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Representations of the fundamental groups of surfaces appear so often in geometry that it's tempting to see them primarily as geometric structures. In recent years, however, researchers have uncovered beautiful new features of these representations by thinking of them instead as dynamical systems. As an invitation to the dynamical point of view, I'll describe how geometric tools from the study of billiards can be used to build invariants of surface group representations.

Positive representations are infinite-dimensional bimodules for the quantum group and its modular dual where both act by positive essentially self-adjoint operators. Fifteen years ago Ponsot and Teschner showed that positive representations are closed under taking tensor products in the case g = sl(2), however similar conjecture remains open for all other types. I will outline its proof for g = sl(n) based on a joint work in progress with Gus Schrader. I will also argue that this conjecture is the key step towards the proof of the modular functor conjecture for quantized higher Teichmuller theories.

We study the effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. The generating functions are proven to be quasi-modular forms. As an application, we construct an exact solution of quantum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves. The talk is based on arXiv: 1612.01292[math.QA]

We will describe a formulation of the Batalin-Vilkovisky formalism using derived symplectic geometry.  In this setting, the classical master equation of the BV formalism describes a space of coisotropic structures.  Using this approach, we resolve a conjecture of Felder-Kazhdan regarding BRST cohomology.  Time permitting, we will also describe applications of these ideas to more general quantization problems.

Motivated by the question about reconstructing a Conformal field theory from the data of a subfactor of finite index, Jones studied the continuous limit of the periodic quantum spin chain which Thompson group acts on. Based on planar algebras, the topological axiomatization of subfactors, we will illustrate the idea of the correspondence between the skein theory of planar algebras and presentation of Thompson groups.

The affine Grassmannian is the analog of the Grassmannian for the loop group. They are very important objects in mathematical physics and the Geometric Langlands program. In this talk, I will explain my recent work on the central degeneration of semi-infinite orbits, Iwahori orbits and Mirkovic-Vilonen cycles in the affine Grassmannian. I will also use lots of convex polytopes to illustrate my results. In addition, I will explain the connections between my work and other parts of geometric representation theory and combinatorial algebraic geometry.  

About a decade ago Eynard and Orantin proposed a powerful computation algorithm
known as topological recursion. Starting with a ``spectral curve" and some ``initial data"
(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction
a collection of symmetric meromorphic differentials on the spectral curves parametrized by
pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).
Despite of many applications of the topological recursion (matrix integrals, WKB expansions, TFTs, etc.etc.)
the nature of the recursive relations was not understood.

Recently, in a joint work with Maxim Kontsevich we found a simple underlying structure of the recursive relations of Eynard and Orantin. We call it  ``Airy structure". In this talk I am going to define this notion and explain how the recursive relations of Eynard and Orantin follow from the  quantization of
a quadratic Lagrangian subvariety in a symplectic vector space.

Integral values of zeta functions are important not only for what they say about other values of their respective functions, but also for what they say about transcendence degree questions for appropriate extensions of the rationals or other number fields.  They  also appear in some recent computations relevant to particle physics.
In this talk we will give a quick introduction to the theory of periods and motives, relate said theory to special values of zeta functions, and discuss a graphical definition of the associated category of motives.
Any original work discussed in this talk is joint with Susama Agarwala.

We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.