Geometric Representation Seminar
Student No.:20
Time:Thu 09:30-10:30
Instructor:Shan Peng  
Place:Conference Room 2, Jin Chun Yuan West Bldg.
Starting Date:2018-3-9
Ending Date:2018-6-28



Speaker:刘正伟 Liu Zhengwei [Havard University]

Title: Antisymmetric characters and Fourier duality

Abstract: Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce antisymmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notations and prove such a correspondence for a more general case. This answers an old question posed by Victor Kac and a recent comment by Terry Gannon.


Speaker:范祐维 Yu-Wei Fan [Havard University]

Title: Systoles, Special Lagrangians, and Bridgeland stability conditions

Abstract: Loewner's torus systolic inequality states that the least length of a non-contractible loop in a two-torus can not be too large compare to its volume. We attempt to generalize the inequality from the viewpoint of Calabi-Yau geometry. This naturally leads to the notions of categorical systole and systolic ratio of a Bridgeland stability condition. We then study the systolic problem in the case of K3 surfaces. Relevant backgrounds on mirror symmetry and Bridgeland stability conditions will be provided.



Speaker: 李毅强 Li Yiqiang [University at Buffalo]

Title: Quiver varieties and symmetric pairs

Abstract: To an ADE Dynkin diagram, one can attach a simply-laced complex simple Lie algebra, say g, and a class of Nakajima’s quiver varieties. The latter provides a natural home for a geometric representation theory of the former. If the algebra g is further equipped with an involution, it leads to a so-called symmetric pair (g,k), where k is the fixed-point subalgebra under involution. In this talk, I’ll present bridges at several levels between symmetric pairs and Nakajima varieties, aiming towards a geometric theory of real simple groups.


Speaker: 翁达平Weng Daping [Yale University]

Title: Cluster Duality of Grassmannian and Cyclic Sieving Phenomenon of Plane Partitions

Abstract: Fix two positive integers $a$ and $b$. Scott showed that a homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to integer multiples of the fundamental weight $w_a$. By proving the Fock-Goncharov cluster duality conjecture for the Grassmannian using a sufficient condition found by Gross, Hacking, Keel, and Kontsevich, we obtain bases parametrized by plane partitions for these irreducible representations. As an application, we use these bases to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen.


Speaker: Andrea Appel [University of Edinburgh]

Title: Quantum groups and monodromy

Abstract: The monodromy of linear differential equations can be thought of as an analytic map generalizing the exponential map of a Lie group. Its computation may be rather cumbersome, but, in the case of certain very special equations arising in representation theory and mathematical physics it is possible to obtain an algebraic and seemingly more explicit description in terms of quantum groups. In the first part of this talk, I will describe several examples involving both differential and difference equations, while in the second part I will mainly focus on the monodromy of the rational Casimir connection, following joint works with V. Toledano Laredo.


Speaker: Michael McBreen [University of Toronto]

Title: Quantization and Quantum Cohomology for Symplectic Dual Spaces

Abstract: I will discuss joint work with Joel Kamnitzer and Nick Proudfoot on a relation between the quantization of a symplectic resolution and the quantum cohomology of the symplectic dual resolution. No prior experience of symplectic resolutions and quantum cohomology will be assumed.


Speaker: Eric Vasserot [Université Paris Diderot]

Title: Cohomological Hall algebras of quivers, curves and surfaces

Abstract: Cohomological Hall algebras are a new class of algebras which are closely related to quantum affine algebras and are attached to several abelian categories such as representations of preprojective algebras or coherent sheaves on curves and surfaces. We'll review a few basic facts concerning them and explain some open problems, in particular in the case of curves and surfaces.




Speaker: 罗栗 Luo Li [华东师范大学]

Title: The q-Schur algebras and q-Schur dualities of finite type


Abstract: We formulate a q-Schur algebra associated to an arbitrary W-invariant finite set X_f of integral weights for a complex simple Lie algebra with Weyl group W. We establish a q-Schur duality between the q-Schur algebra and Hecke algebra associated to W. We then realize geometrically the q-Schur algebra and duality, and construct a canonical basis for the q-Schur algebra with positivity. With suitable choices of X_f in classical types, we recover the q-Schur algebras in the literature. Furthermore, we will establish its connection to the BGG category O. 

This is joint work with Weiqiang Wang.





Speaker: Will Donovan [Kavli IPMU]

Title: Perverse sheaves of categories and birational geometry

Abstract: Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).





Speaker: Dylan Allegretti (Yale University)

Title: Quantum cluster varieties and quantization of canonical bases


Abstract: Distinguished bases for the coordinate rings of various algebraic spaces have been the subject of intense research in representation theory since the pioneering work of Lusztig. In a famous paper from 2003, Fock and Goncharov introduced a moduli space parametrizing PGL(2,C)-local systems on a punctured surface and showed that its coordinate ring possesses a canonical basis similar to the bases studied in representation theory. The moduli space considered by Fock and Goncharov is interesting for a variety of reasons. In particular, it is birationally equivalent to a cluster variety and its coordinate ring can be canonically q-deformed. In this talk, I will explain the basic theory of cluster varieties and describe work with Hyun Kyu Kim which gives a q-deformation of Fock and Goncharov's canonical basis.





Speaker: 邱宇 Yu Qiu [香港中文大学]

Title: q-Stability conditions on Calabi-Yau-X categories for quivers with superpotential

Abstract: We set up the framewrok of q-stability conditions (sigma,s) on a Calabi-Yau-X category D, where sigma is a Bridgeland stability condition and s a complex parameter. We show that they form a complex manifold QStab(X), and when fixing s, the fiber QStab_s(D) gives a Bridgeland type space of stability conditions. The motivating examples are Calabi-Yau-X categories for quivers with superpotential from Riemann surfaces, where QStab_s(D) can be identified with moduli spaces of quadratic differentials on the surfaces. This is a project with Akishi Ikeda.





Speaker: 余世霖 Yu Shilin [Texas A&M University]

Title: Quantization and representation theory

Abstract: In this talk, I will talk about a geometric way to construct representations of noncompact semisimple Lie groups. Kirillov's coadjoint orbit method suggests that (unitary) irreducible representations can be constructed as geometric quantization of coadjoint orbits of the group. Except for a lot of evidence, the quantization scheme meets strong resistance in the case of noncompact semisimple groups. I will give a new perspective on the problem using deformation quantization of symplectic varieties and their Lagrangian subvarieties. This is joint work in progress with Conan Leung.





Speaker: 董志杰 Dong Zhijie [University of Massachusetts Amherst]

Title: A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE

Abstract: The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I will talk about a partial proof of this conjecture and discuss some relation to Hikita Conjecture and Coulomb branch.