Research

My research is primarily in number theory and related areas.  A common theme in my work is to render theoretical topics more concrete by explicitly working out detailed examples. With coauthor John Jones and recently others, I have built extensive tables of number fields and p-adic fields. I have pursued more geometric analogs in my studies of Belyi covers and Hurwitz covers.  Some of my works are focused not on the finite-degree field extensions, but more concretely on the polynomials which define them.

My research generally has Galois theory as its setting with ramification playing a central role. Much of it is in a motivic context where the main groups are infinite rather than finite. When the varieties under study are curves, one can work with abelian varieties instead, but to study higher-dimensional varieties analogously, the language of motives is essential. I have worked primarily on the motives underlying classical hypergeometric functions, but have also studied other similar motives. Starting from my thesis on Shimura curves, I have been interested in modular forms.  I have computationally studied L-functions from number fields and hypergeometric motives, and more recently have focused on L-functions which are transcendental in that they don’t come from motives.

I have also developed a side interest in game theory.

Browse the complete Works archive →

Papers

The papers posted in the Works archive often differ slightly from the published versions. The published versions can often be accessed online, either directly or through MathSciNet.

Presentations

Presentations often correspond to papers, in which case they usually provide an easier path into the material. Presentations sometimes contain material not yet in a paper.