Research interest: 

Harmonic Analysis, Geometric Measure Theory, Incidence Geometry, Discrete Geometry, and related areas.

I am interested in problems at the intersection of Fourier analysis, geometric measure theory, and combinatorics. A central question driving my research is: how large does a set in Euclidean space need to be to guarantee that it determines a rich collection of geometric patterns, such as distances, triangles, or general k-point configurations?

My primary tools are Fourier integral operators and microlocal analysis, which I use to obtain sharp dimensional thresholds for configuration set problems, generalizing the Falconer distance conjecture to a broad class of k-point configurations. 

A second direction in my work concerns dimension expansion phenomenon for analytic functions, in the spirit of the Elekes-Rónyai theorem from combinatorics: if a bivariate or trivariate analytic function is not of a special degenerate form (e.g. f(x,y)=x+y), then it must expand the Hausdorff dimension of any product set of sufficiently large dimension, and in fact can produce a set of positive Lebesgue measure.