Rigidity of positive scalar curvature

We discuss a classical problem in global Riemannian geometry:
Loosely speaking, it asks: "how round can one make a given manifold"?

More concretely: if one avoids the obvious scaling trick: given a metric g with non-negative scalar curvature on a smooth manifold M,
can one find g' such that distances are not decreased when measured with g'
instaed of g, but such that the scalar curvature increases?

A classical result by Llarull states that this is not the case if g is the round metric on the sphere S^n (n>1);
Goette and Semmelmann generalize this to further classes of manifolds, in particular symmetric spaces of compact type with
non-vanishing Euler characteristic.

We discuss the possible approaches to study this question and a number of important generalizations/variations:

a) the condition is purely metric: therefore also the conclusion should hold under low assumptions on the regularity (this is joint work with Simone Cecchini, Bernhard Hanke, Lukas Schoenlinner
b) certain instances where the manifold has Euler characteristic zero (but is not a sphere): this is joint work with Georg Frenck, Lukas Schoenlinner, Thomas Tony.