Mathematics

The study of properties of smooth spaces which are maximal in some sense has had its natural development in the investigation of the stability of such properties, i.e. what being "close'' to a maximal space implies. This, in turn, has led - for compactness reasons - to enlarge the domain under investigation in order to admit the presence of nonsmooth geometric objects, which may appear, for instance, as limits of smooth manifolds. The starting point of this - by now long - journey of discoveries has been the realization that the class of (compact) manifolds with bounded dimension and negative part of the Ricci curvature is compact with respect to the measured Gromov-Hausdorff (mGH) topology.If the properties under investigation are dimension invariant, however, mGH convergence of extremizing sequences cannot be guaranteed.A promising alternative for these situations is the so called concentration topology, introduced by Gromov. This choice allows for great compactness properties, but is known to produce unintuitive phenomena, such as the one point space being the limit of the N-sphere as N approaches infinity.In this talk I will give a brief overview of the notions of mGH and concentration topologies and introduce the natural quantities (Cheeger energy, heat flow, Ricci curvature bounds) which we are interested in studying. I will proceed by recalling some known stability statement and, finally, I will present some new stability results which extend what is known in the mGH case to the realm of convergence in concentration.This results come from a work (in preparation) in collaboration with N. Gigli.