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

Ohio State University, The, Columbus OH

Investigators

Abstract

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.