Understanding the kind of geometry (or curvature) a given manifold will admit is a central theme in modern geometry. The earliest result in this subject is the classical theorem of Gauss-Bonnet which implies for example, that a torus cannot admit a positive curvature geometry.
Attention has recently shifted to a more general problem, namely: given a manifold which admits geometries of a particular type, say with positive curvature, what can we say about the "space of all such geometries"? This is a difficult object to imagine but it may help to think of a path in this space as an animation of a geometric manifold over time, maintaining positive curvature at every stage. The easiest curvature to work with is the scalar curvature, yet even here relatively little is known about the topology of this space. Also, until recently it was thought that it's higher homotopy groups may all be trivial.
In this talk, I will discuss a very surprising result which shows non-triviality of the higher homotopy groups of this space (joint work with Botvinnik, Hanke and Schick), as well as a brand new theorem which opens the door for many more exciting results.