Professor Sperber will study exponential sums using methods from p-adic analysis and p-adic cohomology. He intends to produce better congruence estimates for these sums and relate them to singularities and Hodge theory. The goal is to obtain sharp effective estimates on the size of exponential 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.
