Mathematical Sciences: Stark's Conjectures on Number Fields and Theory Fields

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Hayes 9626381 Stark's conjectures provide a major new insight into class field theory. They are an ingenious and highly original answer to the questions raised by the famous 12th problem of Hilbert. Hayes has given constructive proofs of the abelian Stark conjectures in function fields using the theory of Drinfield modules. He hopes to apply insights gained from the function field setting to number fields. Hayes has shown that the Brumer-Stark units over a number field satisfying Leopoldt's conjecture can be aligned into a Hecke character. In this project, he will attempt to prove the existence of further Hecke characters analogous to the well-known Jacobi sum characters introduced by A. Weil. The existence of such character is intimately related to the invariance under base change of the Brumer-Stark axioms, which is as yet unproved. Since Stark's conjectures are based upon the special value of partial zeta functions at zero, it is important to understand what such special values mean. Haye has given one interpertation of this special value over a real quadratic number field. He will try to extend this interpertation to the special values at negative integers and to arbitrary totally real number fields. Finallly, Hayes will also continue his work with Drinfield modules of rank one using the insights gained from the computation of Drinfield modules associated to hyperelliptic functions fields. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.