This series consists of talks in the area of Mathematical Physics.
An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. You can study local systems of vector spaces over this local system of fields. On a 3-manifold, they are are rigid, and the rank one local systems are counted by the Alexander polynomial. On a surface, they come in positive
We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), provided k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). Our results can be translated to statements about clusters in Fock-Goncharov configuration spaces of affine flags, provided the number of flags is even.
The Feynman diagram expansion for a Wilson loop observable in Chern-Simons gauge theory generates an infinite series of topological invariants for framed knots. In this talk, I will describe a new perturbative formalism which conjecturally generates the same invariants for Legendrian knots in the standard contact R^3. The formalism includes a `perturbative' localization principle which drastically simplifies the structure of calculations. As time permits, I will provide some examples and applications. This talk is based upon joint work with Brendan McLellan and Ruo
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.
I will discuss how Costello's inductive renormalization
procedure for the construction of effective field theories can be
extended to manifolds with boundary.
Positive representations are infinite-dimensional bimodules for the quantum group and its modular dual where both act by positive essentially self-adjoint operators.
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.
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.
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).