Contemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Insights from ergodic theory have led to dramatic progress in old questions concerning the distribution of primes, geometric representation theory and deformation theory have led to new techniques for constructing Galois representations with prescribed properties, and the study of automorphic forms and special values of L-functions have been revolutionized by developments in both p-adic and arithmetic geometry as well as in pure representation theory.
The ideas emerging from the Langlands Program (in its many modern guises) and from the developments that grew out of Wiles’ proof of Fermat’s Last Theorem continue to guide much of the ongoing research on the algebraic and geometric sides of the subject, and in the analytic direction the synthesis of additive combinatorics and harmonic analysis continues to lead to breakthroughs in many directions.
In addition to specialized graduate courses, the number theory group has a weekly research seminar with outside speakers from across all areas of number theory. There are also a variety of learning seminars aimed at helping students and postdocs to acquire familiarity with important techniques and results that are generally not available in textbooks (with notes posted for wider dissemination when possible).