Mathematical Sciences: Refined Conjectures of Birch and Swinnerton-Dyer Type, and Regular Algebras

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Professor Tate will work on refined conjectures of Birch and Swinnerton-Dyer type. In particular, he will study a geometric analogue of this conjecture and its possible connections with two dimensional class field theory. He will also study regular graded associative algebras in dimensions greater than three. 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.