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 |
|---|---|---|
| 07.03. | scheduling | |
| 14.03. | Alessio Vardabasso | Distributional and synthetic curvature bounds for non-smooth Riemannian metrics AbstractThe Cartan-Alexandrov-Toponogov Theorem (sometimes just Toponogov's Theorem) is a cardinal theorem in comparison theory for smooth Riemannian manifolds. In short, it relates uniform bounds on the sectional curvature of a Riemannian manifold to specific inequalities on the internal distances and angle widths of geodesic triangles when compared to similar triangles in two-dimensional "model spaces" - spheres, hyperbolic spaces and the euclidean plane. While most of the theory of smooth Riemannian geometry comfortably works for metrics of regularity C^2, below that threshold one must resort to distributional derivatives and other non-smooth tools to define important objects such as curvature tensors. A natural question then arises: which theorems still hold at these lower regularities? In this talk, after an introduction on distribution theory on manifolds, we will see how we can partially generalize the C.A.T. Theorem to metrics of C^1 and Lipschitz regularity. |
| 21.03. | Luca Benatti | Taming geometric inequalities using partial differential equations Abstract"The ball is the only domain in Euclidean space that minimizes area for a given volume." This simple statement encapsulates the full essence and power of geometric inequalities: the (scale-invariant) ratio between an object's area and volume is always bounded from below by a dimensional constant. By measuring it, I can determine whether the object is a ball.But what if the space around us is curved? And what if we consider other geometric quantities, such as the mean curvature? Through some selected examples, I will introduce a powerful PDE-based technique that has been successfully applied to address these questions. |
| 28.03. | Inés Vega | Integrated local energy decay estimates for solutions to the wave equation in the black hole exterior of sub-extremal Reissner-Nordström-de Sitter spacetimes - part 2 AbstractConsider a non-rotating spherically symmetric charged black hole with mass $M > 0$ and a charge $Q\neq0$, in a de Sitter background of positive curvature ($Λ > 0$). Taking solutions to geometric wave equations on the exterior region of this black hole, we use a physical-space-based method for deriving the leading-order late-time behaviour of integrated local energy decay estimates of solutions.These estimates could be used for deriving the precise leading-order late-time behaviour of asymptotics and energy decay. Our method relies on exploiting the spatial decay properties of time integrals of solutions. With them, we are able to derive the existence and precise genericity properties of energy of the solutions and obtain uniform decay estimates of local energy in time. |
| 04.04. | Omar Zoghlami | Isoperimetric characterization of geodesics in the sub-Lorentzian Heisenberg group and failure of the $\mathrm{TMCP}$ condition. AbstractAfter a brief overview of the Heisenberg group and its sub-Lorentzian structure, I will give a description of how to compute the geodesics of this space by means of an isoperimetric problem in the Minkowski plane. Such problem is completely analogous to Dido's problem in the Riemannian setting and allows us to find geodesics without relying on the Pontryagin's Maximum Principle.I will also discuss (time permitting) how one can use the expression of the geodesics given by the exponential map to disprove the validity of the $\mathrm{TMCP}(0,N)$ condition for any choice of dimension $N$. Such phenomenon does not occur in the Riemannian Heisenberg group and thus provides a relatively well-behaved example of a geometric property which holds in the Riemannian setting but does not in its Lorentzian counterpart. |
| 11.04. | Miguel Manzano | Null matching spacetimes |
| 02.05. | tba | |
| 09.05. | Samuël Borza | Sub-Lorentzian Martinet distribution |
| 16.05. | Davide Carazzato | tba |
| 23.05. | Sebastian Gieger(+Tobias Beran?) | Lorentzian Cartan-Hadamard |
| 30.05. | Miguel Prados | causal completions |
| 06.06. | Karim Mosani | tba |
| 13.06. | Darius Erös | tba |
| 20.06. | Chiara Rigoni | tba |
| 27.06. | Luca Mrini | tba |