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 |
|---|---|---|
| 02. Oct. 2015 | Scheduling | |
| 16. Oct. 2015 | Sebastian Fischer | An introduction to the theory of distributions, part 1 |
| 23. Oct. 2015 | Markus Sobotnik | An introduction to the theory of distributions, part 2 |
| 30. Oct. 2015 | Denise Schmutz | An introduction to the theory of distributions, part 3 |
| 06. Nov. 2015 | Paolo Giordano | Is a student with a free mind able to create a theory more beautiful than Schwartz distributions? AbstractThis is the imaginary story of an imaginary student in the searching for beauty in mathematics. The more the math she sees is beautiful and the more she is glad. The less the math she sees is beautiful and the more she asks herself: “Well, it would be more beautiful if...”. So, after taking the first course in physics, she was disappointed by that use of scalars and defined a very natural way to extend the real field to get infinitesimal and infinite numbers. Then she took a course in history of calculus, where she discovered that Cauchy, Heaviside and Dirac invented the Dirac delta simply by taking a Gaussian with an infinitesimal standard deviation. Following this viewpoint, generalized functions are ordinary set-theoretical functions defined on a new ring of scalars that contains (also) infinitesimals and infinities. In this way, learning Schwartz distributions, she understood that L. Schwartz substituted the effect (they are functionals) with the cause (they are functions on another ring of scalars). She recognized that Schwartz theory is surely beautiful, but very far from the original powerful intuition. So, she formalized in a very simple way the original viewpoint and invented "generalized smooth functions" (GSF).GSF are set theoretical functions defined and with values in the ring $\widetilde{\mathbb{R}}$ of Robinson-Colombeau generalized numbers. They share several properties with ordinary smooth functions, but include 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, inverse function 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 $\widetilde{\mathbb{R}}$-valued norms. We have Cauchy complete spaces of GSF containing all Schwartz distributions, a related Banach-like fixed point theorem and a Picard-Lindeloef theorem. Using the words above, we can say that GSF are a possible formalization of the original intuition of Cauchy, Heaviside and Dirac. In other words, they are a formalization of the cause, not of the effect. So, can we state that it is possible to develop a theory which is more beautiful than that of Schwartz distributions? Are you free minds? This is (really) a joint work with M. Kunzinger, H. Vernaeve, L. Luperi Baglini and R. Steinbauer. |
| 13. Nov. 2015 | Eduard Nigsch | Whitney's extension theorem & a nonlinear Peetre theorem |
| 18. Nov. 2015 | Alexander Lecke | An introduction to calculus of variations in the setting of generalized smooth functions |
| 20. Nov. 2015 | Lorenzo Luperi Baglini | A new characterization of distributions among GSF AbstractIn this talk we try to answer the following question: when is a given GSF $f$ equal to the embedding of a distribution? A characterization of such GSF has already been given by Paolo in a Diana seminar last year; we will present a different (but, of course, equivalent) characterization based on a well-known structural theorem for distributions, and we will show with plenty of examples how this characterization can be used for some applications. We do not assume any prerequisite: all the results that we need to arrive to our characterization will be (shortly) recalled. The results that we present have been obtained in a joint work with Paolo Giordano. |
| 27. Nov. 2015 | Roland Steinbauer | Null Hypersurfaces AbstractGeneral Relativity, Albert Einstein's theory of space, time and gravitation --- whose centerpiece was published on Nov. 25, 1915 --- is first of all a geometric theory. It is built on Lorentzian geometry which is the differential geometry of manifolds equipped with a nondegenerate scalar product. In this talk I will review some basic notions of Lorentzian geometry focussing on the geometry of hypersurfaces which are null, i.e., generated by light rays. This will enable us to discuss (aspects of) the Penrose singularity theorem which says that the gravitational collapse of massive objects leads to a spacetime singularity (and by the way appeared in 1965 as the first true post-Einsteinian contribution to general relativity exactly half way to the present centenary).As an aside: The low regularity version(s) of the Penrose theorem currently lie in the research focus of several people in our group. |
| 04. Dec. 2015 | Melanie Graf | The $C^0$-inextendibility of the Schwarzschild spacetime, Part 1 AbstractWhile it is well-known that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a twice continuously differentiable metric, it has only been shown very recently (J. Sbierski, 2015, arXiv:150700601) that it is even inextendible as a Lorentzian manifold with a continuous metric. In this first part of a two part series discussing Sbierski's paper I will start with a general introduction to the Schwarzschild spacetime and the concept of ($C^k$-)extensions of Lorentzian manifolds, before talking a bit about causality theory for continuous metrics. In the end I plan on discussing how the proof of the $C^0$-inextendibilty of the Schwarzschild spacetime can be split into two distinct parts and, if time permits, I will sketch the proof of the $C^0$-inextendibility of the Minkowski spacetime which serves as a preparation for the first part of the proof for the Schwarzschild metric. |
| 11. Dec. 2015 | Clemens Sämann | The $C^0$-inextendibility of the Schwarzschild spacetime, Part 2 |
| 08. Jan. 2016 | Zorana Matić | The Ekeland variational principle and some applications to fixed point theorems AbstractWe present the Ekeland variational principle and Caristi fixed point theorem, their proofs and relation between these two theorems. The Ekeland variational principle (and its equivalent formulations) have been one of the main subject in many fields of non-linear functional analysis, convex analysis and optimization. |
| 15. Jan. 2016 | Michael Koch | General Relativity |
| 22. Jan. 2016 | Eduard Nigsch | An introduction to Hairer's theory of regularity structures AbstractMartin Hairer won the fields medal in 2014 for his groundbreaking work on nonlinear stochastic partial differential equations (SPDEs). He created a theory of regularity structures which not only enables one to formulate previously ill-defined SPDEs in a mathematically rigorous way, but also to obtain meaningful solutions to them. One main feature of the theory is that it is able to provide products of distributions, encoding some renormalization procedure adapted to the problem at hand. Although we will only be able to scratch the surface, I will give an overview of the concepts and methods involved in this theory. |
| 29. Jan. 2016 | Eduard Nigsch | An introduction to Hairer's theory of regularity structures, part 2 |