Geometry and topology of hyperbolic manifolds

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The main goal of this proposal is to transfer into higher dimensions some of the techniques and intuition gained from recent spectacular progress in low dimensional hyperbolic geometry. Current powerful tools include Perelman's work on the Ricci flow, and over thirty years of insight gained from developing Thurston's theory of hyperbolic 3-manifolds. While these techniques usually do not transfer literally into higher dimensions, the PI thinks it is an opportune moment to study the geometry and topology of higher-dimensional hyperbolic manifolds, and benefit from lower-dimensional intuition and insights. This work will build on an existing collaboration with Steven Kerckhoff. With the natural motivation of understanding the 3 (or 4) dimensional world around us, most research in geometry focuses on "low dimensions", usually meaning less than 5. Nonetheless, in the modern world, higher dimensional geometric objects are abundant: contemporary computer science employs abstract "simplicial complexes" of very high dimension to model concrete systems, studying financial markets often requires estimating integrals over high dimensional spaces, and internet search engines rely on efficient linear algebra algorithms for very large vector spaces. This proposal will focus on studying a specific type of geometry in high dimensions, namely hyperbolic geometry. Traditionally, approaching this subject was difficult without a powerful computer. This is no longer a major obstacle, and it is possible to understand concrete, yet complex, examples. The goal is to build on our low dimensional intuition to better understand high dimensional objects.