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 |
|---|---|---|
| 08.03. | Scheduling | |
| Tue 12.03. 14:00 | Moriz Frauenberger | Master defense: The Ehlers-Kundt conjecture and its failure in the impulsive case |
| 15.03. | Miguel Manzano | Geometry of abstract null hypersurfaces AbstractIn most of the literature, hypersurfaces are considered as embedded in an ambient space. This approach, however, turns out to not be advantageous in all situations. An example of this are initial value problems in General Relativity, where one needs to prescribe some data in a hypersurface that, a priori, must be considered as not embedded in any space. In this talk I will present some results concerning non-embedded hypersurfaces. In particular, I will show that one can codify curvature information at a purely non-embedded level. I will also introduce new non-embedded notions of Killing horizons of order zero and one. These notions happen to generalize the standard concepts of non-expanding, weakly isolated and isolated horizons to general topology of the horizon and to horizons admitting fixed points. Finally, I will derive the so-called generalized master equation for general null hypersurfaces admitting a null and tangent privileged vector field. |
| Mon 18.03. 09:00 | Felix Rott | PhD defense: Fundamental constructions in Lorentzian length spaces |
| 22.03. | Celvin Stanko | Fornberg–Whitham equation AbstractThis talk will give an introduction to dispersive water waves and the phenomena of wave breaking. In particular, we are talking about the Fornberg-Whitham equation, which is also known under the name of Burgers-Poisson equation. The focus will be on developing conditions for wave breaking and presenting various solution concepts. The results are based on the paper ”Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation” by Günther Hörmann. |
| 12.04. | Sebastian Gieger | Areas and Volumes for Null Cones AbstractThis talk will be based on the paper “Areas and Volumes for Null Cones” by James D. E. Grant.We will examine slices of null cones and their properties. In particular we will find a connection between the area and the curvature of these manifolds. Using Riccati techniques we can then find bounds for the null-shape operator as well as the null-mean curvature and subsequently get bounds for the area of slices of null cones. As a result we obtain area and volume comparison theorems reminiscent of the comparison theorems of Bishop-Gromov and Günther, known from Riemannian Geometry. Finally, we will apply these results to estimate the null mean curvature in Ricci-flat four-manifolds. |
| 19.04. | Chiara Rigoni | An introduction to the theory of RCD spaces and their differential structure AbstractIn the first part of this talk, I will recall the two main approaches to the theory of synthetic Ricci bounds: the Lagrangian approach, based on the theory of Optimal Transport, and the Eulerian one, based on the theory of Dirichlet forms. Then I will present in which sense a general metric measure spaces naturally carries a first order differential structure and how we can develop a second order calculus on spaces with Ricci curvature bounded from below, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature. |
| 26.04. | Tobias Beran | The four-point condition and realizability AbstractIn this not well-prepared talk I will discuss brand new results on the different versions of the four-point condition for Lorentzian length spaces for (sectional) curvature bounds from below and the relation to realizability in three-dimensional model spaces. Maybe I will also talk about the metric four-point condition and curvature bounds from above. |
| 03.05. | Eduard Nigsch | Bits and Pieces of Hypocoercivity AbstractIn the context of dissipative evolution equations, hypocoercivity is the phenomenon of exponential convergence to the equilibrium state despite degenerateness of the dissipative part. In my talk I will give a introduction to various results and concepts surrounding this notion, explain the notion of hypocoercivity index for finite-dimensional ODEs, and discuss its extension to the Hilbert space setting. |
| 10.05. | Lara Lichtnecker | Blowup for $L^2$-critical Schrödinger equation AbstractThe first part of this talk will be an introduction about the power-nonlinear Schrödinger Equation and general results on well-posedness of the problem.The main result will be M. Weinstein’s variational characterisation of the ground state in the case of critical exponent in H1, where global well-posedness is no longer given. This characterisation gives a sufficient condition for global well-posedness and on the other hand provides a construction used by Merle and Raphael, who showed an upper bound on the blow-up rate of solutions. |
| 17.05. | Matteo Calisti | Nonsmooth D'Alembertian comparison AbstractThis talk is about an ongoing work with many other people where we introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the weak subslope of an arbitrary time function, which plays the role of the (Lorentzian) modulus of its differential and it satisfies certain chain and Leibniz rules. Moreover, we introduce an object that will lead to a nonlinear yet elliptic q-d’Alembertian, in terms of which we establish a comparison theorem in a weak form under the timelike measure-contraction property. |
| 24.05. | cancelled | |
| 31.05. | Alessio Vardabasso | Cheeger-Gromoll splitting for $C^1$ Riemannian metrics AbstractIn 1971 J. Cheeger and D. Gromoll proved that a smooth Riemannian manifold with non-negative Ricci curvature that contains a geodesic line is isometric to the product manifold between the real line and another smooth Riemannian manifold with non-negative Ricci curvature. This result is commonly known as the Splitting Theorem. After a brief introduction on distributions and local Sobolev spaces on manifolds, we will see how these tools allow us to generalize the aforementioned result to lower regularities of the metric tensor, namely $C^1$. |
| 07.06. | Argam Ohanyan | An elliptic proof of the Lorentzian splitting theorem AbstractThe Lorentzian splitting theorem is an important rigidity result in Lorentzian geometry and was established in the 80's by Eschenburg, Galloway and Newman. When compared to the Riemannian splitting theorem of Cheeger-Gromoll, significant additional difficulties arise due to the lack of ellipticity of the d'Alembertian (=Lorentzian Laplacian) operator. We propose a major simplification in the proof of the Lorentzian splitting theorem by considering a nonlinear analogue of the d'Alembertian, the so-called p-d'Alembertian (p<1), which turns out to be elliptic when considered on functions which are monotone in future time directions. Up to technicalities, one may apply the same elliptic arguments to the p-d'Alembertian as in the Riemannian case to the Laplacian, bringing the proof of the Lorentzian splitting theorem much closer to the Riemannian one.This is based on joint work together with Mathias Braun (U. Toronto), Nicola Gigli (SISSA), Robert McCann (U. Toronto) and Clemens Sämann (U. Oxford). |
| 14.06. | Darius Erös | Riemann curvature tensor on RCD-spaces AbstractBuilding on the framework of a Sobolev calculus and the theory of normed modules, Nicola Gigli proposed a distributional approach to defining a Riemann curvature tensor on any metric measure space satisfying an RCD-condition.Upon recalling some important facts and constructions pertaining to the first order calculus on such spaces, we will introduce second order differential objects such as the Hessian and the covariant derivative for vector fields, relying heavily on the existence of a space of suitably regular test functions. To be able to further give a meaning to a "second" covariant derivative, we moreover define a notion of a distributional covariant derivative in duality with the space of test vector fields. This will finally allow us to write down the Riemann curvature tensor in full generality. |
| 21.06. | Stephan Bornberg | The free Schrödinger Equation and its solutions AbstractIn this talk I will present the Schrödinger Equation applied to the free Potential $V=0$.Starting with that the expectation values of the momentum and position operator exactly follows classical trajectories, we will then solve the Schrödinger Equation by means of Fourier Transform and Convolution. Motivated by this, one will see how a wave packet defined by an amplitude- and phase-function evolves, by deriving closed form expressions. At the end of the talk I will show that this wave packet spreads in position, meaning that the variance of the position operator diverges. |
| 28.06. | Eric Ling | Rigidity aspects of Penrose's singularity theorem AbstractIn this talk, we present some rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship. This is joint work with Greg Galloway. |
| 10.07. | Tobias Beran | Defense: Timelike curvature comparison in Lorentzian length spaces AbstractThis thesis deals with constructions in the theory of Lorentzian (pre-)length spaces, properties of timelike curvature bounds and rigidity statements in that context. Lorentzian (pre-)length spaces are the Lorentzian analogue of metric and length spaces. First, hyperbolic angles are introduced and basic properties of them are discussed. Second, a globalization result for upper curvature bounds is given, akin to the Alexandrov patchwork, as well as a bound on the diameter for negative lower curvature bounds, mimicking the Bonnet-Myers theorem. Third, two rigidity results are proven for spaces with a lower curvature bound. |