Challenges in Negative and Nonpositive Curvature

Suny At Binghamton, Binghamton NY

Investigators

Abstract

These projects will study some fundamental questions about negatively curved and nonpositively curved metrics. For instance, questions about the existence of such geometric structures on spaces, that is, how to construct nonpositively or negatively curved metrics. This includes not just general geodesic metrics but also the more regular smooth versions, i.e, Riemannian metrics. The principal investigator plans to continue his study of the topology of the space of negatively curved metrics, and its variations: the Teichmuller space of negatively curved metrics and the moduli space of negatively curved metrics, together with its generalizations to nonpositive curvature. He also plans to continue his work on some fundamental questions about CAT(0) spaces, the existence of geodesics in spaces of curvature less than infinity, and continue his research on the general area of manifold topology and its applications to geometry. 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.