Rings of Algebraic Differential Operators in Mathematical Physics and Geometry

提出的研究是数学和数学物理几个领域的交叉，包括几何分析、表示理论、非交换代数和代数几何。它是由偏微分方程理论中老的（尚未解决的）问题所激发的。一个这样的问题（称为空洞问题）涉及无扩散波的传播，另一个是有限渐近黎曼流形上的热核问题。尽管这些问题本质上是解析的，但它们表现出一个显著的特征：它们有代数解。PI和他的合作者的早期工作将这些经典的分析问题与代数几何和非交换代数的各种问题联系起来，这些问题涉及代数变异上的微分算子及其具体表示（射影d模）。本文的主要目标之一是系统地研究这些模、它们的同构环和模空间的性质。迄今为止得到的结果（主要是在曲线的情况下）揭示了与非交换几何、表示理论、组合学和可积系统的许多有趣的联系。因此，该提案的另一个目标是调查其中的一些联系。在一个不同的方向上，PI打算开发一种基于同局部代数思想的计算热核渐近的新方法。这个项目解决的数学问题源于一个实际问题：当一个短信号（比如光或声音）从a点遗漏，通过介质到达B点时会发生什么？有很多可能性：可能有聚焦、衍射、微弱回波的持续，或者在b点仍然有一个短的“清晰”信号。最后一种可能性特别有趣，但它只在非常特殊的条件下才会发生。用精确的数学术语描述这些条件是该提议的目标之一。在这个方向上寻求的结果是我们理解波动现象的基础，并可能在相关的物理科学中有应用，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，PI期望这项工作的跨学科性质将促进各相关领域专家之间的交流与合作。

BerestThe proposed research lies at the intersection of several areas ofmathematics and mathematical physics, including geometric analysis,representation theory, noncommutative algebra and algebraic geometry.It is motivated by old (yet unsolved) problems in the theory of partialdifferential equations. One such problem (known as the problem of lacunas)deals with propagation of waves without diffusion, another with heatkernels on Riemannian manifolds with finite asymptotics. Despite beinganalytic by their nature, these problems exhibit a remarkable feature:they have algebraic solutions. The earlier work of the PI and hiscollaborators links these classical problems of analysis to variousquestions of algebraic geometry and noncommutative algebra, concerningrings of differential operators on algebraic varieties and their specificrepresentations (projective D-modules). One of the primary goals of thisproposal is to systematically study the properties of such modules, theirendomorphism rings and moduli spaces. The results obtained so far (mostlyin the case of curves) reveal many interesting connections with noncommutative geometry, representation theory, combinatorics and integrable systems. Another goal of the proposal is thus to investigate some of these connections. In a different direction, the PI intends to develop a new method for computing asymptotics of heat kernels based onideas of homotopical algebra.The mathematical problems addressed in this project arise from the practicalquestion: what happens when a short signal (light or sound, say) is omittedfrom a point A, travels through a medium and arrives at point B? There aremany possibilities: there could be focusing, diffraction, persistence of faintechoes, or still a short `clean-cut' signal at B. This last possibility is ofparticular interest, but it may occur only under very special conditions.To describe these conditions in precise mathematical terms is one of the goalsof the proposal. The results sought in this direction are fundamental for ourunderstanding of wave phenomena and may have applications in related physicalsciences, including the theory of electromagnetic and acoustic waves, space communication technologies, magneto-hydrodynamics, crystal optics, etc. As a broader impact, the PI expects that the interdisciplinary nature of this work will stimulate communication and collaboration between experts in the various areas involved.