Harmonic Maps between Hyperbolic Spaces, Realizing Number Fields as Invariant Trace Fields, and Constructing Surface Subgroups in Hyperbolic Groups

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Pure mathematics fosters the development of ideas that are later utilized in natural sciences such as physics and biology. In physics, the universe is described as a 3-dimensional space; studying the geometry and topology of 3-manifolds may prove important in answering fundamental physical questions. This project investigates and develops techniques that involve an interplay between coarse hyperbolic geometry and statistical properties of various geometric flows, which are becoming indispensable tools in proving results from a range of fields. Although geometry and dynamics of such flows are not always necessary to state or prove these breakthroughs, they give the 'right' way of thinking about them and are likely to catalyze further progress. Through participation in this research project, the next generation of graduate students will be introduced to these concepts. The PI will study questions about geometry of hyperbolic groups and negatively curved manifolds. In connection with the Cannon Conjecture, the question of whether a hyperbolic group whose boundary is the 2-sphere contains an abundance of quasi-convex surface subgroups will be addressed. In a different direction, a Dirichlet type problem of finding harmonic mappings with prescribed quasi-symmetric boundary values will be studied, with particular emphasis on the Schoen Conjecture.