Research
My primary mathematical interests are probability and number theory. In April 2010 I passed my
oral qualifying exam in these two subjects; see my syllabus and a
transcript of what I was asked.
Smallest irreducible of the form x^2-dy^2, Int. J. Number Theory,
5 (2009), pp 449-456.
The crown achievement of my mathematical career thus far, I wrote the bulk of this paper at the 2007
UW-Madison Number Theory REU under the invaluable guidance of
Ken Ono and Jeremy Rouse.
What are the primes of the form x^2+ny^2? When n=1, this is a familiar question from elementary
number theory. For other values of n, the problem is a bit more complicated, but nevertheless such
primes can be characterized, via class field theory, for an infinite set of n. In the paper,
I derived the function field analogue of this characterization, and used an effective version
of the Chebotarev density theorem to bound the degree of the irreducibles in question.
Fourier Analysis in Number Theory, my senior thesis,
under the advisory of Patrick Gallagher.
This thesis contains no original research, but is instead a compilation of results from analytic
number theory that apply Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression (the second non-trivial
result in analytic number theory after the Prime Number Theorem), and Weyl's criterion. I even
threw in a function field counterpart to Fermat's Last Theorem for good measure. The presentation
of the material is completely self-contained.