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.
- Aug 18, 2014 - DMS 0904355 Herman Gluck There were two parts to this project. (1) In the first, we discovered and proved the sense in which Hopf fibrations and Hopf vector fields are optimal. Given a Hopf fibration of a round sphere by parallel great subspheres, we proved that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we viewed a unit vector field tangent to the fibers as a cross-section of the unit tangent bundle of the sphere, and proved that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success. We suspect that many natural geometric maps, such as Riemannian submersions of compact homogeneous spaces, are Lipschitz constant minimizers in their homotopy classes, unique up to composition with isometries of domain and range. (2) In the second part of this project, we studied the geometry and topology of fluid flows and plasma fields in Euclidean, spherical and hyperbolic settings, and their connections with knot theory. In one paper, we obtained integral formulas for twisting, writhing and helicity, and proved the theorem LINK = TWIST + WRITHE on the3-sphere and in hyperbolic 3-space. We then used these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. In two other papers, we associated, to each three-component link in the 3-sphere, a geometrically natural characteristic map from the 3-torus to the 2-sphere, and showed that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy, correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we gave an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural, in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus.
