Number Theory

The integers and prime numbers have fascinated people since ancient times. Recently, the field has seen huge advances. The resolution of Fermat's Last Theorem by Wiles in 1995 touched off a flurry of related activity that continues unabated to the present, such as the recent solution by Khare and Wintenberger of Serre's conjecture on the relationship between mod p Galois representations and modular forms. The Riemann hypothesis, a Clay Millennium Problem, is a part of analytic number theory, which employs analytic methods (calculus and complex analysis) to understand the integers. Recent advances in this area include the Green-Tao proof that prime numbers occur in arbitrarily long arithmetic progressions. The Langlands Program is a broad series of conjectures that connect number theory with representation theory. Number theory has applications in computer science due to connections with cryptography.

The research interests of the Number Theory group involve arithmetic algebraic geometry, or specifically: p-adic analytic methods in arithmetic geometry, p-adic Hodge theory, algorithms, and applications (cryptography, coding theory, etc.), analytic number theory in particular automorphic forms.