Mathematical Sciences: Constraints on Galois Representationswith Applications

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Professor Boston will work on classifying p-adic representations of Galois groups. He hopes to show that in certain cases the deformation space defined by certain mod p representations is empty. This in turn would imply the non- existence of certain elliptic curves. 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 conse- quences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.