Equidistribution in Symmetric Spaces

这个建议是由局部对称空间中的等分布问题。 特别是私家侦探概述了有关封闭测地线的分布问题，算术量子唯一遍历性问题，以及局部对称空间上的强谱隙性质。 量子唯一遍历性问题起源于量子混沌理论，它研究具有混沌动力学的量子系统的高能态的行为。算术表面，沿着与其他算术模型，已被证明是一个非常肥沃的土壤测试的预测，在这个理论。最近有一个很大的进步，在这一领域的应用新技术，从遍历和解析数论解决了（算术）量子唯一遍历猜想算术曲面。然而，对于更高维的系统，我们的知识仍然非常有限;例如，甚至不清楚在这种情况下正确的猜想应该是什么。提出解决这个问题的某些高维对称空间。在光谱的另一侧，强光谱间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，特别是对这些空间上的闭测地线的分布。随着各种数学学科的进步，强谱隙的存在和大小在几乎所有情况下都得到了很好的理解。然而，仍然有几个案件失踪，以充分完成图片，这是P. I。P.I.打算努力缩小这一差距。建议调查有很长的历史，仍然是积极研究的问题。虽然在这个程序中考虑的模型是非常具体和算术的性质，P. I。相信这些模型的研究将导致更好地理解在更一般的设置相关的现象。特别是，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，在频谱间隙问题上的进展可能会导致在密码学中使用的新扩展器的构建。此外，回答这些问题的尝试可能会导致新工具的发展，这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.