Rings of Differential Operators and the Hadamard Problem

Cornell University, Ithaca NY

Investigators

Abstract

Berest proposed research stands at a crossroads of various questions of mathematics and math- ematical physics. A unifying principle is that most of the problems we study involve the global algebraic and geometric properties of rings of differential operators. The complex Weyl algebra A1 (defined by the relation [p; q] = ih) is the prototypical example of such a ring which will be of special concern in the present project. It is perhaps the simplest and the most important example of a noncommutative algebra that finds applications in many areas of mathematics, physics and natural sciences. (For example, it plays a fundamental role in quantum mechanics underlying the famous Heisenberg Uncertainty Principle.) His work so far has obtained essentially new results about this algebra. These mostly concern the structure of projective modules (ideals) and the automorphism group of A1. In the proposed research we intend to address still deeper questions about A1 and to give generalizations of our results, especially to higher dimensions. The mathematical results sought in the main part of the project are motivated by and applied to the theory of wave propagation which is one of the fundamental problems in classical mathematical physics. Of special interest (both from practical and theoretical point of view) is the question of when the waves may propagate without diffusion to allow the possibility of transmitting `clean-cut' (sharp) signals. Well studied in homogeneous spaces this question remains wide open in general. One of the goals of this project is to develop new mathematical tools and techniques to investigate this difficult problem in inhomogeneous and anisotropic media. The results sought in this direction are of fundamental interest and significance in mathematical theory of wave propagation and may have applications in related physical disciplines including the theory of electromagnetic and acoustic waves, space communication technologies, magnetohydrodynamics, crystal optics, etc. As a broader impact, it is expected that the interdisciplinary nature of this work will stimulate communication and collaboration between specialists in the various areas involved, as indeed this work so far has already begun to do. Moreover, several students, both graduate and undergraduate, as well as a postdoctoral fellow will collaborate in this work.