RUI: Biquotients, Cohomogeneity-One Manifolds, and Double Disk Bundles

The Principal Investigator seeks to increase our understanding of mathematical shapes, called manifolds, which are of particular interest to both physicists and mathematicians. One important class of manifolds are those that are everywhere positively curved or flat. The term "positive curvature" refers to curving like a sphere; a very large sphere is approximately flat, and so has a small but positive curvature.

On the other hand, a very small sphere has a very large curvature. This project will focus on these objects through international collaborations, as well as through undergraduate research projects.

The PI's work is motivated by the recent breakthrough of Goette, Kerin, and Shankar on finding metrics of non-negative sectional curvature on so-called codimension one biquotient foliations. These spaces generalize the well known cohomogeneity one manifolds, and all have a decomposition as a union of two disk bundles glued together along their common boundary.

The PI will work on developing a method of classifying such objects. This includes studying the classification of non-simply connected biquotients, as well as developing obstructions at the level of cohomology for a manifold to admit such a structure.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.