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.
One way of studying infinite groups is by analysing
their actions on classes of interesting spaces. This is the case
for Kazhdan's property (T) and for Haagerup's property (also called a-T-menability),
formulated in terms of actions on Hilbert spaces and relevant in many areas
(e.g. for the Baum-Connes conjectures, in combinatorics, for the study of expander graphs, in ergodic theory, etc.)
Recently, these properties have been reformulated for actions on Banach spaces,
with very interesting results. This talk will overview some of these reformulations
and their applications. Part of the talk is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.