Mathematical Sciences: Hypergeometric Functions over Finite Fields

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This research concerns character sum analogues for hypergeometric series over finite fields. The principal investigator will systematically study an analogue of the Bailey transform in an attempt to derive analogues for higher order hpergeometric series identities. He will also study "pseudo" hypergeometric series, that is, functions which due to the nature of the finite field can not be written as a character sum hypergeometric series but are closely related. Also the principal investigator will investigate the relationship between hypergeometric series and representation theory. The purpose of the first two problems is to complete the theory of hypergeometric series over finite fields and to increase its computational power as a tool for character sum evaluation. The goal of the last problem is to develop an interplay between the two areas of mathematics, to bring representation theoretic tools to bear on character sum problems and to determine representation theoretic interpretations for known character sum identities.