Abstract: The cohomology rings of moduli spaces often have distinguished classes called tautological classes. This talk is about the special situation when all cohomology classes on a moduli space are tautological. I will start with the example of projective space. Then I'll introduce the moduli spaces M_{g,n}-bar of n-poined, stable genus g curves, using the example M_{2,0}-bar as a guide. At the end, I'll present several new small values (g, n) where we have proven that all classes on M_{g,n}-bar are tautological. This is joint work with Samir Canning.

Abstract: Shift of argument subalgebras is a family of maximal commutative subalgebras in the universal enveloping algebra U(g) parametrized by regular elements of the Cartan subalgebra of a reductive Lie algebra g. According to Vinberg, the Gelfand-Tsetlin subalgebra in U(gl_n) is a limit case of such family, so one can regard the eigenbases for such commutative subalgebras in finite-dimensional g-modules as a deformation of the Gelfand-Tsetlin basis (which is more general than Gelfand-Tsetlin bases themselves because exists for arbitrary semisimple Lie algebra g). I will define a natural structure of a Kashiwara crystal on the spectra of the shift of argument subalgebras of U(g) in finite-dimensional g-modules. This gives a topological description of the inner cactus group action on a g-crystal, as a monodromy of an appropriate covering of the De Concini-Procesi closure of the complement of the root hyperplane arrangement in the Cartan subalgebra. In particular, this gives a topological description of the Berenstein-Kirillov group (generated by Bender-Knuth involutions on the Gelfand-Tsetlin polytope) and of its relation to the cactus group due to Chmutov, Glick and Pylyavskyy. 

Abstract: Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We will report on recent results that produce such character bounds, and discuss some applications of them, in and outside of group theory.

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Abstract: Cluster algebras from marked surfaces can be interpreted as skein algebras, as functions on decorated Teichmüller space, or as functions on certain moduli of SL2-local systems.  These algebras and their quantizations have well-known collections of special elements called "bracelets" (due to Fock-Goncharov and Musiker-Schiffler-Williams, and due to D. Thurston in the quantum setting).  On the other hand, Gross-Hacking-Keel-Kontsevich used ideas from mirror symmetry to construct canonical bases of "theta functions" for cluster algebras, and this was extended to the quantum setting in my work with Ben Davison.  I will review these constructions and describe recent work with Fan Qin in which we prove that the (quantum) bracelets bases coincide with the corresponding (quantum) theta bases.

Video Slides 

Abstract: The classical Schur duality admits a q-deformation due to Jimbo, which is a duality between a quantum group and Hecke algebra of type A. A new quantum Schur duality between an i-quantum group (arising from quantum symmetric pairs) and Hecke algebra of type B was formulated by Huanchen Bao and myself. In this talk, I will explain these dualities, their geometric incarnation, and applications to super Kazhdan-Lusztig theories of type ABC.

Abstract: The pioneering work of Gross-Hacking-Keel studied the mirror symmetry for log Calabi-Yau surfaces proved that there exists a natural superpotential defined on the mirrors. The key intermediate product of the mirror construction are some combinatorial data called scattering diagrams. In this talk, I will explain the symplectic heuristic of the construction and mathematically how we retrieve the superpotentials and the scattering diagram from Lagrangian Floer theory. As corollaries, we prove a version of cluster mirror symmetry of rank two, a real analogue of 27 lines on cubic surfaces and a folklore conjecture "open Gromov-Witten invariants=log Gromov-Witten invariants". This is a joint work with Bardwell-Evans, Cheung and Hong. 

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Abstract: In this talk I will discuss an equivalence of categories relating SL(2)-equivariant vector bundles on the nilpotent cone for sl(2) and the annular Bar-Natan category (this latter category appears in the context of Khovanov homology for links in a thickened annulus).  Indeed, both categories admit a diagrammatic description in terms of the same "dotted" Temperley-Lieb diagrammatics, as I will explain.  Under this equivalence, Bezrukavnikov's quasi-exceptional collection on the nilcone (in the SL2 case) has an elegant description in terms of some special annular links.  In recent joint work with Dave Rose and Paul Wedrich, we constructed a very special Ind-object in the annular Bar-Natan category which is a categorical analogue of a "Kirby element" from quantum topology; I will conclude by sketching a neat "BGG resolution"  afforded by our categorified Kirby element.  This is based on joint work with Rose and Wedrich.

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Abstract: Local Whittaker functions for reductive groups play an integral role in number theory and representation theory, and many of their applications extend to the metaplectic case, where reductive groups are replaced by their metaplectic covering groups. We will examine these functions for covers of GLr through the lens of a solvable lattice model, or ice model: a construction from statistical mechanics that provides a surprising bridge between spaces of Whittaker functions and representations of quantum groups. This story has been well studied before for the case of one particularly nice cover of GLr, which eliminates all complications arising from the center of the group. In this talk, we will see that the same types of connections hold for any metaplectic cover of GLr, as well as examine how different choices of covering group interact with the center of GLr to change the story.

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Abstract: In this talk, we study the problem of characterizing the set of G-invariant measures on a space of infinite-dimensional matrices over a finite field. The groups G being considered are inductive limits of the finite general linear groups GL(n, q) and the finite even unitary groups U(2n, q^2) over a finite field; our proposed problem is still open in the latter even unitary case and the talk focuses on it. One partial result translates the problem to the classification of positive harmonic functions on branching graphs that are Hall-Littlewood versions of the Young graph. A second partial result is the construction of a large class of invariant measures by means of the Hopf-algebra structure on the ring of symmetric functions. The talk is based on joint work with Grigori Olshanski.

Abstract: The cactus group is a finitely presented group analogous to the braid group. It acts on combinatorial objects, especially tensor products of crystals. It is also the fundamental group of the moduli space of marked real genus 0 stable curves. The virtual cactus group contains both the cactus group and the symmetric group with some natural relations (here "virtual" is in the sense of virtual knot theory). I will explain how the virtual cactus group appears combinatorially and topologically.

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