Rings of Differential Operators and the Hadamard Problem

贝瑞斯特提出的研究站在数学和数学物理的各种问题的十字路口。一个统一的原则是，我们研究的大多数问题都涉及微分算子环的整体代数和几何性质。复Weyl代数A1（由关系[p； q] = ih定义）是这种环的典型例子，它将在本项目中特别关注。它可能是非交换代数中最简单和最重要的例子，在数学、物理和自然科学的许多领域都有应用。（例如，它在著名的海森堡测不准原理背后的量子力学中起着重要作用。）到目前为止，他的工作已经获得了关于这个代数的基本新结果。这些主要涉及到A1的射影模（理想）的结构和自同构群。在提议的研究中，我们打算解决关于A1的更深层次的问题，并对我们的结果进行概括，特别是在更高的维度上。该项目主要部分所寻求的数学结果是由经典数学物理中的基本问题之一波传播理论所激发并应用的。从实践和理论的角度来看，特别令人感兴趣的问题是，波何时可以不扩散地传播，以允许传输“清晰”（锐利）信号的可能性。这个问题在齐次空间中得到了很好的研究。该项目的目标之一是开发新的数学工具和技术来研究非均匀和各向异性介质中的这一难题。在这个方向上所寻求的结果在波传播的数学理论中具有根本的兴趣和意义，并可能在相关的物理学科中有应用，包括电磁波和声波理论、空间通信技术、磁流体动力学、晶体光学等。作为更广泛的影响，预期这项工作的跨学科性质将促进各有关领域的专家之间的交流和合作，事实上这项工作迄今已经开始这样做。此外，几名研究生和本科生以及一名博士后将共同参与这项工作。

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 globalalgebraic 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.