This award supports research in the field of arithmetic algebraic geometry. One direction of this research involves the l-adic cohomology of varieties over finite fields, the l-adic theory of exponential sums over finite fields, the arithmetic theory of differential equations, Fourier transforms (both l-adic and for D-modules), the theory of perverse sheaves, and the relations among them. The remainder of the project centers around the conjectures about special values of L-functions attached to varieties over number fields. The mathematical area of arithmetic algebraic geometry is an ultramodern research area which combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though arithmetic geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.
