This award provides funding for a project concerned with the theory and application of the Hardy-Littlewood circle method. The investigator is pursuing an improved, detailed understanding of the circle method both from the analytic viewpoint, through estimates for mean values of exponential sums, and from an arithmetic viewpoint, through an understanding of the role played by the trivial solutions of diophantine systems lying on special subvarieties. The analytic viewpoint will be considered in through further investigation and development of the proposer's new method for estimating fractional moments of smooth Weyl sums and through detailed investigations of low moments of exponential sums. The arithmetic viewpoint will be considered by continuing the investigator's research on the number of non-diagonal solutions of systems of symmetric diagonal equations using slicing methods and also perhaps by sieve methods. The latter viewpoint will also be pursued through the testing of the "Quasi- Hardy-Littlewood" model suggested recently by Vaughan and Wooley. 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.
