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Number Theory

Number theory studies some of the most basic objects of mathematics: integers and prime numbers. It is a huge subject that makes contact with most areas of modern mathematics, and in fact, enjoys a symbiotic relationship with many. The last fifty years in particular have seen some dramatic progress, including Deligne's proof of the Weil conjectures (giving optimal asymptotics for the number of solutions of polynomial equations over finite fields), Faltings' proof of the Mordell conjecture (establishing finiteness of rational points on hyperbolic curves), Wiles' proof of Fermat's last theorem (which brought a 400-year-long quest to completion) and Zhang's proof of the boundedness of prime gaps (taking a huge step towards the twin prime conjecture). The solutions to these problems rely on techniques from many areas, including algebraic and complex geometry, representation theory and modular forms, differential and algebraic topology, and real and complex analysis. Moreover, grand new vistas (such as the Langlands program) have been uncovered, which will surely keep mathematicians busy for decades.

The research interests of our group are diverse and reflect the breadth of the subject. They include arithmetic as well as classical algebraic geometry; automorphic, geometric and p-adic representation theory; Shimura varieties and Galois representations; p-adic Hodge theory; harmonic analysis and analytic number theory; representation stability and commutative algebra; and algorithmic and computational number theory.