Counts of special trajectories of quadratic differentials on Riemann surfaces are an example of Donaldson-Thomas invariants, they exhibit wall-crossing phenomena over the moduli space of differentials. I will describe a combinatorial construction of the Kontsevich-Soibelman invariant for this counting problem based on ribbon graphs of degenerate Jenkins-Strebel differentials. Time permitting, I will also sketch generalizations to motivic wall-crossing invariants, and to collections of k-differentials in the context of Hitchin systems.

I will talk about recent work with Jesse Gell-Redman, Jacob Shapiro and Junyong Zhang on

solving the semilinear Helmholtz equation on Euclidean, or more generally asymptotically conic spaces, 

and ongoing work on extending this result to the semilinear Klein-Gordon and time dependent Schr\"odinger

equations. 

The proof relies on finding suitable pairs of Hilbert spaces between which the

operator acts invertibly, such that the range space has suitable algebra (multiplicative) properties. 

Similar work has been done recently by Hintz-Vasy and Gell-Redman-Haber-Vasy. The difference in the

present work is that the analysis at the boundary (`infinity') takes place in the scattering calculus, rather than

the b-calculus. 

In this talk, I study a PDE that models interface effects between insulators. It is a Schrodinger equation with periodic asymptotics (the bulk), away from a strip (the interface). I will prove the bulk-edge correspondence. At the physical level, this index-like theorem predicts that the interface between two topologically distinct insulators behaves like a conductor. 

The proof relies on a semiclassical trace expansion, though the original problem is not semiclassical. It suggests that energy propagates along the interface in the form of waves  microlocalized at the singularities of an underlying (Bloch) bundle. This draws an apparently new connection between microlocal analysis and condensed matter physics.

This will be a conference on the wide range of topics aimed to inspire future directions and new applications of microlocal methods.