Rings of Algebraic Differential Operators in Mathematical Physics and Geometry

建议的研究涉及数学和数学物理的几个领域，包括几何分析、表示论、非对易代数和代数几何，其动机是偏微分方程式理论中的老问题(尚未解决)。一个这样的问题(称为空隙问题)涉及没有扩散的波的传播，另一个问题是具有有限渐近的黎曼流形上的热核。尽管这些问题本质上是分析性的，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，涉及代数簇上的微分算子环及其特殊表示(投射D-模)。这一建议的主要目的之一是系统地研究这类模及其自同态环和模空间的性质。到目前为止所得到的结果(主要是在曲线的情况下)揭示了与非对易几何、表示论、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一些联系。在一个不同的方向上，PI打算开发一种基于同伦代数的思想来计算热核的渐近性的新方法。这个项目中所讨论的数学问题源于一个实际问题：当一个短信号(比方说光或声音)从A点被省略，穿过介质并到达B点时会发生什么？有很多种可能性：可能有聚焦、绕射、微弱的余辉，或者在B处仍有一个很短的“清晰”信号。最后一种可能性特别令人感兴趣，但只有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应用，包括电磁波和声波理论、空间通信技术、磁流体力学、晶体光学等。作为更广泛的影响，国际和平研究所预计，这项工作的跨学科性质将促进所涉各个领域的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.