Terry Harris

Harmonic analysis and geometric measure theory

My research mostly uses tools from "Fourier restriction theory", which considers functions with Fourier transform supported near a curved surface such as a sphere, cone, or paraboloid. These functions arise naturally when solving the wave and Schrödinger equations, but they also come up in many geometric problems related to incidence counting. One such problem is: does projection typically decrease the dimension of an object? This area is known as projection theory. A related question is the Kakeya problem, which asks whether a family of lines pointing in different directions is approximately disjoint. I am also interested in versions of the above questions in settings where Euclidean methods no longer work. The "Fourier slice theorem" is the main tool connecting Fourier analysis to projection theory. This theorem breaks down in the Heisenberg group, and finding ways around this is a difficult problem.