The fundamental problem of interest is that of acquisition of rational points on algebraic varieties: given a variety V over a number field K, describe the set of finite field extensions L/K such that V has L-rational points, or has points everywhere locally over L. The P.I. seeks to understand what sort of acquisitive behavior is possible -- e.g., what are the possible indices of a curve of genus g over K? Can elements of all orders occur in the Shafarevich-Tate group of an elliptic curve over K, or over a small degree field extension? -- and also what happens generically -- e.g. what is the Zariski-closure of the locus of curves having points everywhere locally but not globally on the moduli space of all curves of genus g? Particular attention is paid to modular curves, including Atkin-Lehner twists and Shimura curves, which provide a natural proving ground for more general conjectures. These investigations are fueled by tools from Galois cohomology, field arithmetic, Fuchsian groups, spectral theory, p-adic geometry and deformation theory, and thus provide a welcome incentive to interact with and learn from many other researchers.
The P.I.'s research is in the general area of mathematics known as Diophantine geometry: given a system of polynomial equations, one wishes to find all integer solutions. The evident first question -- to determine whether there exist any solutions at all -- has proven to be quite daunting (in fact, algorithmically impossible) in general, so until recently most research has assumed the existence of at least one solution and then attempted to study the structure of the set of all solutions. However, it seems that "most" systems of equations do not have any integral solutions, so the greater part of Diophantine diversity has gone unstudied. These "pointless varieties" have a rich intrinsic geometry -- there are geometric phenomena that can only exist in the absence of integral solutions -- and their study is closely connected to many other problems in related fields. The P.I.'s research program consists both in a general study of pointless varieties -- including working towards several conjectures on their ubiquity -- and in pursuing connections and applications, for instance to the structure of Shafarevich-Tate groups of abelian varieties.
