Spaces of geometric structures via geometric transitions

University Of Texas At Austin, Austin TX

Investigators

Abstract

A geometric manifold is an abstract mathematical object designed to model the space we live in. The concept also encompasses the notion of a space-time in the theory of relativity, as well as phase spaces, configuration spaces, and other useful structures in physics. While empirical physics is dedicated to measuring the precise features (e.g. shape, size, etc.) of our universe, the question of what possible features could the universe have, subject to certain laws, is a mathematical one. The data comprising the answer to such a question is called a moduli space; it is a topological space whose points are all possible geometric manifolds of a certain type and whose topology organizes those geometric manifolds into families whose features vary continuously. This project addresses important questions about many different classes of low-dimensional geometric manifolds and their moduli. The PI will study these questions by further developing and applying an emerging new mathematical framework which describes interaction between moduli spaces of different types of geometric manifolds through a mechanism called geometric transition. The research to be conducted in this project weaves together ideas from a wide cross-section of mathematics and physics, including geometry, topology, group theory, dynamics, and relativity. Progress on these problems will be of significance to many researchers across these disciplines and, more broadly, will contribute to the growing base of foundational knowledge on which many innovations in science and engineering are built. Recent progress by the PI and collaborators, in the setting of complete affine Lorentzian three-manifolds called Margulis spacetimes, suggests a rubric to approach difficult questions in higher dimensional affine geometry. By further developing and generalizing the geometric transition technology used to study Margulis spacetimes as limits of curved spacetimes, the PI will address important questions in higher dimensional affine geometry surrounding the Auslander Conjecture. A second main theme is the study of the relationship between hyperbolic and anti de Sitter (AdS) geometry in dimension three. Work of the PI on the hyperbolic-AdS transition establishes an explicit and natural connection between the two geometries with the potential to expedite progress in the study of geometric structures in both settings. In particular, recent joint work gives new evidence for the bending measure conjecture for convex AdS spacetimes, a counterpart to Thurston's conjecture characterizing convex hyperbolic three-manifolds in terms of bending data on the convex core. The PI will work toward the resolution of these conjectures by further examining the interaction between convex structures in both settings. Additionally, the PI will begin a new project to develop applications of geometric transitions in the setting of representation theory of hyperbolic groups into higher rank Lie groups, a growing subject known as higher Teichmüller theory.