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

Time: 2:00 pm -- 3:30 pm, Friday

Place: Lecture Hall B627, Shuangqing Complex Building（双清综合楼B627报告厅), Yau Mathematical Sciences Center,  Tsinghua University

双清综合楼地址：北京市海淀区逸清南路西延6号院1号，双清公寓马路对面、清华附小（双清校区）西侧。

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

• 03/01 Yehao Zhou (Kavli IPMU)

Chiralization of Nakajima quiver varieties

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.

• 03/08 Yevgen Makedonskyi (BIMSA)

Duality theorems for Iwahori Lie algebras and Highest weight categories

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.

• 03/15 Jianrong Li (University of Vienna)

Tropical geometry, quantum affine algebras, and scattering amplitudes 

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.

• 03/22 Quan Situ (YMSC)

Principal block of quantum category O

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.

• 03/29 Pengcheng Li (YMSC)

Categorical action for finite classical groups and its applications

 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.

• 04/12 (2024 Algebra Geometry and Supersymmetry)

• 04/19 Mingzhi Yang (Sun Yat-sen University)

A Plucker coordinate mirror for flag varieties and quantum Schubert calculus 

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.

• 04/26 Fang Yang (Tsinghua University)

Quantum cluster algebras associated to weighted projective lines

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.

• 05/08 2:00 pm -3:00 pm, C548, Shuangqing Complex Building

Rui Xiong (University of Ottawa)

Motivic Lefschetz Theorem for twisted Milnor Hypersurfaces

In this talk, I will discuss the motivic decomposition of a smooth hyperplane section in twisted Milnor Hypersurfaces. The key feature of our result is the appearance of a spectrum of a particular field in the decomposition. A critical ingredient is in the non-triviality of the (monodromy) Galois action on the equivariant Chow group. The steps of our proof can be likened to several theorems in Hodge theory of complex algebraic geometry. This is a joint work with Kirill Zainoulline.

• 05/10 10:00 am-11:30 am, Changlong Zhong (State University of New York at Albany)

Elliptic classes via the periodic module 

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.

• 05/17 Weideng Cui (Shandong University)

Cells in modified iquantum groups of type AIII and related Schur algebras

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.

• 05/24 Hao Li (YMSC)

Higher rank $N=1$ triplet vertex superalgebras

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.  

• 05/31 Taiwang Deng (BIMSA)

A geometrization of Zelevinsky's derivatives

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.

• 06/07 4:00 pm - 5:30 pm, ONLINE

Tommaso Maria Botta (ETH Zurich)

Maulik-Okounkov Lie algebras and BPS Lie algebras I

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.

• 06/14 4:10 pm - 5:30 pm, ONLINE

Tommaso Maria Botta (ETH Zurich)

Maulik-Okounkov Lie algebras and BPS Lie algebras II

• 07/25 9:30 am --10:30 am, Zheng Hua (University of Hong Kong)

A modular construction of Positroid varieties 

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)

Affine Laumon spaces and the dual Verma module of quantum affine algebra

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)

A geometric perspective on generalized weighted Ehrhart theory

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.)

• 07/29, 11 am -12 pm, Jiuzu Hong (University of North Carolina at Chapel Hill)

Line bundles on moduli stack of parahoric bundles

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.

• 08/27, 10 am -11:30 am, Gufang Zhao (University of Melbourne)

An introduction to Cohomological Hall algebras I

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.

• 08/29, 10 am - 11:30 am, Yaping Yang (University of Melbourne)

An introduction to Cohomological Hall algebras II

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.

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