Equidistribution in Symmetric Spaces

这一建议由局部对称空间中的均匀分布问题组成。具体地说，P.I.概述了关于闭测地线的分布、算术量子唯一遍历性以及局部对称空间上的强谱隙性质的问题。量子唯一遍历性的问题起源于量子混沌理论，它研究具有混沌动力学基础的量子系统高能态的行为。算术表面，以及其他算术模型，已经证明是测试这一理论中所作预测的非常肥沃的土壤。近年来，这一领域取得了很大的进展，应用遍历和解析数论的新技术解决了算术曲面的量子唯一遍历猜想。然而，对于高维系统，我们的知识仍然非常有限；例如，在这种情况下，甚至不清楚正确的猜想应该是什么。P.I.提出在某些高维对称空间中解决这个问题。在光谱的另一边，强光谱带隙的概念与最低能态有关。这一概念在许多应用中是至关重要的，特别是对于闭测地线在这些空间上的分布而言。随着各种数学学科的进步，强光谱间隙的存在和大小几乎在所有情况下都得到了很好的理解。然而，仍有一些案件失踪，以全面完成这一画面，这是公安部S的意图，努力缩小这一差距。公安部提出调查的问题由来已久，仍在积极研究中。尽管本计划中考虑的模型是非常具体和算术性质的，但P.I.相信，对这些模型的研究将导致在更广泛的环境中更好地理解相关现象。特别是，关于算术模型的量子唯一遍历性问题的结果将为其他物理系统的行为提供有价值的见解。此外，光谱间隙问题的进展可能会导致构建新的扩展器，这些扩展器在密码学中有用途。此外，回答这些问题的尝试可能会导致新工具的开发，这些工具将有助于研究数论中的基本问题。

This proposal is composed of problems on equidistribution in locally symmetric spaces. Specifically, the P.I. outlines questions regarding the distribution of closed geodesics, the question of arithmetic quantum unique ergodicity, and the the strong spectral gap property on locally symmetric spaces. The question of quantum unique ergodicity originates in the theory of quantum chaos, which studies the behavior of high energy states of quantum systems with underlying chaotic dynamics. Arithmetic surfaces, along with other arithmetic models, have proven a very fertile ground for testing predictions made in this theory. Recently there has been a great advancement in this field; application of new techniques from ergodic and analytic number theory resolved the (arithmetic) quantum unique ergodicity conjecture for arithmetic surfaces. However, for higher dimensional systems our knowledge is still very limited; for example, it is not even clear what the correct conjecture should be in this setting.The P.I. proposes to address this problem for certain higher dimensional symmetric spaces. On the other side of the spectrum, the notion of a strong spectral gap is related to the lowest energy state. This notion is crucial in many applications and in particular to the distribution of closed geodesics on these spaces. Following advancements in various mathematical disciplines, the existence and magnitude of the strong spectral gap is well understood in almost all cases. However, there are still a few cases missing in order to fully complete the picture, and it is the P.I.'s intention to work on closing this gap.The problems the P.I. proposes to investigate have a long history and are still matters of active research. Although the models considered in this program are very specific and arithmetic in nature, the P.I. believes that the study of these models will lead to a better understanding of related phenomena in more general settings. In particular, results on the question of quantum unique ergodicity for arithmetic models will provide valuable insights into the behavior of other physical systems. Also, progress on the spectral gap question may lead to construction of new expanders which have uses in cryptography. Moreover, the attempts to answer these questions could lead to the development of new tools that will facilitate studying fundamental questions in number theory.