The Spectral Theory of Difference Operators

Gieseker DMS 96-22905 该奖项支持D教授的研究。Gieseker工作的代数几何差分方程。他希望将代数几何的技巧应用于各种非线性方程，这些方程通常来自物理学。在这个建议中，他打算集中在KP和Davey-Stewartson方程。 他希望做的是离散连续线性算子，使相关的线性问题承认等谱变形。 这项研究是在代数几何领域，这是现代数学最古老的部分之一，但在过去的10年里，它已经发展到解决了几个世纪以来的问题。最初，它处理由最简单的方程，即多项式在平面上定义的图形。今天，该领域使用的方法不仅来自代数，而且来自分析和拓扑学，相反，它被广泛用于这些领域。 此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人学等不同领域都很有用。

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.