This seminar is an informal forum where members of the DIANA group meet to discuss topics of interest. We meet on a weekly basis. The programme for these meetings will be advertised below, and by email.
If you wish to be added to (or removed from) our email list, please contact tobias.beran@univie.ac.at: subscribe or unsubscribe.
The the seminar takes place every Friday at 09:45 am in SE 07 and streamed via moodle and will be announced by email weekly.
Anyone interested is welcome to attend.
| Date | Speaker | Title |
|---|---|---|
| 03.10. | Scheduling | |
| 10.10. | David Lenze | Isometric rigidity of the Ebin metric AbstractIn 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. |
| 17.10. | (Lausanne) | |
| 24.10. | Tobias Beran | Coordinates for Lorentzian CBB – an overview Abstract(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. |
| 31.10. | Paul Haberger | Null Geometry Abstract(Smooth) 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". |
| 07.11. | Davide Manini | On the geometry of synthetic null hypersurfaces and the Null Energy Condition AbstractIn the talk, 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 ($NC^e(N )$), 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 it applies to low-regularity spacetimes. A key property of $NC^e(N )$ is the 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 celebrated Penrose singularity theorem to continuous spacetimes and we prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic null energy condition. |
| 12.11. (extra) | Roland Steinbauer | Colloquium ? |
| 14.11. | Mauricio Adrian Che Moguel | Topological data analysis and spaces of persistence diagrams AbstractIn 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. |
| 21.11. | Stefano Saviani | Wasserstein gradient flows Abstract"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 meant to be an overview of the key aspects of this theory, starting with gradient flows in Hilbert spaces as a motivation and heuristics. The presentation will focus on the three necessary ingredients to define Wasserstein gradient flows: the notion of "tangent" to an (absolutely continuous) curve of measures, the displacement convexity of the functional and the "Wasserstein" subdifferential calculus. The aim is to revisit Otto’s estimates from this abstract framework. Time permiting, we will discuss the Benamou-Brenier formula, and its link to the characterization of tangent vectors in the Wasserstein space." |
| 28.11. | Leonardo Garcia-Heveling | Isometry groups of spacetimes AbstractThe 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 will 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. |
| 05.12. | tba | |
| 12.12. | Miguel Manzano | Conformal something |
| 19.12. | Waiho Yeung | Laplacian of the distance function |
| 09.01. | Joe Barton, Samuël Borza, Jona Röhrig | Causal set something |
| 16.01. | Marta Sálamo Candal | tba |
| 23.01. | Karim Mosani | C^0 extensions ? |
| 30.01. | Peter Michor |