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

这个项目研究代数数论中的问题。代数数是整系数多项式的根。 这些数所固有的对称性是数论的基础，数论是数学中专门研究整数的分支。 朗兰兹的深度解析预言，代数数的对称性在非常特殊的几何空间中是明显的，称为局部对称空间，它与大群整数矩阵相关联。 在每一点附近，局部对称空间都是异常对称的，并且它们在任意两点附近的形状是不可区分的。 然而，这些空间的大尺度几何是无序和混乱的，这是由于一个不协调的几何功能：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，后一个属性使得这些数论世界难以绘制。 本项目旨在系统地组织由算术产生的局部对称空间，以提取其内在结构。 该项目融合了概率、几何和算法方法，非常适合PI为初中和高中学生开展的推广计划，旨在培养开箱即用的数学思维。本计画将从多个角度探讨数论起源的正基本秩局部对称空间的拓扑、几何与算术。 首先，它将使用扩展球算法构建点网格来绘制这些空间，类似于通过逐步构建手机信号塔网络并定期发送信号以检测附近的其他信号塔来绘制世界地图。 其次，通过谱与圈复杂度的关系研究双曲流形的低音。 第三，它将试图通过系统地使用相关的p-adic对称空间来克服覆盖阿基米德对称空间的复杂分析结构的缺乏。 伴随着刚性亚纯上循环相关的p-算术群的建设给予可能的进展希尔伯特的第十二个问题，关于显式类域理论，在新的背景下超越CM数域和经典理论的复杂multiply.This奖项反映了NSF的法定使命，并已被认为是值得通过使用该基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.