Asymptotic and Variational Problems in Nonpositive Curvature

提议：DMS-9972425PI：Christopher Judge摘要：我们考虑非正曲率流形上测地流和拉普拉斯特征值的渐近行为，重点讨论与度量变化有关的方面。例如，有理多边形的台球流属于流的自然模空间，系统的周期渐近行为可以从该模空间的知识中推导出来。虽然已知测地线流和拉普拉斯谱通过迹公式联系在一起，但物理学家长期以来一直相信“对应原理”提供了一种更直接的关系。这种关系将在模空间理论的背景下考虑。研究的对象是物理系统的模型。特别是，我们在这里考虑粒子的渐近行为，无论是经典的还是量子的，在不同的条件下。我们认为的系统表现出高度的混乱行为，就像现实世界中经常出现的情况一样。尽管这些系统是简单的抽象，但它们代表了进一步理解真实世界系统的试验场。最终的“下游”应用可能包括高能物理和密码学。

Proposal: DMS-9972425PI: Christopher JudgeAbstract: We consider the asymptotic behavior of the geodesic flow and Laplace eigenvalues on manifolds of nonpositive curvature with an emphasis on aspects related to metric variation. For example, the billiard ball flow of a rational polygon belongs to a natural moduli space of flows, and the system's periodic asymptotic behavior can be deduced from knowledge of the moduli space. Although, the geodesic flow and the Laplace spectrum are known to be related via trace formulae, physicists have long believed in a more direct relationship provided by the `correspondence principle'. This relationship will be considered in the context of the moduli space theory.The objects under study are models for physical systems. In particular, we consider here the asymptotic behavior of particles, both classical and quantum, under varying conditions. The systems we consider exhibit a high degree of chaotic behavior as is often the case in the real world. Although these systems are simple abstractions, they represent a testing ground for further understanding of real world systems. Eventual 'downstream' applications could include high energy physics and cryptography.