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

职业：拓扑、谱几何和局部对称空间算术

• 批准号: 2338933
• 负责人: Michael Lipnowski
• 金额: $ 50万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-05-01 至 2029-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2338933&HistoricalAwards=false
• 关键词: CAREER; Topology; Spectral; Geometry; Arithmetic

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.

该项目研究了代数数理论中的问题。代数数是具有整数系数的多项式的根。 这些数字固有的对称性在数量理论上是基本的，这是用于整数研究的数学分支。 Langlands的深层猜想预测，代数数的对称性在非常具体的几何空间（称为局部对称空间）中显而易见，这些空间与大型整数矩阵有关。 在每个点附近，局部对称空间是异常对称的，它们的形状附近任何两个点都是无法区分的。 然而，由于一个刺耳的几何特征，这些空间的大规模几何形状是无序的，并且混乱的几何特征：在不同方向上从同一点发出的直线往往会以指数的速度彼此不同。 此外，后一种属性使这些数字理论世界难以绘制。 该项目的目的是系统地组织算术中产生的本地对称空间，以使其固有的结构提炼。 该项目基础的概率，几何和算法方法的融合非常适合中学和高中生的外展计划，PI驾驶了该计划，旨在促进开箱即用的数学思维。从几个角度来看，该项目将探测正基本等级的拓扑，几何形状和算术局部对称性空间的数字理论来源。 首先，它将使用扩展的球算法来绘制这些空间来构建点网格，类似于通过逐步构建细胞塔网络并定期传输信号以检测附近的其他塔楼来绘制世界。 其次，它将通过光谱和循环复杂性之间的关系研究双曲线歧管的低音音符。 第三，它将试图通过系统地使用相关的P-ADIC对称空间来克服上覆的Archimedean对称空间上的复杂分析结构。 在CM数字领域以外的新环境和复杂乘法的经典理论中，为希尔伯特的第十二个问题，关于明确的班级理论的建设，为希尔伯特的第十二个问题，关于希尔伯特的第十二个问题的构建可能会涉足希尔伯特的第十二个问题。该奖项通过评估了Infellitia的支持，这反映了NSF的法定任务，并反映了构成构成的构成群体的支持。