CAREER: p-adic and mod p Galois representations

Johns Hopkins University, Baltimore MD

Investigators

Abstract

The PI's research is in number theory and representation theory; its goal, broadly, is to understand the Galois representations associated (sometimes conjecturally) to automorphic forms and automorphic representations. The PI will undertake several projects related to the conjectures of Serre, Fontaine-Mazur, and Breuil-Mezard (and their generalizations) as well as to the emerging p-adic and mod p Langlands correspondences. The PI will study the weight part of Serre's conjecture for reductive groups over number fields, with the goal of giving an explicit Serre weight recipe in considerable generality. The investigator will produce evidence for generalizations of the Breuil-Mezard conjecture, and will prove some cases of such a generalization, with applications to the Langlands program. Another component of the project involves the explicit reduction modulo p of certain p-adic Galois representations. Number theory is one of the oldest branches of mathematics. At its most fundamental, number theory is the study of whole number solutions to equations, although sophisticated modern techniques can sometimes give the appearance of being far-removed from this goal. Number theory has many important practical applications; for instance, most cellular telephone calls are protected by a code based on elliptic curves, one of the primary objects of study in the investigator's field. In addition to these broader impacts, this award will support several educational initiatives, including a transition program for minority calculus students at the University of Arizona, the development of an undergraduate problem-solving curriculum, and the recruitment of underrepresented minority students to high school mathematics summer programs. The PI is one of the directors of Canada/USA Mathcamp, a summer program for mathematically talented high school students.