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 |
|---|---|---|
| 06. Mar. 2015 | Scheduling | |
| 27. Mar. 2015 | Roman Popovych | Noether theorems AbstractAfter recalling classical Noether's first and second theorems on conservation laws of differential equations, we enhance and generalize Noether's second theorem. More specifically, we prove that a system of differential equations is abnormal, i.e., it has an identically vanishing differential consequence, if and only if it possesses a trivial conserved vector corresponding to a nontrivial characteristic. Moreover, the above properties are also equivalent to that the system admits a family of characteristics that is parameterized by an arbitrary function of all independent variables in a local way and whose Fréchet derivative with respect to the parameter-function does not vanish on solutions of the system for a value of the parameter-function. The theorem is illustrated by physically relevant examples. |
| 17. Apr. 2015 | Melanie Graf | The Hawking and Penrose Singularity Theorem AbstractAs all singularity theorems, the singularity theorem of Hawking and Penrose proves that under certain (mainly curvature and causality) assumptions a spacetime has to be singular, i.e., there exists an incomplete causal geodesic. In this talk I will present this theorem and briefly discuss some of its assumptions and its relation to the other singularity theorems. The second part of the talk will be dedicated to sketch some parts of the proof that highlight the difficulties one faces when trying to show an analogous theorem for $C^{1,1}$ metrics. |
| 24. Apr. 2015 | Eduard Nigsch | The theory of vector valued distributions and its applications AbstractWhile L. Schwartz' theory of distributions is widely used today, its extension to vector valued distributions often remains somewhat arcane. I will highlight the main points and difficulties of this theory and mention some concrete applications. |
| 08. May. 2015 | Enxin Wu | An overview of the technique of diffeology AbstractSmooth manifolds are central objects of mathematics. However, the category of smooth manifolds is not well-behaved. Since the 1960's, mathematicians are trying to generalize the concept of smooth manifold. Diffeology is one such generalization proposed by French Mathematician J.M. Souriau. It provides a good framework to do differential geometry on more general spaces than smooth manifolds, including infinite dimensional spaces and singular spaces. In this talk, I will introduce the general theory and give some examples to show the usefulness of this new field. |
| 15. May. 2015 | Lorenzo Luperi Baglini | A fixed-point iteration method for arbitrary generalized ODE AbstractGeneralized Smooth Functions (GSF) are a minimal extension of Colombeau generalized functions to arbitrary domains of generalized points. A key property of GSF is their conceptual analogy with smooth functions: they are set-theoretical maps, they are closed by composition, they generalize all classical theorems of calculus and they have a good notion of being compactly supported. In this talk we show that this analogy holds also in the study of first order ODEs y'=F(t,y) where F is a GSF. In fact, we prove an existence and uniqueness theorem for local solutions of such ODEs in spaces of compactly supported GSF that is analogous to the classical Picard-Lindel\"{o}f theorem. Our proof is based on a generalized version of Banach fixed point theorem for GSF. If time allows, we will also discuss related arguments such as epsilonwise-solutions, changes in the growth conditions and characterization of distributions in GSF spaces. |
| 22. May. 2015 | Paolo Giordano | Universal properties of Colombeau algebras AbstractThis seminar is, more or less, a pretext to talk of universal properties and their interpretation as the simplest/natural way to solve a problem. After a brief general introduction, we will see some universal properties of the special Colombeau algebra. These properties will also give characterizations up to isomorphisms of these algebras. This approach permits to focus on some important properties and on the corresponding choices, highlighting possible generalizations. Concerning category theory, the seminar requires no more than the knowledge of the definitions of category, functor and natural transformation. |
| 29. May. 2015 | Roland Steinbauer | Geodesics in non-expanding impulsive gravitational waves with $\Lambda$ AbstractImpulsive gravitational waves are simple exact models of short but violent bursts of gravitational radiation in general relativity. They are described by spacetime metrics which are either of regularity $C^{0,1}$ or even distributional and hence provide a relevant test bed for any theory of low regularity Lorentzian geometry. In this talk we report on recent progress describing the geodesics in non-expanding impulsive gravitational waves propagating in a cosmological background. |
| 12. Jun. 2015 | Walter Simon | Initial data for rotating cosmologies AbstractI revisit the "conformal method" for solving the initial value problem for Einstein's vacuuum equations with "cosmological constant". I show in particular how to obtain certain "cosmologies" whose initial data are axially symmetric, compact manifolds of topology S^2 x S, with a second fundamental form which can be interpreted as "rotation". The construction of such data amounts to solving a semilinear elliptic ("Lichnerowicz-") equation. I discuss and apply recent results of Hebey, Pacard and Pollack and of Premoselli to this problem. (Joint work with P. Bizon and S. Pletka, http://arxiv.org/abs/1506.01271) |
| 19. Jun. 2015 | Norbert Ortner | On the space $\dot{\mathcal{B}}'$ of Laurent Schwartz |
| 26. Jun. 2015 | Clemens Sämann | Geodesic completeness and global hyperbolicity for non-smooth spacetimes AbstractGeneral relativity as a geometric theory has been developed for smooth Lorentzian metrics and therefore extending notions and methods to non-smooth settings is a mathematically challenging endeavor. Nonetheless, physically relevant models of spacetimes (shock waves, matched spacetimes, conical singularities, impulsive waves, etc.) and the initial value formulation of general relativity demand this extension to settings of low regularity. In this thesis we extend two geometric notions, whose purpose is to exclude genuine singularities, to non-smooth settings. In fact, we are interested in regularity classes below C (the first derivative of the metric exists and is Lipschitz continuous), which is the most general class where the bulk of classical Lorentzian geometry remains valid. The first of these conditions is geodesic completeness. We prove a completeness result for so-called impulsive N-fronted waves with parallel rays (NPWs), which are generalizations of impulsive pp-waves and give a precise definition of geodesic completeness in the framework of the geometric theory of generalized functions. We conclude our treatment of geodesic completeness by investigating geodesics in non-expanding impulsive gravitational waves propagating in spaces of constant curvature. One purpose of the second condition - global hyperbolicity - is to exclude so-called "naked singularities". We study the globally hyperbolic metric splitting for a class of nonsmooth NPWs. Furthermore we extend the usual notion of global hyperbolicity (based on the compactness of the causal diamonds) to spacetimes with continuous metrics and show that several classically valid equivalences and implications still hold. |