Mathematical Sciences: Analytic and Algebraic Methods in Geometry and Number Theory

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This research will concentrate on various problems related to arithmetic algebraic geometry. Professor Dwork will work on the theory of p-adic differential equations. More specifically he will work on the p-adic properties of generalized hypergeometric functions, or more exactly upon the p-adic properties of the Frobenius matrix associated with the corresponding cohomology. Professor Shimura will work on arithmeticity properties concerning zeta functions and automorphic forms. These include special values of Eisenstein series, certain combinations of zeta functions and periods of automorphic forms associated with quaternion algebras. Dr. Bien, a post doctoral associate, will work on the representations of Kac-Moody algebras. Another post doctoral associate, Dr. Chai, will work on the moduli schemes of abelian varieties. The theory of arithmetic algebraic geometry has benefited greatly in the past from the important contributions of the two principle investigators. They are one of the strongest pairs in arithmetic in the world today. The subject uses geometric (and in their case analytic, as well) tools to delve deeply into number theoretic questions. Certain types of special functions play important roles in number theory. These two researchers attack problems on these types of functions in complementary ways. Dwork looks at p-adic analogues while Shimura concentrates on their transformation properties. The post doctoral associates work in similar areas. Much of great interest will surely result from this work.