Moduli spaces of compact RCD(0,N)-structures.

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of -structures. First, we relate the convergence of -structures on a space to the associated lifts' equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity.

Sharp Cheeger-Buser Type Inequalities in Spaces.

The goal of the paper is to sharpen and generalise bounds involving Cheeger's isoperimetric constant and the first eigenvalue of the Laplacian. A celebrated lower bound of in terms of , , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on in terms of was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below.