Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, May 1, 2017 - 12:00 to 13:00

Location:

Stag 163

Guest Speaker:

Institution:

University of Oregon

Abstract:

For a fixed compact, smooth manifold M, one can ask if M admits a Riemannian metric for which the sectional curvatures, Ricci curvatures, or scalar curvature have a constant sign. The sign of the sectional curvatures places a significant restriction on the topology of M. If positive, Gromov shows that the sum total of Betti numbers of M is bounded by a dimensional constant. If negative (and simply connected), M is contractible. For Ricci curvature, Lohkamp proved that any manifold admits a metric with negative Ricci curvatures. In the simply connected case, positive scalar curvature has been completely determined by Gromov-Lawson and Stolz in terms of a topological invariant. The remaining case, when Ricci curvature is positive, remains incomplete, and in many situations existence is known only through explicit constructions of examples. To this end, I will explain Perelman's construction of Ricci-positive metrics on connected sums of complex projective planes and how it can be generalized to every complex projective space, every quaternion projective space, the Cayley plane, and any combination of connected sums of the these three classes of spaces.

Host: