My primary area of research is algebraic number theory. Much of my work involves Artin representations and their relationship to automorphic representations. In my dissertation, I show the existence of an icosahedral Hilbert modular form modulo GRH, and I give an expression for the number of roots of a quintic polynomial modulo primes in terms of the coefficients of this form.

Explicit computations of representations and modular forms have been very important in my research. I make much use of the wonderful number theory calculator PARI/GP. Feel free to contact me regarding explicit computations of Artin representations or theta series attached to quaternion algebras.

I am also interested in realizing class groups as cohomology groups. Doing so reveals connections between class groups and other arithmetic objects, such as Selmer groups of elliptic curves, giving bounds or explicit expressions for the size of the n-torsion subgroup of class groups for fixed values of n. Furthermore, we hope that such connections might reveal techniques for showing that Tate-Shafarevich groups are finite.

I am also very interested in mathematics education. At Rutgers University, I was a fellow with Metromath, an NSF-funded teaching and research group based at Rutgers, CUNY, and UPenn. The primary goal of this group was the research and promote leadership in urban mathematics classrooms. I was a member of the Affect and Motivation group, the purpose of which was to study the ways in which emotions and attitudes affect learning in the classroom. My own behavior in the classroom is very much informed by my experiences with this research group.