The Dirac equation describes the relativistic evolution of electrons and positrons.  We consider the (time-dependent!) Dirac equation in three dimensions coupled to a potential with Coulomb-type singularities.  We prove a propagation of singularities result for this equation and show that singularities are diffracted by the singularities of the potential.  We finally compute the symbol of the diffracted wave and show it is typically non-zero.  This talk is based on joint work with Jared Wunsch.

We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols. This is joint work with L. Coburn, M. Hitrik, and J. Sjöstrand.

We study the trapped set of spacetimes whose metric decays to a stationary Kerr metric at an inverse polynomial rate. In the first part of the talk, I will focus on the dynamical aspects of this problem and show that the trapped set is a smooth submanifold which converges to that of the stationary metric at the same rate. In the second part, I will explain how to use this to prove microlocal estimates at the trapped set for solutions of wave equations on such spacetimes.

The study of pseudo-Anosov homeomorphisms of surfaces can be understood via train tracks carrying the stable and unstable laminations. After Hamenstadt, one has a splitting complex, which gives rise to a dual triangulation of the punctured mapping torus known as a veering triangulation. I'll discuss the combinatorial properties of these triangulations, and survey some of the literature on the topic. Applications include the conjugacy problem, describing short geodesics in moduli spaces, and relations with hyperbolic geometry. 

When a 3-manifold fibers over the circle it often does so in infinitely many ways, and this gives a kind of laboratory for studying Teichmuller spaces and mapping class groups in many surfaces at once. Farb-Leininger-Margalit proved an influential theorem showing that fibered 3-manifolds organize the mapping classes of "short" Teichmuller translation length in all genera. We prove an analogous theorem for Weil-Petersson translation length. The proof uses recent theorems on renormalized volume as well as good old fashioned 3-manifold topology from the 1980s.  Quadratic differentials come into the story in a couple of different ways that I will try to point out. Joint work with Leininger, Souto and Taylor.

In this talk I will discuss geometric generalizations of Euclidean resolvent estimates, such as estimates for the resolvent of the Laplacian of an asymptotically conic metric plus a decaying potential, in a Fredholm framework that focuses on capturing the outgoing asymptotics of the resolvent applied to a Schwartz function (outgoing waves); this is different from even the usual treatment of the Euclidean problem. More precisely, the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$ on asymptotically conic spaces $(X,g)$ of dimension at least $3$ (with the asymptotic behavior at the `large end’ of the cone), and the main results are the limiting absorption principle, as well as uniform estimates for $P(\sigma)^{-1}$ as $\sigma\to 0$, on function spaces between which $P(\sigma)$ is Fredholm even for real $\sigma\neq 0$ and which correspond to finite regularity Lagrangian distributions (second microlocal spaces) associated to the, conic in the base, Lagrangian given by the outgoing radial set of the Hamilton flow. Such results have immediate applications to the behavior of the wave equation on black hole spacetimes.