Mathematical Sciences: Kloosterman Sums and Traces of Lisse Sheaves as Algebraic Integers

Professor Fisher will work on the equidistribution theorem for Kloosterman sums. He also hopes to generalize Katz's equidistribution theorem to lisse sheaves with traces in a number field. He also hopes to study numerically the distinctness of Kloosterman sums. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.