Maximal representations form unexpected connected components of the character variety of the fundamental group of a hyperbolic surface in a semisimple Lie group, that only consist of injective homomorphsims with discrete image. They thus generalize the Teichmüller space, and can be thought of as parametrizing certain locally symmetric spaces of infinite volume. After a general introduction to character varieties and maximal representations, I will discuss joint work with Andres Sambarino and Anna Wienhard in which we prove a sharp upper bound for the exponential orbit growth rate of the associated actions on the symmetric space.

I describe a program, in work with T. Dimofte, to associate 2d supersymmetric field theories to triangulated 4-manifolds, complementary to recent advances associating field theories to smooth 4-manifolds initiated by Gadde-Gukov-Putrov. BPS observables in the field theory capture topological invariants of the 4-manifold. In particular, this field theoretic prescription can be used to associate chiral algebras, and corresponding graded traces, to triangulated 4-manifolds with boundary. I will propose the 2d theory that should be associated to the simplest triangulated 4-manifold, the 4-simplex, and discuss how elementary field theory dualities encode 4d Pachner moves.

The subject of this talk is the moduli space of Riemann surfaces, which is the set of isometry classes of constant curvature metrics on a surface. The cotangent space of the moduli space is given by holomorphic quadratic differentials, and there is a natural Weil-Petersson metric defined by an $L^2$-type pairing. I will discuss the behavior of the moduli space when approaching the boundary of the Deligne-Mumford compactification, and show how to use tools from microlocal analysis to understand the degeneration of the Weil-Petersson metric. 

I will talk about some joint work with Andrew Neitzke, where we introduce a new "invariant" (with possible wall-crossing) for framed links in a three-manifold M=C \times R with C being an oriented surface. This invariant, denoted as F(L) for a framed link L, is valued in the GL(1) skein algebra of another three-manifold M'=C' \times R, where C' is an N-fold cover of C. 

Under various special limits, F(L) turns into more familiar objects. When L is contained in a 3-ball in M, F(L) reproduces certain one-variable limit of the HOMFLY polynomial of L. When the projection of L to C has no crossings and the homology class of L is nontrivial, F(L) becomes a generating function encoding the spectrum of framed BPS states associated with certain half-BPS line defect in a 4d N=2 supersymmetric theory. In general, F(L) is a "hybrid" of the above two quantities. 

The construction of F(L) is realized via a homomorphism from the GL(N) skein algebra of M to the GL(1) skein algebra of M'. In my talk I will first review the notion of skein algebras. Then I will describe this homomorphism for the special case of N=2, followed by some examples.