A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. Currently, we know only a very small percentage of the total cohomology. In this talk I will explain some of what is known and discuss some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups. 

In the first half of the talk, we introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in spacetime or, equivalently, foliations of a null cone. We also show how certain, fairly generic, convexity assumptions on the null cone allows for a proof of the Penrose Conjecture. On the Black Hole Horizon, we find that the convexity assumptions become sharp; therefore, the second half of the talk will explore the existence of a class of Black Hole Horizons admitting such convexity. From this, building upon the work of S. Alexakis, we will show that the Schwarzschild Null Cone--the case of equality for the Penrose Conjecture--is also critical in light of recent work on the perturbation of stable, weakly isolated Horizons.

Sheel Ganatra (USC): Liouville sectors and localizing Fukaya categories
We introduce a new class of Liouville manifolds-with-boundary, called Liouville sectors, and show they have well-behaved, covariantly functorial Fukaya/Floer theories. Stein manifolds frequently admit coverings by Liouville sectors, which can then be used to study the Fukaya category of the total space. Our first main result in this setup is a local criterion for generating (global) Fukaya categories. One of our goals, using this framework, is to obtain a combinatorial presentation of the Fukaya category of any Stein manifold. This is joint work with John Pardon and Vivek Shende.

Nathan Dunfield (UIUC): An SL(2, R) Casson-Lin invariant and applications
When M is the exterior of a knot K in the 3-sphere, Lin showed that the signature of K can be viewed as a Casson-style signed count of the SU(2) representations of pi_1(M) where the meridian has trace 0. This was later generalized to the fact that signature function of K on the unit circle counts SU(2) representations as a function of the trace of the meridan. I will define the SL(2, R) analog of these Casson-Lin invariants, and explain how it interacts with the original SU(2) version via a new kind of smooth resolution of the real points of certain SL(2, C) character varieties in which both kinds of representations live. I will use the new invariant to study left-orderability of Dehn fillings on M using the translation extension locus I introduced with Marc Culler, and also give a new proof of a recent theorem of Gordon's on parabolic SL(2, R) representations of two-bridge knot groups. This is joint work with Jake Rasmussen (Cambridge).
 

A framework to systematically decouple high order elliptic equations into combination of Poisson-type and Stokes-type equations is developed using the tools of differential complexes and Helmholtz decompositions. The key step is to systematically construct the underling commutative diagrams involving the complexes and Helmholtz decompositions in a general way.

Discretizing the decoupled formulation leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but not limit to: the primal formulations and mixed formulation of biharmonic equation, fourth order curl equation, and triharmonic equation etc. As a by-product, Helmholtz decompositions for many dual spaces are obtained.