Quantitative implications of positive scalar curvature.

Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.