Geometry of scalar curvature, that is comparable in scope to symplectic geometry, mediates between two worlds: the domain of rigidity, one sees in convexity and the realm of softness, characteristic of topology, such as the cobordism theory.
The aim of this course is threefold:
1. An overview of old and new results, mostly, but not exclusively, on the rigidity side, of manifolds X with positive and, more generally, bounded from below scalar curvatures Sc(X), along with a brief introduction to main techniques.
2. Proof of new geometric comparison type inequalities for Riemannian manifolds X with lower bounds on Sc(X) and on mean curvatures of the boundaries of X.
3. Discussion of open problems concerning Sc>0.
Emmanuel Ullmo & Pierre Pansu