Mathematical Sciences: Research in Number Theory and Automophic Forms

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In this research Harris will continue to study the arithmetic of holomorphic and non-holomorphic automorphic forms and relations with special values of L-functions. It will concentrate on the conjectures of Deligne on the special values of L-functions attached to the cohomology of algebraic varieties in the only cases in which these L-functions are known to be tractable: namely, when they can be realized as automorphic L- functions. Automorphic functions are complex analytic functions that encode number theoretic information and allow one to bring the powerful machinery of analysis to bear on the number theory problems. In this case the area of application is in arithmetic algebraic geometry, namely in the area of number theory connected with the solutions of systems of polynomial equations. This then becomes a sweeping area of study encompassing most areas of mathematics in its tools. Harris is a master of this and much exciting mathematics will result from this research.