Number theory is one of the oldest of all branches of mathematics, with its first important results dating back more than two millennia. While whole numbers are in a sense discrete objects, in modern times they are often studied using techniques of continuous mathematics, such as calculus. The PI's work focuses in particular on the use of p-adic numbers, which were introduced early in the 20th century as a number system analogous to the real numbers, but recording information about divisibility of integers. In addition to theoretical results, the p-adic numbers give rise to computational techniques with some relevance in computer science, especially in modern cryptography.
Building on recent breakthroughs in p-adic Hodge theory, we extend the field in several different directions. We extend the theory of p-adic representations to allow coefficients in larger rings, with an eye towards applications to the p-adic interpolation of automorphic forms. We also allow the replacement of p-adic Galois groups with etale fundamental groups; this is expected to shed new light on the p-adic Langlands correspondence. Finally, we consider ways to replace the p-adic numbers with the full ring of integers; this involves systematic use of Witt vectors over arbitrary rings.
