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.