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. Oct. 2022 | Scheduling | |
| 14. Oct. 2022 | cancelled | |
| 21. Oct. 2022 | Michael Kunzinger | Synthetic vs. distributional lower Ricci curvature bounds AbstractWe compare and relate the two main approaches to defining Ricci curvature bounds for Riemannian manifolds with metrics of regularity below $C^2$. On the one hand, weak derivatives and the notion of positive distributions can be applied to metrics of low regularity. On the other hand, optimal transport theory gives a characterization of Ricci bounds via displacement convexity of an entropy functional, which carries over even to the general setting of metric measure spaces. Both approaches are compatible with the classical definition via the Ricci tensor (and hence with each other) in the case of a smooth metric. We show that distributional bounds imply entropy bounds for $C^1$ metrics and that the converse is true for metrics of regularity $C^{1,1}$. |
| 28. Oct. 2022 | Chiara Schindler | A Hitchhiker's Guide to the Disintegration Theorem AbstractThe Disintegration Theorem is a result in measure theory that is very useful e.g. in optimal transport. It allows the decomposition of a measure on a product space $X \times Y$ into a family of measures on $Y$. While the full details of its proof are somewhat lengthy, the underlying structure turns out to follow an approachable, clear and actually really beautiful path. In my talk, I will guide you through the proof of this crucial theorem. |
| 04. Nov. 2022 | Argam Ohanyan | The splitting theorem for Lorentzian length spaces AbstractSplitting theorems are a class of results found in various fields of geometry which state that the relationship between positive curvature and (non-)realization of distance is rigid. In this talk, we present such a result which was recently obtained in the setting of Lorentzian length spaces, a synthetic analogue of spacetimes. This is based on joint work with Tobias Beran, Felix Rott and Didier A. Solis. |
| 11. Nov. 2022 | Tobias Beran | Equivalent formulations of timelike curvature bounds in Lorentzian length spaces AbstractI discuss different versions of timelike curvature bounds in Lorentzian pre-length spaces: Triangle comparison, monotonicity comparison, angle comparison and the four-point condition. I prove them all to be equivalent. |
| 18. Nov. 2022 | Felix Rott | Gluing of Lorentzian length spaces and the causal ladder AbstractGluing operations for Lorentzian length spaces were recently introduced, which provide a very general way of constructing new spaces out of old ones. After a brief introduction into the general theory of Lorentzian length spaces, we will discuss Lorentzian gluing constructions in more detail. Then we showcase some compatibility results concerning the preservation of various properties of Lorentzian pre-length spaces under gluing, in particular steps of the causal ladder. We conclude with a discussion about properties which seem not so easily transferable. Talk is based on https://arxiv.org/abs/2209.06894 |
| 25. Nov. 2022 | Matteo Calisti | Timelike Ricci bounds for low regularity spacetimes by optimal transport AbstractWe prove that a globally hyperbolic smooth spacetime endowed with a $C^1$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions obeys the timelike measure-contraction property $TMCP_p$. This embeds the class of spaces of lowest regularity for which classical singularity theorems are known into the synthetic Lorentzian setting of Cavalletti-Mondino. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $C^{1,1}$, in fact the stronger timelike curvature-dimension condition $TCD_p$ holds. In this regularity, we also obtain uniqueness of chronological $\ell_p$-geodesics and $\ell_p$-optimal couplings. |
| 02. Dec. 2022 | Chiara Rigoni | Lie brackets of nonsmooth vector fields and commutation of their flows AbstractIt is well-known that the flows generated by two smooth vector fields commute, provided that the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this result cannot be generalize to a.e. differentiable vector fields admitting a.e. unique flows, but however this extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (and admitting a regular Lagrangian flow). This is a joint work with E. Stepanov and D. Trevisan. |
| 09. Dec. 2022 | Carla Mladek | Global hyperbolicity through the eyes of the null distance AbstractIn their paper from 2016 C. Sormani and C. Vega introduced a "null distance" on a spacetime equipped with a time function. It is a pseudometric and in many relevant cases it even is a metric that induces the manifold topology. A. Burtscher and L. Heveling proved that global hyperbolicity of a spacetime is equivalent to the existence of a time function such that the manifold equipped with the null distance (with respect to this time function) is metrically complete. They also obtained a Bi-Lipschitz estimate for the null distance and proved that for globally hyperbolic spacetimes that admit a Cauchy temporal function the null distance encodes causality. |
| 16. Dec. 2022 | Carl Rossdeutscher | Time evolution of merging marginally trapped surfaces |
| 13. Jan. 2022 | Benedict Schinnerl | Hawking-type singularity theorems for worldvolume energy inequalities |
| 20. Jan. 2022 | Kevin Islami | Fixed point Theorems for generalized smooth functions AbstractIn this talk we will first introduce the ring of Robinson-Colombeau, the sharp topology and define the basic notions associated with generalized smooth functions. Then we will discuss the finite Banach fixed point Theorem with its proof. The second fixed point Theorem that will be presented is Brouwer's fixed point theorem and how it is needed in the proof of Pontryagin's maximum principle. |
| 27. Jan. 2022 | Sekar Nugraheni | Natural definition of generalized holomorphic functions AbstractIn this talk, we will introduce and discuss how to define a generalized holomorphic function (GHF) without already assuming the Cauchy-Riemann equations but using a natural definition of complex differentiability. This goal can be accomplished using two gauges to define a new notion of hyperlimit function and of little-oh. |