Number theory is one of the oldest branches of mathematics. At its most fundamental, number theory is the study of whole number solutions to polynomial equations, and especially to equations motivated by geometry. For example, the lengths of the three sides of a right triangle are related by the Pythagorean theorem. While it is straightforward to find all right triangles whose side lengths are rational numbers, it perhaps surprisingly remains an unsolved problem to determine which whole numbers can be the area of a right triangle with rational sides. 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 Langlands program, the principal investigator's area of research.
Significant progress has been made in recent decades towards understanding Langlands's conjectural reciprocity between automorphic forms and Galois representations. Techniques for proving reciprocity laws generally require the consideration of p-adic families of Galois representations. Three projects to investigate p-adic families of Galois representations will be undertaken. The first project will produce moduli stacks of certain Galois representations in which the residual representation is allowed to vary. This will open up the possibility of proving results in the Langlands program by establishing them generically on these spaces. The second project, on Breuil's local-global compatibility conjecture for types in the p-adic Langlands program, will be a demonstration of this technique. A third project is to study the weight part of Serre's conjecture for GL(n), for n larger than 2, in non-generic cases.
