My thesis at U.C. Berkeley dealt with Galois representations associated to modular forms. Since then, however, my work has gravitated more toward the p-adic geometry of modular curves. In particular, my primary focus for the first several years out was on computing the stable reduction of the modular curve X

Recently, others have made significant progress on the stable reduction problem. Most notably, Jared Weinstein has in some sense solved the problem for all n, but from the "local Langlands" perspective. Takahiro Tsushima has also made significant advances using the methods of Coleman-McMurdy. This has prompted me to shift gears somewhat into the area of overconvergent modular forms. My background in rigid-analysis and stable reduction of modular curves gives me a slightly different perspective on this field, and I'm hopeful that as a result I will be able to bring some new insight. A first paper on the U

Other interests include the arithmetic of supersingular elliptic curves (see recent paper and earlier joint work with K. Lauter below), and special properties of Galois representations associated to rational points on certain modular curves. I also have an enduring interest in explicit equations for modular curves, which has been the basis for the computational aspects of much of my work. At some point I hope to publish a semi-expository paper on the various methods for obtaining explicit equations which I have come across and/or developed over the years. My lecture notes on eta product models from the Heilbronn Workshop (see below) have some very useful formulas which were actually coded up by David Loeffler for a SAGE module which comes with the standard download. Here is a table with some Equations for X

- "Explicit Representation of the Endomorphism Rings of Supersingular Elliptic Curves" (preprint, Sage Code)
- "Applying a Galois Transformation to the Roots of a Polynomial" (preprint)
- "Slopes of the U
_{7}Operator Acting on a Space of Overconvergent Modular Forms" (with L. J. P. Kilford), LMS J. Comput. Math.**15**(2012), 113-139 (download, Sage Code) - "Stable Reduction of X
_{0}(p^{3})" (with R. Coleman), Algebra and Number Theory**4 (4)**(2010), 357-431 (preprint) - "Stable Reduction of X
_{0}(81)", Contemp. Math.**463**(2008), 91-109 (preprint) - "Do the Coefficients of a Modular Form Really 'Encode Arithmetic Data?'" (with Hari Ravindran), Rose-Hulman MSTR 08-01 (download)
- "Fake CM and the Stable Model of X
_{0}(Np^{3})" (with R. Coleman), Documenta Math. Extra Volume: John H. Coates' Sixtieth Birthday (2006), 261-300 (download) - "Explicit Generators for Endomorphism Rings of Supersingular Elliptic Curves", joint with Kristin Lauter, preprint posted to online proceedings for Conference in Honor of Harold Stark
- "Stable Model of X
_{0}(125)," LMS J. Comput. Math.**7**(2004), 21-36 (download) - "Explicit Parameterizations of Ordinary and Supersingular Regions of
X
_{0}(p^{n})," Modular Curves and Abelian Varieties, Progress in Mathematics**224**, 165-179 (Birkhauser, 2004) (download) - "A Splitting Criterion for Galois Representations Associated to Exceptional Modular Forms," Ph.D. thesis, U.C. Berkeley (2001) (download)

- "Elliptic Curves with Non-abelian Entanglements", Maine-Quebec Number Theory Conference (F '15) (slides)
- "How to Transform the Roots of a Polynomial (without finding them first)", Ramapo Math Club, 2014 (slides)
- "A New Algorithm for Computing Endomorphism Rings of Supersingular Elliptic Curves", AMS-MAA Joint Meetings (San Diego, 2013) (slides)
- "Stable Models and Up Slope Calculations," Maine-Quebec Number Theory Conference, U.C. Berkeley Number Theory Seminar (F '11) (slides)
- "Slopes of the U
_{7}Operator Acting on a Space of 7-adic Overconvergent Modular Forms," 24^{th}AFW (Hawaii, 2010) (slides) - "Stable Reduction of X
_{0}(625), with Implications," U. C. Berkeley Number Theory Seminar (S '09) (slides) - "Eta Products and Models for Modular Curves," Heilbronn Workshop on Computations with Modular Forms (Bristol, UK, F '08) (slides)
- "Explicit Verification of a Theorem of Shimura," 22
^{nd}Annual Workshop on Automorphic Forms and Related Topics (Texas A&M, S '08) (slides) - "Heegner Points on X
_{0}(p^{n})," AMS (sectional) Special Session on Computational Arithmetic Geometry (San Francisco, 2006) (slides) - "Stable Reduction of X
_{0}(p^{3})," AMS Special Session on Arithmetic Algebraic Geometry (Atlanta, 2005) (slides) - "Why Number Theorists Care About Elliptic Curves," RHIT Math Colloquium (notes: part 1, part 2)
- "Explicit Generators for Endomorphism Rings of Supersingular Elliptic Curves," Conference in Honor of Harold Stark (Minneapolis, 2004) (slides)
- "p-adic Properties of CM j-invariants." University of Rochester Number Theory Seminar (notes: part 1, part 2)