CAREER: Topology, Spectral Geometry, and Arithmetic of Locally Symmetric Spaces

本课题研究代数数论中的问题。代数数是系数为整数的多项式的根。这些数字固有的对称性是数论的基础，数论是研究整数的数学分支。朗兰兹的深度猜想预测，代数数的对称性在非常特定的几何空间中是明显的，被称为局部对称空间，它与大的整数矩阵群有关。在每一点附近，局部对称空间都是异常对称的，它们在任意两点附近的形状都是不可区分的。然而，由于一个不和谐的几何特征，这些空间的大规模几何结构是无序和混乱的：从同一点向不同方向发出的直线往往以指数速率彼此发散。此外，后一种性质使得这些数论世界难以绘制。该项目旨在系统地组织由算术产生的局部对称空间，以提炼其内在结构。这个项目的基础是概率、几何和算法方法的融合，它很好地适用于PI已经试点的初高中学生的推广计划，旨在培养创造性的数学思维。本课题将从几个角度探讨数论起源的正基秩局部对称空间的拓扑、几何和算术。首先，它将使用扩展球算法来绘制这些空间，以构建点网格，类似于通过逐步建立蜂窝塔网络并定期发送信号以探测附近的其他塔来绘制世界地图。其次，通过谱复杂度和周期复杂度之间的关系研究双曲流形的低音音符。第三，通过系统地使用相关的p进对称空间，试图克服叠加在阿基米德对称空间上的复杂解析结构的缺失。相关p算术群的刚性亚纯环的构造，在超越CM数域和经典复乘法理论的新背景下，为希尔伯特关于显式类场论的第十二个问题提供了可能的进展。该奖项反映了美国国家科学基金会的法定使命，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。

This project investigates problems in algebraic number theory. Algebraic numbers are the roots of polynomials with integer coefficients. Symmetries inherent to these numbers are fundamental in number theory, the branch of mathematics devoted to the study of integers. Deep conjectures of Langlands predict that the symmetries of algebraic numbers are apparent in very specific geometric spaces, known as locally symmetric spaces, which are associated to large groups of integer matrices. Near every point, locally symmetric spaces are exceptionally symmetric, and their shapes nearby any two points are indistinguishable. However, the large-scale geometry of these spaces is disordered and chaotic owing to one jarring geometric feature: straight lines emanating from the same point in different directions tend to diverge from each other at an exponential rate. Additionally, the latter property makes these number theoretic worlds difficult to chart. This project aims to systematically organize locally symmetric spaces arising from arithmetic in order to distill inherent structure thereon. The blend of probabilistic, geometric, and algorithmic methods underlying this project lends itself well to an outreach program for middle school and high school students which the PI has piloted, designed to foster outside-the-box mathematical thinking. From several perspectives, this project will probe the topology, geometry, and arithmetic of positive fundamental rank locally symmetric spaces of number theoretic origin. First, it will chart these spaces using an expanding ball algorithm to construct point grids, akin to mapping the world by progressively building a network of cell towers and regularly transmitting signal to detect other towers nearby. Second, it will study the bass notes of hyperbolic manifolds via relationships between spectrum and cycle complexity. Third, it will attempt to overcome the absence of complex analytic structure on the overlying archimedean symmetric space by systematic use of associated p-adic symmetric spaces. Construction of attendant rigid meromorphic cocycles for associated p-arithmetic groups give possible inroads to Hilbert's twelfth problem, regarding explicit class field theory, in new contexts beyond CM number fields and the classical theory of complex multiplication.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.