We investigate the local concavity of time separation functions on Finsler spacetimes as a Lorentzian counterpart to Busemann’s convexity condition in metric geometry. We show that a Berwald spacetime is locally concave if and only if its flag curvature is nonnegative in timelike directions. This is joint work with Tobias Beran, Darius Erös and Felix Rott. 

Wasserstein spaces over Riemannian manifolds inherit a surprisingly rich geometry from their base. We study this geometry and prove that all isometries of the quadratic Wasserstein space arise from isometries of the Riemannian base if and only if that base admits no Euclidean de Rham factor; as a consequence, the Wasserstein space determines the manifold up to isometry. Along the way we survey some foundational work on the geometry of Wasserstein spaces from the last two decades. 

While for many metric spaces we can find a subset for which the Hausdorff-distance to the surrounding space is significantly larger than the Gromov-Hausdorff distance, for metric graphs it turns out that there are conditions to guarantee equality for a broad class of subsets. In this talk we discuss different conditions for this equality to hold, among them a condition involving the Hausdorff-distance being at most 1/12*sys(G), and we focus on the question if and by how much the constant factor 12 can or cannot be improved.

Wasserstein spaces over Riemannian manifolds inherit a surprisingly rich geometry from their base. We study this geometry and prove that all isometries of the quadratic Wasserstein space arise from isometries of the Riemannian base if and only if that base admits no Euclidean de Rham factor; as a consequence, the Wasserstein space determines the manifold up to isometry. Along the way we survey some foundational work on the geometry of Wasserstein spaces from the last two decades. 

Wasserstein spaces over Riemannian manifolds inherit a surprisingly rich geometry from their base. We study this geometry and prove that all isometries of the quadratic Wasserstein space arise from isometries of the Riemannian base if and only if that base admits no Euclidean de Rham factor; as a consequence, the Wasserstein space determines the manifold up to isometry. Along the way we survey some foundational work on the geometry of Wasserstein spaces from the last two decades. 

Gromov conjectured that the integral of the mean curvature of the boundary of a compact Riemannian manifold is bounded above by a constant that depends only on the induced metric on the boundary and a lower bound for the scalar curvature in the interior. We prove Gromov’s conjecture in various cases, including spin and non-spin manifolds, with an upper bound that also depends on a lower bound for the mean curvature. To achieve this, we use surgery to reduce the statement to the case when the boundary is a sphere. This case can be addressed using instances of the positive mass theorem.

This is joint work with Georg Frenck (University of Augsburg) and Sven Hirsch (Columbia University).

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.

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.

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: 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.

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.

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.