Mathematical Sciences: Exponential Sums and Toric Varieties

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This award supports the research of Professor Adolphson to work in number theory. This project will explore the connections between exponential sums over finite fields and toric varieties. Two major topics will be pursued. The first topic is the study of the p-adic differential equations that arise from the variation of cohomology of families of exponential sums. The second topic is the study of the p-adic cohomology of complete intersections. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.