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

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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.