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 |
|---|---|---|
| 01. Oct. 2021 | Scheduling | |
| 08. Oct. 2021 | cancelled | |
| 15. Oct. 2021 | Anton Baldinger | Penrose Diagrams |
| 22. Oct. 2021 | Argam Ohanyan | Lorentzian Length Spaces and Time Functions AbstractI will begin the talk by introducing Lorentzian length spaces as a synthetic analogue to classical spacetimes. Then, following the recent paper by Annegret Burtscher and Leonardo Garcia-Heveling, time functions on Lorentzian length spaces will be introduced and their connection to various levels of the causal ladder will be discussed. The talk will conclude by a discussion of a result which characterizes globally hyperbolic Lorentzian length spaces by means of Cauchy time functions and Cauchy sets. |
| 29. Oct. 2021 | Tobias Beran | Hyperbolic angles in Lorentzian pre-length spaces AbstractI will define hyperbolic angles in Lorentzain pre-length spaces, and do some properties (triangle inequality of angles, semi-continuity), and some applications (space of directions, future timelike tangent cone, exponential and logarithmic map). We probably won't be able to cover all. |
| 05. Nov. 2021 | Benedict Schinnerl | On the causal hierarchy of Lorentzian Length Spaces AbstractFollowing the paper titled "On the causal hierarchy of Lorentzian Length Spaces" by Hau, Pacheco and Solis, I will introduce several causality notions for Lorentzian Length Spaces and establish a causal hierarchy similar to the case of spacetimes. The talk will mainly focus on the upper levels of the causal ladder (i.e. stable causality, causal continuity and causal simplicity) as these have not been previously studied in the setting of Lorentzian Length Spaces. |
| 12. Nov. 2021 | Florian Lang, Stephan Schneider | Jump formulas in distributions |
| 19. Nov. 2021 | Gunter Wirthumer | On cosmic censorship AbstractLargely following the review paper "Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose" by Klaas Landsman (Jan. 2021), I will present the concept's key notions and ideas, and also address some of the difficulties in defining and proving it. |
| 26. Nov. 2021 | Clemens Sämann | A Lorentzian analog for Hausdorff dimension and measure AbstractWe define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that distinguishes between e.g. spacelike and null subspaces of Minkowski spacetime. The volume measure corresponding to its geometric dimension gives a natural reference measure on a synthetic or limiting spacetime, and allows us to define what it means for such a spacetime to be collapsed (in analogy with metric measure geometry and the theory of Riemannian Ricci limit spaces). As a crucial tool we introduce a doubling condition for causal diamonds and a notion of causal doubling measures. Moreover, applications to continuous spacetimes and connections to synthetic timelike curvature bounds are given. |
| 03. Dec. 2021 | cancelled | |
| 10. Dec. 2021 | Djamel Kebiche | Hahn-Banach theorem in Colombeau spaces AbstractI'll present in the seminar a suitable version of the Hahn-Banach theorem in the framework of Colombeau generalized functions and some of its applications like "separation of convex sets" and "Krein-Milman theorem". |
| 17. Dec. 2021 | Milos Vujicic | de Sitter and anti de Sitter space |
| 07. Jan. 2021 | Cezary Zaboklicki | Rademacher's Theorem AbstractI will present a proof of Rademacher's Theorem (if U is an open subset of $R^n$ and $f: U \rightarrow R^m$ is Lipschitz continouous then $f$ is differentiable almost everywhere in $U$) and in the process look more closely at some properties of Lipschitz functions concerning their differentiability. |
| 15. Jan. 2021 | Kevin Islami | Pontryagin’s Maximum Principle AbstractPontryagin’s maximum principle (PMP) is a Theorem that arises in the area of optimal control. In this talk we will first talk about how PMP arises and what the motivation respectively the intuition behind it actually is. Then we will discuss an example of an optimal control problem that is trying to put a particle that is moving on a line with a given bounded control force to rest at the origin in least time. Another point that makes PMP interesting is that it is closely related to Huygen’s principle. We will talk about how Huygen’s principle implies that optical rays maximize their normal velocity to the front and how PMP comes into play. The last part will discuss some of the background of the linearized and adjoint equations that arise in the formulation of PMP. |
| 21. Jan 2021 | Felix Rott | Gluing constructions in Lorentzian length spaces AbstractWe introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for $CAT(k)$ spaces, which roughly states that gluing is compatible with upper curvature bounds. We formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces. |
| 28. Jan 2021 | Matteo Calisti | Polar factorization of maps on Riemannian manifolds AbstractWe show that on a compact Riemannian manifold a Borel map can be factorized uniquely by the composition of the Brenier-McCann map which is the solution to the Monge optimal transportation problem in case the cost is the distance (squared) and a volume-preserving map. As said the proof follows directly from the solution to the optimal transport problem on Riemannian manifolds. We show then a linearization of this decomposition yields the Helmoltz/Hodge decomposition of vector fields. |