[See also my list of publications and schedule of upcoming talks.]
I was first exposed to number theory as a seventh grader in the Boston area Math Circle, and have been hooked ever since. I've worked in several areas of number theory -- algebraic number theory, multiple Dirichlet series, random matrices and random graphs, probabilistic number theory, and the burgeoning theory (pioneered by Bhargava) of higher composition laws. However, the bulk of my research has focused on analytic number theory and, in particular, on character sums.
Character sums lie at the heart of many classical questions in number theory, from determining the least non-square modulo a prime to understanding the behavior of Dirichlet L-functions in the critical strip. They are notoriously mysterious, and despite nearly a century of intense scrutiny are still poorly understood. In particular, the classical Pólya-Vinogradov inequality has stood as the state-of-the-art upper bound for long character sums since 1918, and no one has substantially improved our understanding of short character sums (except in special cases) beyond Burgess' work of 1957. Inspired by a breakthrough of Granville and Soundararajan, my colleagues and I have made some inroads into the field; click here to see my publications.
Before I decided to pursue research in math, I explored a couple of other fields. My first experience with research was in fluid dynamics. Working under the guidance of Sergey Buldyrev in the lab of H. Eugene Stanley at Boston University, I studied the flow of a liquid through a non-homogeneous medium. The resulting paper appeared in Physical Review E. I spent the following summer working in the Page lab at the Whitehead Institute, MIT as a programmer for the Human Genome Project; I developed subroutines which were used in the analysis of the Y chromosome.
I'm always interested in hearing about cool problems; feel free to email me if you have questions or ideas.