Research

Professional background

I am a Professor of Mathematics in the Georgia Tech College of Sciences. My formal training is in the field of arithmetic geometry, a branch of number theory which arose out of the study of integer solutions to polynomial equations. It is now well-established that in order to study such solutions, one should often look at the underlying geometry of the equations, over not just the real numbers and complex numbers but also the p-adic numbers for each prime number p. Ideas from seemingly unrelated parts of mathematics often find applications in number theory, and my work is no exception: I use tools from analysis (especially potential theory), algebraic geometry, dynamical systems, graph theory, and matroid theory.

Algebraic curves, torsion points on abelian varieties, and canonical heights

My early work centered around torsion points on abelian varieties and modular curves.  These are central objects in number theory; for example, they play key roles in the proof of Fermat's Last Theorem. Through Robert Coleman's work applying p-adic integration to torsion points on curves, I became familiar with rigid analytic geometry in graduate school and my interest in the subject was rekindled when I learned several years later about Berkovich's theory of non-archimedean analytic spaces.  My work has been heavily influenced by the Szpiro-Ullmo-Zhang equidistribution theorem, which gives a powerful way to study torsion points, and more generally points of small canonical height, on subvarieties of abelian varieties.  In thinking about which properties of torsion points extend to points of small canonical height, I was led into the fascinating realm of algebraic dynamics. The interplay between potential theory, canonical heights, dynamics, and Berkovich spaces has turned out to be very fruitful.

Connections between Berkovich spaces, complex dynamics, tropical geometry, and combinatorics

In a larger context, my research on Berkovich spaces is meant to address a broad unifying principle within number theory, the idea that all completions of a number field should be treated in a symmetric way.  My research monograph "Potential Theory and Dynamics on the Berkovich Projective Line", written with Rumely, is a good illustration of this theme; we develop p-adic theories of harmonic and subharmonic functions, iteration of rational functions, and much more in a way which emphasizes the similarities between real and p-adic theories. These results are not only aesthetically pleasing, they have a lot of applications, too, sometimes unexpected ones.  For example, in my work with Laura DeMarco, we use potential theory on non-archimedean Berkovich spaces to prove results about complex dynamics. My work on canonical heights over function fields is another illustration of the unexpected power of non-archimedean potential theory.

I am also fascinated by connections between algebraic geometry and combinatorics, especially involving algebraic curves and graphs. In thinking about what kind of information about a Berkovich curve is retained by its skeleton, I stumbled upon the Riemann-Roch theorem for graphs (which I developed in collaboration with Serguei Norine). Riemann-Roch theorem for graphs is closely related to ideas in tropical geometry, and I have spent a lot of time thinking about connections between arithmetic geometry, combinatorics, and tropical geometry. 

Matroid theory

In addition to studying the deep link between graphs and algebraic curves, in recent years I have begun investigating connections between tropical geometry, matroids, and geometry over the field of one element. Matroids are beautiful and important objects which lie at the interface of combinatorics and algebraic geometry, and in collaboration with Nathan Bowler and Oliver Lorscheid I have developed new algebraic and algebro-geometric tools for understanding them.