I’m interested in the application of algebraic techniques to number theoretic questions

My research focuses on using the techniques of $p$-adic Hodge theory and $(\phi, \Gamma)$-modules to study a number of arithmetic principles. In particular, there is a wonderful conjecture of Bloch and Kato which generalizes both the analytic Class Number Formula and the Birch and Swinnerton-Dyer conjecture.   My work with Matthias Flach has strengthened the evidence for this conjecture in the case of Tate motives over number fields.

I am currently most interested in the properties of supercharacters and their applications to number theory.  This is a new area with connections to representation theory and number theory, and presents a number of interesting problems which are accessible to undergraduate researchers and produce interesting graphical representations.  I also hope to some day return to the study of non-unique factorization problems in numerical monoids.

Papers:

### Doctoral Thesis:

• Determining unitary equivalence to a $3 \times 3$ complex symmetric matrix from the upper triangular form (pdf), advised by Stephan Garcia (2008).