This award supports the research in Algebraic Geometry of Professor Steven Sperber of the University of Minnesota. Dr. Sperber plans to continue his work on the properties of L- functions associated with character sums defined over finite fields. One of the specific projects planned is to continue joint work with A. Adolphson of Oklahoma State University relating properties of exponential sums to the shape of certain Newton polygons; another is to study the p-adic analytic behavior of families of exponential sums, and relations with the monodromy of the associated deformation equation. This research is work in the algebraic geometry of positive characteristic. Although the field originated with notions of continuously varying geometric structures like lines and planes, in this context the discrete takes over, and methods akin to those from the theory of whole numbers are most useful. Reciprocally, the algebraic geometry of positive characteristic is now having great influence on the Theory of Numbers, and is finding application in Computer Science and Coding Theory.
