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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.

Abstract: Various concepts from smooth differential geometry allow generalisations to metric measure spaces. This includes a generalised first order calculus leading to a synthetic definition of the Sobolev space W^{1,2}.  A metric measure space is called infinitesimally Hilbertian if the associated Sobolev space W^{1,2} is a Hilbert space. A vast class of metric measure spaces arises from manifolds equipped with non-smooth Riemannian metrics. 

In this talk, we compare the classical differential calculus on manifolds with the synthetic first order calculus on metric measure spaces in the setting of a manifold M endowed with a low regularity Riemannian metric g. We then study under which regularity assumptions on g the induced metric measure space is infinitesimally Hilbertian. Part of the results are joint work with Andrea Mondino.

Abstract: A conjecture of Berger states that on a simply connected manifold $M$ in which all geodesics are closed, all geodesics have the same length. In this thesis, we prove that this conjecture holds for three dimensional manifolds. The proof is carried out in two parts. In the first part, we use the Morse theory of the free loop space of $M$ to deduce that the geodesic flow acts freely on the unit tangent bundle $T^1M$, or that it has at least two distinct singular orbit types.

In the second part, we apply methods from symplectic geometry and topology to investigate the topology of the symplectic orbifold $T^1M / \mathbb{S}^1$, which arises as the quotient of the unit tangent bundle $T^1M$ by the $\mathbb{S}^1$-action induced by the geodesic flow.

Combining these two approaches ultimately yields a proof of Berger’s conjecture for three-dimensional manifolds.

I review a recent approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective proved extremely fruitful in the Riemannian case (Alexandrov-, CAT(k)- and CD-spaces). I will try to convey the basics without any prerequisites of Lorentzian geometry and only in the end report on recent progress.