Event Type:

Department Colloquium

Date/Time:

Monday, October 4, 2010 - 09:00

Local Speaker:

Abstract:

A dominant theme in modern geometry is the relationship between curvature

and topology. The earliest result in this subject is the classical theorem

of Gauss-Bonnet which relates the Gaussian curvature of a 2-dimensional

Riemannian manifold with its Euler characteristic. This theorem implies,

for example, that the torus cannot be given a metric whose curvature is

everywhere strictly positive or negative.

In dimensions greater than 2, these problems become much more complicated.

In particular, there are now three notions of curvature which are commonly

studied: the sectional, Ricci and scalar curvatures. In this talk, we will

restrict our attention to the scalar curvature (this has proven to be the

easiest to analyze). We will discuss the well-studied problem of whether

or not a manifold admits a metric with strictly positive scalar curvature,

as well as some more recent problems where far less is known.