-Moody algebra on it. Furthermore, the categorical double quantum Heisenberg action gives rise to some new invariants. We show that those new invariants and the uniform projection can distinguish all irreducible characters of finite classical groups. We also show that the theta correspondence can explicitly determine the Kac-Moody action on the Grothendieck group of the whole category. If time permits, I will also discuss its application in some problems of modular representations of finite classical groups. This is a joint work with Peng Shan and Jiping Zhang.
Title: Principal block of quantum category O
Speaker: Quan Situ (YMSC)
Time: 2024/03/22 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
In this talk, we will introduce a quantum analogue of the BGG category O, which is the category O for the hybrid quantum group introduced by Gaitsgory. We show that the principal block of quantum category O is a version of the affine Hecke category. More precisely, the principal block is abelian equivalent to a category of coherent sheaves involving the Springer resolution and its non-commutative counterpart.
Title: Tropical geometry, quantum affine algebras, and scattering amplitudes
Speaker: Jianrong Li (University of Vienna)
Time: 2024/03/15 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
In this talk, I will talk about a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We propose a generalization of Grassmannian string integrals in physics, in which the integrand is a product indexed by prime modules of a quantum affine algebra. We give a general formula of u-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type A) and Auslander-Reiten quivers of Grassmannian cluster categories. This is joint work with Nick Early.
Title: Duality theorems for Iwahori Lie algebras and Highest weight categories
Speaker: Yevgen Makedonskyi (BIMSA)
Time: 2024/03/08 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
There are various classical duality theorems such as Schur-Weyl duality, Howe duality etc. We prove the versions of these theorems for current and related Lie algebras. I will explain how these theorems follow from the highest weight structure on the category of representations.
Title: Chiralization of Nakajima quiver varieties
Speaker: Yehao Zhou（Kavli IPMU）
Time: 2024/03/01 2:00-3:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
Chiralization is a procedure that quantizes the jet scheme of a given scheme. In the first part of this talk I will introduce chiralization of a Nakajima quiver variety, which produces a sheaf of hbar-adic vertex algebras on an extended Nakajima quiver variety, following the construction in the recent work of Arakawa-Kuwabara-Moller. I will also introduce a global version of the above construction, which assigns a vertex algebra to a quiver. The latter global version is closely related to what physicists called “boundary vertex algebra of a H-twisted 3d N=4 quiver gauge theory”. It turns out that there exists a natural global to local map, whose injectivity or surjectivity is not clear in general. In the second part of this talk I will explain an idea of the proof of injectivity for a class of quivers.
Title: Higher representation theory of gl(1|1)
Speaker: Raphaël Rouquier (UCLA)
Time: 2024/1/26 3:30-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
The notion of representations of Lie algebras on categories ("2-representations") has proven useful in representation theory. I will discuss joint work with Andrew Manion for the case of the super Lie algebra gl(1|1). A motivation is the reconstruction of Heegaard-Floer theory, a 4-dimensional topological field theory, and its extension down to dimension 1.
Title: The FLE and the W-algebra
Speaker: Gurbir Dhillon (Yale)
Time: 12/29 3:30-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
The FLE is a basic assertion in the quantum geometric Langlands program, proposed by Gaitsgory-Lurie, which provides a deformation of the geometric Satake equivalence to all Kac-Moody levels. We will report on a proof via the representation theory of the affine W-algebra, which is joint work in progress with Gaitsgory.
Title: Pure braids and group actions
Speaker: Caroline Namanya (Makerere University)
Time: 3:30 pm - 4:30 pm, Friday 12/08
Place: online / watching together in 双清 Shuangqing B627
Zoom Meeting ID: 271 534 5558 Passcode: YMSC
Abstract:
The first part of the talk will be about a new and simplified presentation of the classical pure braid group. Motivated by twist functors from Algebraic geometry, the generators are given by the squares of longest elements over connected subgraphs, and the relations are either commutators or certain length 5 palindromic relations.
In the second part of the talk, I will summarise a construction of derived autoequivalences associated to an algebraic flopping contraction. These functors are constructed naturally using bimodule cones, and these cones are locally two-sided tilting complexes. The autoequivalences combine to give an action of the fundamental group of an associated infinite hyperplane arrangement on the derived category.
https://us06web.zoom.us/j/2715345558?pwd=eXRTTExpOVg4ODFYellsNXZVVlZvQT09&omn=87944709298
Title: Good position braids, transversal slices and affine Springer fibers
Speaker: Chengze Duan (University of Maryland-College Park)
Time: 12/01 8:30 am-9:30 am
Venue: Zoom Meeting ID: 271 534 5558 Passcode: YMSC
Abstract:
Let G be a reductive group over an algebraically closed field and W be its Weyl group. Using Coxeter elements, Steinberg constructed cross-sections of the adjoint quotient of G which also yield transversal slices of regular unipotent classes. In 2012, He and Lusztig constructed transversal slices using minimal length elements in elliptic conjugacy classes in W, which yield transversal slices of basic unipotent classes. In this talk, we generalize minimal length elements to good position braids in the associated braid monoid of W and use these elements to construct transversal slices of all unipotent classes in G. We shall see these new elements also appear in many other aspects of representation theory, such as affine Springer fibers and the partial order on unipotent classes, etc.
Title: Fusion product and Global modules
Speaker: Huanhuan Yu (BICMR)
Time: 11/24 3:30 pm-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
Let g be a finite dimensional simple Lie algebra. In this talk, I will start with the definition of fusion product of g[t]-modules, and then introduce several approaches to study it, such as the global modules and Borel-Weil type theorem. In the meantime, I will talk about some partial results on the fusion product and applications including my work on twisted global Demazure modules joint with Jiuzu Hong.
Title: Non-integral Kazhdan-Lusztig algorithm and an application to Whittaker modules
Speaker: Qixian Zhao (YMSC)
Time: 11/17 3:30 pm-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
Let g be a complex semisimple finite dimensional Lie algebra, and consider a category of representations of g where a Kazhdan-Lusztig algorithm exists for integral regular infinitesimal characters. In this talk, we will discuss a potential approach for extending the integral algorithm to arbitrary non-integral regular infinitesimal characters, using intertwining functors. We will then apply this approach to Whittaker modules and demonstrate the non-integral algorithm there using an explicit example.
Title: Quantum difference equations for affine type A quiver varieties
Speaker: Tianqing Zhu (YMSC)
Time: 11/10 3:30 pm-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. Its construction relies on the K-theoretic stable envelope and it is conjectured that the construction is related to the quantum affine algebra of the corresponding quiver type.
In this talk, I will focus on the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ and give an analog of the qde. We explicitly construct the qde and give the explicit formula for the case of instanton moduli space and Hilbert scheme of A_n-singularities. We also discuss its connection to the Dubrovin connection of the quantum cohomology with the example of the instanton moduli space.
Title: On the Feigin-Tipunin's construction
Speaker: Shoma Sugimoto (YMSC)
Time: 10/27 3:30 pm-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
The induced G-module of a VOA with a B-action again has a VOA structure (Feigin-Tipunin's construction). In the original preprint of FT and my papers, we used this technique to construct the (1,p)-VOA from a lattice VOA with a good B-action and proved some basic results. As a matter of fact, it is the good B-module structure, not the VOA structure, that plays an essential role in the proof of these results. Therefore, It is expected that we can construct and study interesting VOAs from other VOAs with similar B-actions using the FT construction. In this talk, I will outline the FT construction and my results, and then discuss future works relating the FT construction to plumbed 3-manifolds.
Title: Cylindrical structures for Drinfeld-Jimbo quantum groups and the origin of trigonometric K-matrices
Speaker: Bart Vlaar (BIMSA)
Time: 10/20 3:30 pm-4:30 pm
Venue: Shuangqing Complex Building B627
Abstract:
The study of spectral R-matrices, matrix solutions of the (parameter-dependent) Yang-Baxter equation, was a major motivation for the discovery of quantum groups. The quasitriangular structure of these bialgebras is the origin of large classes of spectral R-matrices. The Yang-Baxter equation has a type-B/cylindrical counterpart: the reflection equation. Its matrix solutions, spectral K-matrices, have been studied since the 1980s. Do they have a similar origin?
To answer this, in joint works with Andrea Appel we develop a general framework, in terms of braided tensor categories with additional structures. Concretely, take any Letzter-Kolb quantum symmetric pair: a Drinfeld-Jimbo quantum group together with a suitable coideal subalgebra (also known as i-quantum group). Building on works by Bao & Wang and Balagovic & Kolb, we show that a twisted intertwiner of the subalgebra satisfies a twisted reflection equation, acts on (integrable) category O modules, and endows this braided tensor category with a twisted cylinder braiding. In the case of affine quantum groups one can then indeed answer the above question, explaining large classes of so-called trigonometric K-matrices.
Title: Generic Hecke algebra modules in theta correspondence over finite fields
Speaker: Jiajun Ma (Xiamen University)
Time: 09/04, 10:00 am--11:30 am
Venue: Jingzhai 静斋 304
Abstract:
In this talk, we consider the theta correspondence of type I dual pairs over a finite. Aubert, Michel, and Rouquier established an explicit formula for theta correspondence between unipotent representations of unitary groups and made a conjecture for the symplectic group-even orthogonal group dual pair. Shu-Yen Pan recently proved the conjecture. These works are based on Srinivasan's formula for the uniform projection of the Weil representation.
Joint with Congling Qiu and Jialang Zou, we found an alternative approach to solve the problem by analyzing the relevant Hecke algebra bimodules. Joint with Zhiwei Yun, we geometrized the whole picture. Consequently, we obtained a relation between the Springer correspondence and theta correspondence.
Title：Quantum symmetric pairs and $\imath$quantum groups
Speaker: Weinan Zhang (University of Hong Kong)
Time: Aug30, Aug31, and Sep 1st, 10:00 am - 11:30 am
Venue: Jingzhai 静斋 304（updated）
Abstract:
The classical theory of symmetric pairs concerns Lie algebras with involutions. The theory of quantum symmetric pairs, systematically developed by Letzter around 2000, is a quantization of classical symmetric pairs. The $\imath$quantum groups, arising from quantum symmetric pairs, are certain coideal subalgebra of quantum groups. The $\imath$quantum groups can be viewed as a natural generalization of quantum groups, and many results for quantum groups have been generalized to $\imath$quantum groups.
In the first lecture, we will review some basics of symmetric pairs and introduce quantum symmetric pairs and $\imath$quantum groups. We will construct the quasi K-matrix associated to a quantum symmetric pair, which will be a key ingredient in the later construction of relative braid group symmetries.
In the second lecture, we will construct the relative braid group symmetries (associated to the relative Weyl group of the underlying symmetric pair) on $\imath$quantum groups and on their modules. These symmetries generalize Lusztig's braid group symmetries on quantum groups.
In the third lecture, we will construct the Drinfeld type presentation for (quasi-split) affine $\imath$quantum groups, which generalizes the Drinfeld (loop) presentation for affine quantum groups. The relative braid group symmetries will play an important role in this construction. The Drinfeld type presentation for affine $\imath$quantum groups will lead to a Drinfeld type presentation for twisted Yangians.
Title: An irregular Deligne-Simpson problem and Cherednik algebras
Speaker: Zhiwei Yun (MIT)
Time: 2023/8/4, 4:00-5:00 pm
Venue: 1st conference room, West Jin Chun Yuan Building
Abstract:
The Deligne-Simpson problem asks for a criterion of the existence of connections on an algebraic curve with prescribed singularities at punctures. We give a solution to a generalization of this problem to G-connections on P^1 with a regular singularity and an irregular singularity (satisfying a condition called isoclinic). Here G can be any complex reductive group. Perhaps surprisingly, the solution can be expressed in terms of rational Cherednik algebras. This is joint work with Konstantin Jakob.
Title: Eisenstein Series for P^1 with three points
Speaker: Tahsin Saffat (UC Berkeley)
Time: 2023/8/1, 2:00-3:00 pm
Venue: 1st conference room, West Jin Chun Yuan Building
Abstract:
I’ll explain a conjecture about the space of Eisenstein series for the function field of P^1 with three points of tame ramification as a trimodule for the Hecke algebra. It can be thought of as a generalization of the fact that the space of automorphic functions for P^1 with two points is the regular bimodule for the Hecke algebra.
Title: On the Gaiotto conjecture II: sketch of proof
Speaker: Ruotao Yang (Skoltech)
Time: Fri., 10:30-11:30 am, July 7, 2023
Venue: Ning Zhai W11
Abstract:
In this talk, our goal is to introduce the tools used in the proof and sketch the proof of the Gaiotto conjecture for GL(M|N).
Title: On the Gaiotto conjecture I: statement
Speaker: Ruotao Yang (Skoltech)
Time: Wed., 10:30-11:30 am, July 5, 2023
Venue: Ning Zhai W11
Abstract:
A conjecture of Davide Gaiotto predicts a geometrization of the category of representations of the quantum supergroup. This geometrization is given by the category of twisted D-modules on the affine Grassmannian with a certain equivariant condition. In this talk, we will introduce the precise statement of the Gaiotto conjecture and the recent progress. This is based on the joint works with R. Travkin, and also the works of A. Braverman, M. Finkelberg, V. Ginzburg and R.Travkin.
Title: KLR algebras and their representations
Speaker: Peter McNamara (University of Melbourne)
Time:Tues, 16:00-17:00, June 20, 2023
Venue: Conference Room 3, Jin Chun Yuan West Building
Abstract:
KLR algebras are a family of algebras introduced approximately fifteen years ago as part of the programme of categorifying quantum groups. We will discuss various representation-theoretic properties of these algebras. Along the way, we see an interesting mix of geometry, combinatorics and homological algebra.
Title: Brane and DAHA Representations
Speaker: Du Pei (University of Southern Denmark)
Time: Fri., 10:30 am- 11:30 am, June 9, 2023
Venue: Ning Zhai W11
Abstract:
In this talk, we will use "brane quantization" to study the representation theory of Double affine Hecke algebras.
Title: Cylindrical structures for Drinfeld-Jimbo quantum groups and the origin of trigonometric K-matrices
Speaker: Bart Vlaar (BIMSA)
Time: Fri., 10:30 am- 11:30 am, June 2, 2023
Venue: Ning Zhai W11
Abstract:
The study of R-matrices, matrix solutions of the spectral (parameter-dependent) Yang-Baxter equation, was a major motivation for the discovery of quantum groups. The quasitriangular structure of these bialgebras is the origin of large classes of R-matrices. The Yang-Baxter equation has a "twisted type-B/cylindrical" counterpart: the reflection equation. Its matrix solutions, known as K-matrices, have been studied since the 1980s. Is there an analogous origin for these solutions?
To answer this, in joint works with Andrea Appel we develop a general framework, in terms of braided tensor categories with additional structures. Concretely, take any Letzter-Kolb quantum symmetric pair: a Drinfeld-Jimbo quantum group (quantized enveloping algebra of a Kac-Moody algebra) together with a suitable subalgebra (also known as i-quantum group). Further to works by Bao & Wang and Balagovic & Kolb, a twisted intertwiner of the subalgebra satisfies a reflection equation, acts on (integrable) category O modules and endows this braided tensor category with a twisted cylinder braiding. For affine quantum groups one can develop the parallel with R-matrices much further and account for large classes of so-called trigonometric K-matrices.
Title: Lusztig's perverse sheaves for quivers and integrable highest weight modules
Speaker: Yixin Lan (Tsinghua University)
Time: Fri., 10:30 am- 11:30 am, May 26, 2023
Venue: Ning Zhai W11
Abstract:
Lusztig has introduced semisimple perverse sheaves for quivers and the induction and restriction functors to categorify the positive part of the quantum groups and provoided the existence of the canonical basis. Even though one can use an algebraic construction to obtain the canonical basis of irreducible integrable highest weight modules, how to realize the integrable highest weight modules and their canonical bases via Lusztig’s sheaves is still an important problem. We generalize Lusztig’s theory to N-framed quivers and define certain localizations of Lusztig’s perverse sheaves to realize (tensor products of) irreducible integrable highest weight modules. As a byproduct, we give a proof of the Yang-Baxter equation by using the coassociativity of Lusztig’s restriction functor. This is a joint work with Jiepeng Fang and Jie Xiao.
Title: Microlocal Sheaves on Affine Slodowy Slices
Speaker: Michael McBreen (Chinese University of Hong Kong)
Time: Fri., 10:30 am- 11:30 am, May 19, 2023
Venue: Ning Zhai W11
Abstract:
I will describe certain moduli of wild Higgs bundles on the line, and explain why they are affine analogues of Slodowy slices. I will then describe an equivalence between microlocal sheaves on a particular such space and a block of representations of the small quantum group. Joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Zhiwei Yun.
Title: Linkage and translation for tensor products
Speaker: Jonathan Gruber (National University of Singapore)
Time: Fri., 10:30 am- 11:30 am, May 12, 2023
Venue: Ning Zhai W11
Abstract:
Let G be a simple algebraic group over an algebraically closed field of characteristic p>0. The decomposition into blocks of the category of finite-dimensional rational G-modules is described by two classical results of H.H. Andersen and J.C. Jantzen: The linkage principle and the translation principle. We will start by recalling these results and explaining why they are a-priori not well suited for studying tensor products of G-modules. Then we introduce a tensor ideal of 'singular G-modules' and give a linkage principle and a translation principle for tensor products in the corresponding quotient category. This also gives rise to a decomposition of the quotient category as an external tensor product of its principal block with the Verlinde category of G.
Title: Modularity for W-algebras and affine Springer fibers
Speaker: Wenbin Yan (YMSC)
Time: Fri., 10:30 am- 11:30 am, Apr. 28, 2023
Venue: Ning Zhai W11
Abstract:
We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibers. Furthermore, we can also match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras. We will also explain how to extend these relations to representations of W-algebras. This is based on joint work with Peng Shan and Dan Xie.
Title: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
Speaker: Yalong Cao (RIKEN)
Time: Fri., 10:30 am- 11:30 am, Apr. 21, 2023
Venue: Ning Zhai W11
Abstract:
I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.
Title: Functions on commuting stack via Langlands duality
Speaker: Penghui Li (YMSC)
Time: Fri., 10:30 am- 11:30 am, Apr. 14, 2023
Venue: Ning Zhai W11
Abstract:
We explain how to calculate the dg algebra of global functions on commuting stacks using tools from Betti Geometric Langlands. Our main technical results include: a semi-orthogonal decomposition of the cocenter of the affine Hecke category; and the calculation of endomorphisms of a Whittaker sheaf in a diagram organizing parabolic induction of character sheaves. This is a joint work with David Nadler and Zhiwei Yun.
Title: Crystals from the Stokes phenomenon
Speaker: Xiaomeng Xu (BICMR)
Time: Fri., 3:30 pm-4:30 pm, Apr. 7, 2023
Venue: Ning Zhai W11
Abstract:
This talk first gives an introduction to the Stokes phenomenon of meromorphic linear system of ordinary differential equations. It then explains how crystal bases in the representation theory naturally arise from the Stokes phenomenon.
Title: Infinitesimal categorical Torelli
Speaker: Xun Lin (YMSC)
Time: Fri.,10:30am-11:30am, Mar. 31, 2023
Venue: Ning Zhai W11
Abstract:
Motivated from the categorical Torelli theorems, we introduce two types of infinitesimal categorical problems, connecting infinitesimal Torelli problems with a commutative diagram. Our constructions are general, and the main examples in this talk are nontrivial components of derived categories of Fano 3-folds. The infinitesimal categorical Torelli theorems for Fano 3-folds are summarized. I will talk about the unknown cases, and explain how the infinitesimal categorical Torelli theorems apply to Kuznetsov Fano 3-folds conjecture, and the categorical Torelli problems for hypersurfaces. This is based on joint works with J. Augustinas, Zhiyu Liu, and Shizhuo Zhang.
Title: Towards a geometric proof of the Donkin's tensor product conjecture
Speaker: Yixuan Li (University of California, Berkeley)
Time: Fri.,10:30am-11:30am, Mar. 24, 2023
Venue: Ning Zhai W11; Zoom Meeting ID: 271 534 5558 Passcode: YMSC
Abstract:
In the modular (char p) representation theory of algebraic reductive groups, the Frobenius twist is a great self-symmetry of the category of representations. Geometrically this self-symmetry is related to the embedding of the affine grassmannian, which is the based loop space of the reductive group, into itself as based loops that repeat themselves for p times. I'll explain an interpretation of the Donkin's tensor product conjecture as a consequence of this geometry and point out some potential ways to turn this into a proof. I'll also explain how to prove the quantum group version of the Donkin's tensor product theorem using this geometry.
Title：Derived blow ups and birational geometry of nested quiver varieties
Speaker：Yu Zhao (Kavli IPMU)
Time: Fri.,10:30am-11:30am, Mar. 17, 2023
Venue: Ning Zhai W11
Abstract:
Given a quiver, Nakajima introduced the quiver variety and the Hecke correspondence, which is a closed subvariety of Cartesian products of quiver varieties. We introduce two nested quiver varieties which are the fiber products of Hecke correspondences along the projection morphisms. We prove that, after blowing up the diagonal, they are isomorphic to a smooth variety which Negut observed for the Jordan quiver. We also prove that the blow up has an derived enhancement in the sense of Hekking.
Title: Quantum affine algebras and KLR algebras
Speaker: Jianrong Li (University of Vienna)
Time: Fri.,10:30am-11:30am, Mar. 10, 2023
Venue: Ning Zhai W11
Abstract:
Recently, Baumann-Kamnitzer-Knutson introduced a remarkable algebra morphism: \bar{D} from C[N] to the field of rational functions C(a_1, ..., a_n), where N is the unipotent radical of a simply laced complex algebraic group and a_i are simple roots, in their proof of a conjecture of Muthiah about MV basis of C[N]. The algebra C[N] and a larger algebra K_0(C^{\xi}) have monoidal categorifications using representations of quantum affine algebras introduced by Hernandez and Leclerc. We defined an algebra morphism \tilde{D} from K_0(C^{\xi}) to C(a_1, ..., a_n) and proved that when restricts to C[N], \tilde{D} coincides with \bar{D}. Moreover, using \tilde{D} and \bar{D}, we can recover information of q-characters of representations of quantum affine algebras from ungraded characters of modules of KLR algebras and vice versa. This is joint work with Elie Casbi.
Title: Quantum affine algebras and Grassmannians
Speaker: Jianrong Li (University of Vienna)
Time: Fri.,10:30am-11:30am, Mar. 3, 2023
Venue: Ning Zhai W11
Abstract:
In this talk, I will talk about the joint work with Wen Chang, Bing Duan, and Chris Fraser on quantum affine algebras of type A and Grassmannian cluster algebras.
Let g=sl_k and U_q(^g) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism Phi from the Grothendieck ring R_l^g of a certain subcategory C_l^g of finite dimensional U_q(^g)-modules to a quotient C[Gr(k,n, \sim)] of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux. Using the isomorphism, we defined ch(T) in C[Gr(k,n, \sim)] for every rectangular semistandard tableau T.
Using the isomorphism and the results of Kang, Kashiwara, Kim, Oh, and Park and the results of Qin, we proved that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form ch(T) for some real (resp. prime real) rectangular semi-standard Young tableau T.
We translated a formula of Arakawa–Suzuki and Lapid-Minguez to the setting of q-characters and obtained an explicit q-character formula for a finite dimensional U_q(^sl_k)-module. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables. We also give a mutation rule for Grassmannian cluster algebras using semi-standard Young tableaux.
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.
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
Speaker: Gus Schrader (Northwestern)
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
Password: YMSC
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
Password: YMSC
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
Password: YMSC
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
Password: YMSC
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
Password：BIMSA
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
Password：BIMSA
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
Password：BIMSA
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
Password：BIMSA
Speaker: Nicolas Addington
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
Password：BIMSA
Speaker: David Nadler (UC Berkeley)
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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webcal://p44-caldav.icloud.com/published/2/MTAyNjc3ODU1NDUxMDI2N6u0UZbdquLXYBj0BHqibhq2tnAyUQL92ZssPrsxYupRMNYLbinA3gkvQW2gWiMfGim4dkQcHMpsiPnhEkftcUc
