Research

Synopses of some current research topics

Can you hear your location on a manifold?

Suppose you find yourself standing at an unknown point in a familiar room with your eyes closed. Can you determine where you are standing by clapping your hands once and listening to the resulting echos and reverberations?

In a paper with Yakun Xi, we ask this question and determine the answer for "most" rooms that take the form of a compact, boundary-less Riemannian manifold. In a follow-up paper with 

Eigenfunction concentration on manifolds

The study of the behavior of Laplace eigenfunctions on manifolds is both broad and deep, with connections reaching from quantum physics to number theory. I study eigenfunctions of large eigenvalue–in particular how the geometry of the underlying manifold influences their asymptotics.

I have been particularly excited lately about the prospect of obtaining polynomial improvements to the remainder of the Weyl law if the manifold is the Cartesian product of two or more compact manifolds. See my paper with Alex Iosevich, and a recent paper by Michael Taylor.

Fourier integral operators and their applications

Among the main tools used to study high-frequency eigenfunctions is the theory of Fourier integral operators (FIOs). FIOs are also used to attack problems in geometric measure theory (such as Falconer's distance problem) where the key feature of FIOs are their Sobolev mapping properties.

I have encountered with increasing frequency what could be called "Fourier integral multilinear forms." An example arose in my work on eigenfunction triple products, and I am working to develop general Sobolev continuity estimates for such objects.

VC-dimension and geometry

The Vapnik–Chervonenkis dimension measures the complexity of a family of binary classifiers, and is a core part of learning theory. To illustrate, consider the following game of battleship. Alice secretely draws a triangle in the unit square. Bob then starts throwing darts at the square, and each time, Alice tells Bob if the dart lands inside or outside of the triangle. After throwing some number of darts, Bob must then make a guess at what triangle Alice drew. The VC-dimension of the set of triangles determines how many darts Bob must throw to be able to guess the triangle up to a small error with high confidence. (The VC-dimension of the set of triangles in the plane is 7, by the way.) The higher the VC-dimension, the harder it is to guess. This game can be played in a variety of settings and seems to give rise to fascinating combinatorial and geometric configurations (such as in this paper).

Talks

Publications and preprints