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 |
|---|---|---|
| 06. Oct. 2023 | Scheduling | |
| 13. Oct. 2023 | Djamel Kebiche | Singular Sobolev spaces and applications to degenerate elliptic partial differential equations. AbstractThe talk aims to present new Sobolev-type spaces, called singular Sobolev spaces. We highlight that in a singular Sobolev space, elements are not necessarily locally integrable, and hence a new notion of weak derivative has been introduced to replace the notion of distributional derivative. The main aim of the construction is the study of the solvability of degenerate elliptic partial differential equations. Indeed, assumptions under which the solution is not locally integrable are given. |
| 20. Oct. 2023 | Alessio Vardabasso | $L^2$-normed $L^\infty$-modules - How to define vector fields on a metric measure space. AbstractThis talk will be an introduction to first order calculus, in the Sobolev sense, on metric measure spaces. The main focus will be the algebraic structure known as $L^2$-normed $L^\infty$-module, which allows us to define 1-forms/vector fields on a metric measure space, hence in particular weak differentials of Sobolev functions. If time allows, we will discuss the concept of infinitesimally Hilbertian metric measure spaces, which is crucial, for example, in the definition of the RCD(K,$\infty$) class. |
| 27. Oct. 2023 | Zoia Petrovskaia | 2 different approaches to proving the inverse function theorem on $\mathbb{R}^n$ AbstractInverse function theorem as a consequence of the Hopf-Rinow theorem. Inverse function theorem on simple-connected spaces. |
| 03. Nov. 2023 | Tobias Beran | Tau-convexity curvature bounds for Lorentzian pre-length spaces AbstractThe equivalence of triangle comparison curvature bounds and tau-convexity curvature bounds. |
| 10. Nov. 2023 | cancelled | |
| 17. Nov. 2023 | Matteo Calisti | The future is not always open AbstractI present the article of the same name, of which M. Kunzinger and R. Steinbauer are two of the authors, where the breakdown of many fundamentals of Lorentzian causality theory is shown in low regularity. For example, the chronological futures defined via locally Lipschitz curves may be non-open and may be different from the ones defined via piecewise $C^1$ curves. Explicit examples of this odd and interesting behaviour will be presented. |
| 24. Nov. 2023 | Carl Rossdeutscher | High genus CMC surfaces on the 3-sphere and MOTS in de-Sitter AbstractWe will talk about a recent result to construct smooth families of CMC surfaces in the 3-sphere deforming the Lawson surface for high genus. We will also talk about its application to find tubes of MOTS in de-Sitter space. |
| 01. Dec. 2023 | Felix Rott | Toponogov globalization in Lorentzian length spaces AbstractWe present a Toponogov globalisation theorem for Lorentzian length spaces, stating that lower curvature bounds (in the sense of triangle comparison) globalise. This is done via a "cat's cradle" construction used by Lang and Schröder in the metric setting. Applications may include generalisations of previously known results (e.g. splitting theorem), as well as stability of lower curvature bounds under convergence and dimensional homogeneity. This is joint work with Tobias Beran, John Harvey and Lewis Napper. Based on https://arxiv.org/abs/2309.12733 |
| 15. Dec. 2023 | Argam Ohanyan | The Cheeger-Colding almost splitting theorem AbstractGeometric results are often very rigid in their assumptions on curvature, i.e. slight deviations lead to the failing of a given result only in very special cases. An important class of such rigidity results are splitting theorems, which say that a geometric space of nonnegative curvature containing an infinitely long distance-realizing curve must split off that curve isometrically. In Riemannian geometry, this type of result under non-negative Ricci curvature bounds goes back to Cheeger and Gromoll in 1972. Some time later, in 1996, Cheeger and Colding proved a quantitative version of this result, essentially saying that a very long "almost" segment and "almost" non-negative Ricci curvature lead to "almost" splitting locally near the segment. This result has proven to be of great importance in Riemannian (and metric) geometry, leading to the study of Ricci limit spaces which, in our modern understanding, are special examples of spaces covered by the RCD-theory (pioneered by Ambrosio-Gigli-Savaré; expanding the CD-theory of Lott-Sturm-Villani).Historical accounts of the Cheeger-Colding almost splitting theorem (in particular, the original Cheeger-Colding paper) are often difficult to parse, and it will be the main purpose of this talk to provide a guideline to the proof. We will address the most important steps: the Abresch-Gromoll excess estimate, harmonic replacements, the segment inequality, and the almost Pythagoras theorem. We will follow the presentation given in the recent paper by Galloway, Khuri and Woolgar. |
| 12. Jan. 2023 | Phillip Bachler | Optimal Transport on Riemannian Manifolds AbstractIn this talk, we will start with an overview of the Monge and Kantorovich problem in historical perspective based on Monge’s original work. Thereafter, we sketch a modified proof for McCann’s theorem, avoiding more advanced tools of Riemannian geometry. As a by-product, we will see that technics from non-smooth analysis for the proof arise naturally. Finally, we take a glance at Ricci curvature bounds on Riemannian manifolds in the context of the Shannon entropy, based on the work of Sturm. |
| 19. Jan. 2023 | Sebastian Gieger | Area and Volume Comparison in Lorentzian Geometry AbstractThis talk will be based on the paper “Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems” by Jan-Hendrik Treude and James D. E. Grant.We will study certain distance functions and see that the shape operator on the level sets of these functions satisfies a Riccati equation. We can then apply Riccati comparison techniques to get bounds on the mean curvature of these level sets and as a direct consequence bounds on the area. In particular we will derive area and volume comparison theorems similar to the Bishop-Gromov Theorem in Riemannian Geometry. As a direct result we will get an alternative proof for the Hawking Singularity Theorem. |
| 26. Jan. 2023 | Benedikt Miethke | $C^0$-inextendibility of the Schwarzschild spacetime AbstractThis talk aims to give a gentle introduction to $C^0$-extensions of smooth time-oriented Lorentzian manifolds. In particular, we will discuss usual notions and tools used to study $C^0$-(in)extendibility of a given spacetime. Using these we will be able to prove the $C^0$-inextendibility of Minkowski space and sketch the proof for the Schwarzschild spacetime, which is a relatively recent result from Jan Sbierksi (2016/18). |