Much of my work focuses on the effect of background geometry on solutions to dispersive partial differential equations such as the wave equation and the Schrödinger equation. Such geometry may arise from (1) a fixed background geometry such as those arising in general relativity, (2) the introduction of a boundary, or (3) the study of quasilinear equations where one, in essence, has geometry that depends on the solution but the solution in turn depends on the geometry.

A phenomenon of particular interest is trapping. We are used to, for example, light traveling in straight lines. But on curved geometries, it travels along certain paths called geodesics. And on some geometries, some of these geodesics may not escape to infinity, which is called trapping. This happens, for example, on black hole backgrounds, which have a region where the light will orbit the black hole rather than falling into the black hole or escaping to infinity. Trapping is a known obstruction to many typical dispersive estimates, but depending on the nature of the trapping, related estimates with losses can be proved.