I'm interested in the application of algebraic techniques to number theoretic questions
My doctoral 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 study of non-unique factorization problems in numerical monoids. Numerical monoids have a rich theory of factorization that is accessible to students with an undergraduate background in mathematics.
On the local Tamagawa number conjecture for Tate motives over tamely ramified fields, Algebra and Number Theory, 10(6):1221–1275, 2016 (with Matthias Flach)
- Delta sets of numerical monoids using nonminimal sets of generators Comm. Algebra 38 (2010), no. 7, 2622–2634. (with Scott Chapman, Rolf Hoyer, and Nathan Kaplan).
- Sampling Lissajous and Fourier knots Experiment. Math. 18 (2009), no. 4, 481–497 (with Adam Boocher, Jim Hoste, and Wenjing Zheng).
- "On the local Tamagawa number conjecture for Tate motives", advised by Matthias Flach (2014).
- Determining unitary equivalence to a $3 \times 3$ complex symmetric matrix from the upper triangular form (pdf), advised by Stephan Garcia (2008).