LOGOS Seminar

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.

(joint work with John Harvey, Felix Rott and Clemens Sämann) I will define strainers and the corresponding coordinate map, and show it is continuous and open. If this map is not a local homeomorphism, a way of increasing of the dimension of the strainer is presented. This then gives a coordinate theorem for finite dimensional LLS with CBB: near each point there is either an open set homeomorphic to $\mathbb R^n$, or a nested sequence of open sets and corresponding sequence of strainers (which one should interpret as the space being infinite dimensional). If time permits, I will show that the time separation function lies between two flat time separation functions, making the coordinate map weakly bi-Lipschitz.

Null hypersurfaces of spacetimes play a prevalent role in General Relativity, where they describe various kinds of horizons. When it comes to studying the geometry of null submanifolds in general, the degeneracy of the metric causes technical difficulties that must be properly addressed. The goal of the talk is to give an introduction to the geometry of null hypersurfaces and to develop the necessary tools to handle their degenerate structure. In particular, this will be applied by presenting a proof of the Penrose Incompleteness Theorem which makes heavy use of the structure of achronal boundaries. We will closely follow ideas by Prof. Dr. Gregory J. Galloway; see especially https://www.math.miami.edu/~galloway/vienna-course-notes.pdf." "2024-11-07";"Davide Manini";"On the geometry of synthetic null hypersurfaces and the Null Energy Condition";"Winter 2025";"I will present a joint work with Fabio Cavalletti (Milan) and Andrea Mondino (Oxford), where we develop a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple $(H,G,m)$: $H$ is a closed achronal set in a topological causal space, $G$ is a gauge function encoding affine parametrizations along null generators, and $m$ is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the synthetic null energy condition $\mathrm{NC}_e(N)$, defined via the concavity of an entropy power functional along optimal transport, with parameterization given by the gauge $G$. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and applies to low-regularity spacetimes. A key property of $\mathrm{NC}_e(N)$ is stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov–Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking’s area theorem. Moreover, we extend the Penrose singularity theorem to continuous spacetimes and prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic null energy condition.

In this talk, I will give an introduction to topological data analysis (TDA), with an emphasis on the notion of persistence diagrams. These objects, arising in algebraic topology, provide a concise, quantitative way to visualise the homological information carried by filtrations of topological spaces. In TDA, filtrations are often built from data sets using Vietoris–Rips complexes or similar constructions. One can study persistence diagrams from a geometric point of view, by equipping the space of diagrams with metrics inspired by optimal transport. I will discuss this connection and what it reveals about the metric structure of spaces of persistence diagrams." "2024-11-21";"Stefano Saviani";"Wasserstein gradient flows";"Winter 2025";"Felix Otto’s pioneering work 'The geometry of dissipative evolutions' introduced a geometric perspective on the porous medium equation, interpreting it as a gradient flow in the space of absolutely continuous probability measures equipped with the Wasserstein metric. This insight led to the rigorous framework developed by Ambrosio, Gigli, and Savaré, which formalizes Wasserstein gradient flows and extends Otto’s asymptotic estimates to a broader class of dissipative equations. This talk is an overview of key aspects of the theory, starting with gradient flows in Hilbert spaces as motivation and heuristics. The presentation focuses on three ingredients to define Wasserstein gradient flows: the notion of a tangent to an absolutely continuous curve of measures, displacement convexity of the functional, and the Wasserstein subdifferential calculus. The aim is to revisit Otto’s estimates from this abstract framework. Time permitting, we will discuss the Benamou–Brenier formula and its link to the characterization of tangent vectors in Wasserstein space.

The Myers–Steenrod theorem states that the isometry group of a compact Riemannian manifold is a compact Lie group. In Lorentzian signature, however, there are counterexamples: compact manifolds with non-compact isometry group. In this talk, we instead consider (non-compact) globally hyperbolic spacetimes satisfying a 'no observer horizons' condition. Our main result is that the isometry group acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy time function, and a splitting of the isometry group into two subgroups: a compact one corresponding to spatial isometries, and a trivial, $\mathbb{Z}$, or $\mathbb{R}$ factor corresponding to time translations. Time permitting, we will also discuss the conformal groups of these spacetimes. Based on joint work with Abdelghani Zeghib.

In General Relativity, horizons—understood as null hypersurfaces where the deformation tensor $K := \mathcal{L}_{\eta} g$ of a null and tangent vector $\eta$ satisfies certain restrictions—play a core role. They mark the boundary of causally relevant regions of a spacetime and reveal the existence of spacetime symmetries. The importance of horizons raises fundamental questions, such as under which conditions a spacetime admits a horizon, or what data must be prescribed on a horizon so that it induces a specific type of spacetime symmetry. I will present a new formalism for describing general horizons (i.e., characterized by any $K$), which enables the derivation of a generalized form of the well-known near-horizon equation. I will also establish the necessary and sufficient conditions for a non-degenerate totally geodesic horizon to be embeddable in a spacetime satisfying any (not necessarily $\Lambda$-vacuum) field equations. This will be done first in the case of arbitrary horizon topology, and then when the horizon admits a cross-section.

The distance function is crucial for understanding the structure of Riemannian Geometry. In 1958, Calabi proved a notable Laplacian comparison theorem, establishing an upper bound for the Laplacian of the distance function under the assumption of a lower Ricci bound. Cheeger later expanded on this by demonstrating that the upper bound holds globally in a distributional sense, particularly in his proof of the splitting theorem. This theorem forms a foundational element of Cheeger and Colding's work in studying geometry and limit spaces with Ricci lower bounds. In 2015, Gigli extended the concept of the Laplacian to metric measure spaces, thereby broadening the Laplacian comparison theorem to non-smooth contexts. This talk provides an overview of results obtained by Fabio Cavalletti and Andrea Mondino in 2019, who presented an explicit formula for the Laplacian of the distance function and established a lower bound for it. We will explore their construction and discuss some of its applications.

We will present the basics and some new results in causal set theory, a radical approach to quantum gravity in which spacetime is represented by a discrete causal graph. We formulate a new notion of curvature, inspired by Ollivier–Ricci curvature on metric graphs, using optimal transport between causal diamonds. We show that it recovers Ricci curvature on smooth Lorentzian manifolds. If time permits, we will also present numerical examples.

Timelike lower Ricci curvature bounds in smooth spacetimes are known to be characterised by the convexity of a suitable entropy functional along geodesics in the space of probability measures. Such geodesics are defined via optimal transport, in analogy with the Wasserstein distance in the Riemannian setting. Pioneering work in this direction was carried out by McCann and by Mondino–Suhr, who considered cost functions given by the $p$-th power of the time separation function for $0 < p \le 1$. In this talk, I will discuss an approach based on more general cost functions, namely Lorentz–Orlicz costs. I will introduce the associated optimal transport problem and explore classical aspects such as Kantorovich duality. I will then explain how these costs lift to the space of probability measures, inducing a spacetime structure and, in particular, a notion of geodesics. Finally, I will address the characterisation of timelike lower Ricci curvature bounds for this class of costs.

A spacetime $(M,g)$ is $C^0$-extendible if there exists a spacetime $(\tilde M,\tilde g)$ of the same dimension, equipped with a continuous Lorentzian metric, such that $(M,g)$ embeds isometrically as a proper subset of $(\tilde M,\tilde g)$. Otherwise, $(M,g)$ is called $C^0$-inextendible. In this talk, we outline Jan Sbierski’s proof of $C^0$-inextendibility of maximally analytic Schwarzschild spacetimes, and discuss how it can be adapted to more general black hole spacetimes.

This is joint work with M. Bauer and F.-X. Vialard. We present weak Riemannian metrics on six different spaces related to the space of all Riemannian metrics which fit together via Riemannian submersions. Their geodesics are like the unbalanced version of the Brenier–Otto optimal transport, with the most interesting version on the space of Riemannian metrics. Unfortunately, positivity of geodesic distance on this space is still unproved.