CAT(0) spaces are a metric generalization of simply connected Riemannian manifolds with nonpositive sectional curvature. However, in this class there are many non-manifold examples. In this seminar I will address the following question: if a CAT(0) space is quasi-isometric to R^n, is it homeomorphic to R^n? I will show positive answers and interesting negative ones, even in dimension 2.

We consider metric spaces that are homeomorphic to a closed oriented manifold. Under mild assumptions, we show that they admit a top-dimensional integral cycle, i.e., an integral current without boundary. This cycle generates the top-dimensional integral homology, providing an analytic representation of the fundamental class. We also discuss applications to geometric and analytic problems, including isoperimetric inequalities and Lipschitz-volume rigidity. Based on joint works with Basso-Wenger and Soultanis.

Abstract: In 1970, Ebin introduced a natural L2-type metric on the infinite-dimensional space of Riemannian metrics over a given manifold. Though the infinite dimensional geometry of this space has been extensively-studied, a new metric perspective emerged in 2013 when Clarke showed that the completion with respect to the Ebin metric turns out to be a CAT(0) space.

Recently, Cavallucci provided a shorter and more conceptual proof of a strengthened result that in addition to being CAT(0) establishes the completion of the space of Riemannian metrics to depend only on the dimension of the underlying manifold.

In this talk I will sketch some of this recent progress and present new results which provide a complete characterization of the self-isometries of the space of Riemannian metrics with respect to the Ebin metric.