Equidistribution in Symmetric Spaces

对称空间中的均匀分布

• 批准号: 1237412
• 负责人: Dubi Kelmer
• 金额: $ 7.89万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1237412&HistoricalAwards=false
• 关键词: Equidistribution; Symmetric; Spaces

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.

该提议由局部对称空间中的均匀分布问题组成。 具体来说，P.I.概述了有关闭合测地线分布、算术量子唯一遍历性问题以及局部对称空间上的强谱间隙特性的问题。 量子唯一遍历性问题源于量子混沌理论，该理论研究具有潜在混沌动力学的量子系统高能态的行为。算术曲面以及其他算术模型已被证明为检验该理论中的预测提供了非常肥沃的土壤。最近，这一领域取得了很大进展；遍历和解析数论新技术的应用解决了算术曲面的（算术）量子唯一遍历性猜想。然而，对于高维系统，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。提出针对某些高维对称空间解决这个问题。在光谱的另一边，强光谱间隙的概念与最低能态有关。这个概念在许多应用中至关重要，特别是对于这些空间上闭合测地线的分布。随着各种数学学科的进步，几乎在所有情况下，强谱间隙的存在和大小都得到了很好的理解。然而，为了完整地完成整个图景，仍然缺少一些案例，P.I. 打算努力缩小这一差距。提出调查具有悠久的历史并且仍然是积极研究的事项。尽管该程序中考虑的模型本质上是非常具体和算术的，但 P.I.相信对这些模型的研究将有助于更好地理解更一般环境中的相关现象。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，频谱间隙问题的进展可能会导致构建可用于密码学的新扩展器。此外，回答这些问题的尝试可能会导致新工具的开发，从而有助于研究数论中的基本问题。