The Principal Investigator (PI) proposes to work on a series of projects that are centered around the notion of non-positive curvature. Most of the projects in the proposal are interdisciplinary in nature, and will likely require solutions that involve a mix of techniques from various different fields (differential and metric geometry, algebraic and differential topology, etc.). The PI's projects loosely fall into three broad categories: 1. Topology of CAT(0)-manifolds (and more generally, of aspherical manifolds): the PI plans on focusing on the distinction, for manifolds, between Riemannian non-positive curvature and metric non-positive curvature. The PI also proposes some homological criterion for finding lattice subgroups. Another project involves studying some new examples of aspherical manifolds. 2. Riemannian geometry: the PI plans on exploring manifolds of non-positive curvature from the viewpoint of Riemannian geometry. This includes studying the distribution of lengths of geodesics (arithmetic progressions), constructing new phenomena in spectral theory (can one hear the Hauptvermutung?), and studying the structure of almost non-positively curved manifolds. 3. Algebraic K-theory: the PI plans on continuing his research on computational aspects of algebraic K-theory in the presence of non-positive curvature. Some concrete goals include establishing the Farrell-Jones Isomorphism conjecture for classical Kleinian groups, constructing "small" models for classifying spaces for mapping class groups, and finding efficient algorithms for computing the lower algebraic K-theory of 3-orbifold groups.
Geometry is usually focused on understanding objects quantitatively. Topology is focused on understanding objects qualitatively. For example, a sphere and an ellipsoid are quantitatively different: the sphere contains a point (the center) through which all "slices" are identical, while an ellipsoid contains no such point. But qualitatively they are the same: one can "squish" an ellipsoid and turn it into a sphere (without puncturing or tearing). In contrast, a sphere and a donut are qualitatively different; the donut has a "hole", and you cannot turn a sphere into a donut without puncturing. The projects in this proposal study the relationship between geometry and topology. In other words, how do qualitative properties of a space constrain (and in turn, are constrained by) the quantitative properties of the space.
