UNIVERSITY OF ARIZONA
The investigator's research areas are number theory and representation theory. In particular, the investigator studies p-adic Galois representations and p-adic Hodge theory, with an eye towards applications to the modularity of Galois representations and the Langlands program. The PI will undertake several projects organized around the theme of p-adic representations and their reduction modulo p. He will study the weight in Serre's conjecture for arbitrary split reductive groups and arbitrary 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 rather far-removed from this goal. In recent decades, number theory has had revolutionary applications to the fields of cryptography (creating codes) and cryptanalysis (breaking codes). For instance, most cellular telephone communications are protected by a cryptosystem based on elliptic curves, one of the primary objects of study in the investigator's field. The PI is one of the directors of Canada/USA Mathcamp, a summer program for mathematically talented high school students, and believes it is vital that research-active mathematicians participate in such endeavors.