The Spectral Theory of Difference Operators

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Gieseker DMS 96-22905 This awards supports the research of Professor D. Gieseker to work on the algebraic geometry of difference equations. He hopes to apply techniques of algebraic geometry to various nonlinear equations usually arising from physics. In this proposal he intends to concentrate on the KP and Davey-Stewartson equations. What he hopes to do is discretize the continuous linear operator so that the associated linear problem admits isospectral deformations.. This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past 10 years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.