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.
My doctoral research focused 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.
Papers:
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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).
Doctoral Thesis:
- "On the local Tamagawa number conjecture for Tate motives", advised by Matthias Flach (2014).
Undergraduate Thesis:
- Determining unitary equivalence to a $3 \times 3$ complex symmetric matrix from the upper triangular form (pdf), advised by Stephan Garcia (2008).