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，一个多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、几何学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。反射群在数学中普遍存在，与算术群一样，自19世纪以来一直被研究。Poincare关于2维双曲反射群的工作在Klein关于双曲平面的离散等距线群的工作中起着重要的作用，而类似的关于双曲三维空间的结果在瑟斯顿关于三维流形的几何化的工作中起着重要的作用。纵观双曲反射群的历史，对于那些研究更一般的离散等距线群的人来说，双曲反射群一直是激励例子的重要来源。这个项目将利用代数和解析数论的最新进展来加深我们对反射群的了解。作为这个项目的一部分，首席研究者(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。Vinberg在20世纪80年代的开创性工作发起了一个程序，对那些算术性质的双曲反射群进行分类。通过结合双曲流形的谱理论、解析数论和二次型算术理论的工具，PI和他的合作者将朝着同余算术双曲反射群的完全分类取得进展。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最大公开问题之一是短测地线猜想，该猜想断言其收缩几何存在一个普遍的正下界。作为该项目的一部分，PI将证明可公度类算术双曲流形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，并已通过使用基金会的智力优势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.