RUI: The Geometry of Arithmetic Locally Symmetric Spaces

算术群的研究起源于高斯关于二次型的工作，并且在世纪以来一直是一个活跃的研究领域。这个项目将借鉴群论，数论，几何和拓扑学的思想和技术，以研究算术群及其相关的局部对称空间的几何。将特别注意算术双曲反射群。反射群在数学中是普遍存在的，并且像算术群一样，自世纪以来一直在研究。庞加莱的工作双曲反射群在第2维发挥了突出作用的工作克莱因离散群的等距的双曲平面，和类似的结果双曲三空间发挥了重要作用瑟斯顿的工作几何化的三维流形。在整个历史上，双曲反射群一直是那些研究更一般的离散等距群的例子的重要来源。本项目将利用代数和解析数论的最新进展，以促进我们对反射群的认识。作为本项目的一部分，主要研究员（PI）的工作将在两个背景下研究算术局部对称空间：（1）双曲反射群的情况，和（2）收缩几何。种子工作的Vinberg在20世纪80年代发起了一项计划，以分类那些双曲反射群是算术。通过汇集工具从谱理论的双曲流形，解析数论和算术理论的二次形式的PI和他的合作者将取得进展的完整分类的同余算术双曲反射群。PI还将研究算术双曲流形的收缩几何。流形的收缩是流形上闭测地线的最小长度。关于算术双曲流形的收缩几何的一个最大的公开问题是短测地线猜想，它断言它们的收缩有一个普遍的正下界。作为该项目的一部分，PI将证明算术双曲流形的可扩展性类包含收缩小于任何固定阈值的代表的概率为零。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The study of arithmetic groups has its origins in Gauss' work on quadratic forms and has been an active area of research for well over a century. This project will draw upon ideas and techniques from group theory, number theory, geometry and topology in order to study arithmetic groups and the geometry of their associated locally symmetric spaces. Special attention will be paid to arithmetic hyperbolic reflection groups. Reflection groups are ubiquitous in mathematics and, like arithmetic groups, have been studied since the nineteenth century. Poincare's work on hyperbolic reflection groups in dimension 2 played a prominent role in the work of Klein on discrete groups of isometries of the hyperbolic plane, and analogous results for hyperbolic three-space played an important role in Thurston's work on the geometrization of three-dimensional manifolds. Throughout their history hyperbolic reflection groups have been an important source of motivating examples for those studying more general classes of discrete groups of isometries. This project will employ recent advancements in algebraic and analytic number theory in order to further our knowledge of reflection groups.The Principal Investigator's (PI) work as part of this project will study arithmetic locally symmetric spaces in two contexts: (1) the case of hyperbolic reflection groups, and (2) systolic geometry. Seminal work of Vinberg in the 1980s initiated a program to classify those hyperbolic reflection groups which are arithmetic. By bringing together tools from the spectral theory of hyperbolic manifolds, analytic number theory and the arithmetic theory of quadratic forms the PI and his collaborators will make progress towards the complete classification of congruence arithmetic hyperbolic reflection groups. The PI will also study the systolic geometry of arithmetic hyperbolic manifolds. The systole of a manifold is the least length of a closed geodesic on the manifold. One of the biggest open problems concerning the systolic geometry of arithmetic hyperbolic manifolds is the Short Geodesic Conjecture, which asserts that there is a universal positive lower bound for their systoles. As part of this project the PI will prove that the probability that a commensurability class of arithmetic hyperbolic manifolds contains a representative with systole less than any fixed threshold is zero.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.