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 |
|---|---|---|
| 03. Oct. 2014 | Scheduling | |
| 10. Oct. 2014 | No Seminar | |
| 17. Oct. 2014 | No Seminar | |
| 24. Oct. 2014 | Lorenzo Luperi Baglini | An introduction to Nonstandard Analysis with some applications. AbstractWe assume that the audience of this seminar does not know NSA. We will try to explain its basic ideas both from an hystorical point of view and by mean of some simple examples. To avoid the heavy logical apparatus usually needed to construct nonstandard extensions, we will base our approach on some "topological" considerations. In the final part of the talk we will introduce a particular family of generalized functions, called ultrafunctions. We will finally show how ultrafunctions can be used to solve certain differential problems that have no classical solution. |
| 31. Oct. 2014 | Paolo Giordano | It would be beautiful if... Generalized Smooth Functions and their future steps. AbstractCauchy, Heaviside and Dirac invented the Dirac delta by using an intuitive point of view, which is still in use among several physicists and engineers. After the work of L. Schwartz, they can all end up this "definition" by citing the fairy tail: "Of course, this is not an ordinary function... and the important property is that it works as a functional". However, they still get into (temporary) trouble by making some calculations with the Heaviside function... This happens because Schwartz substituted the effect (they are functionals) with the cause (they are not ordinary functions). Generalized smooth functions (GSF) are set theoretical functions defined and with values in the ring $\tilde{\mathbb{R}}$ of Colombeau generalized numbers that share several properties with ordinary smooth functions, but include all Colombeau generalized functions (and hence all Schwartz distributions). E.g. GSF are closed w.r.t. composition, can be differentiated using (generalized) incremental ratios, have chain rule, primitives, integration by substitution, intermediate value theorem, mean value theorems, any form of Taylor theorem, extreme value theorem and a useful sheaf property. We can also define spaces of compactly supported GSF and the corresponding strict inductive limit, which have very good properties and can be easily studied using $\tilde{\mathbb{R}}$ valued seminorms. Using the words above, we can say that GSF are a possible formalization of the cause, not of the effect. If time allows, we will explore some future steps, with the idea that in case I'll say bullshits, we can all have a good laugh: Banach-like fixed point theorem and Picard-Lindeloef theorem, very weak solution of any differential equation, Hahn-Banach theorem for generalized smooth functionals... This is a joint work with M. Kunzinger and H. Vernaeve. A minimal knowledge of Colombeau theory is assumed. |
| 07. Nov. 2014 | Eduard Nigsch | Reduction to trivial bundles AbstractWell-known isomorphisms involving spaces of sections of vector bundles are usually proved by reduction to the case of the trivial line bundle. I will explain the conceptual background of this reduction precedure in easy category theoretical terms using the notion of additive functors and categories. Moreover, I will point out how in principle the respective isomorphisms can be seen to hold also topologically. |
| 14. Nov. 2014 | Shantanu Dave | Controlling geometry and analysis at infinity AbstractThis talk will provide an introduction to large scale geometry of metric spaces specially of Riemannian manifolds. In particular we recall some properties of rigidity of “quasi-regular” mappings and some implication to low regularity geometry. We shall consider examples of Gromov hyperbolic spaces for illustration. Some aspects like conformal stability of asymptotic dimensions will be covered. The main aim to argue that these ideas are also relevant in semi-Riemannian settings and a plan of some ongoing research. |
| 21. Nov. 2014 | Shantanu Dave | Controlling geometry and analysis at infinity, part 2 |
| 28. Nov. 2014 | Clemens Sämann | Global hyperbolicity for spacetimes with continuous metrics |
| 12. Dez. 2014 | David Rottensteiner | Time-Frequency Analysis on the Heisenberg Group AbstractI'll give a summary of my PhD thesis, in which I studied the Weyl quantization and modulation spaces on the Heisenberg group H_n. The connection between these two concepts is very strong in the Euclidean, i.e., \R^n-setting, but it can also be recovered for arbitrary nilpotent Lie groups such as the H_n. Of particular importance in this context is the work of N.V. Pedersen (1949-1996). |
| 09. Jan 2015 | Milena Stojković | Strong differentiability of the exponential map for $C^{1,1}$ metrics. AbstractThis talk is based on the paper "Convex neighborhoods for Lipschitz connections and sprays" by Ettore Minguzzi, in which he proves that the exponential map of a $C^{1,1}$ metric is locally a bi-Lipschitz homeomorphism and strongly differentiable at the origin. We will recall Peano's definition of strong differential and its basic properties, as well as Leach's inverse function theorem. The proof of the strong differentiability of the exponential map will pass through a local analysis based on the Picard-Lindelöf approximation method. It will then follow that the inverse is Lipschitz by means of Leach's inverse function theorem. |
| 16. Jan 2015 | Alexander Lecke | Geodesics in low regularity |
| 23. Jan 2015 | Alexander Lecke | Geodesics in low regularity, part 2 |