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 |
|---|---|---|
| 16. Mar. 2012 | Michael Grosser | Convolution without cut-offs AbstractI would like to volunteeer for a warm-up talk opening this terms sequence of meetings. The topic completely pertains to the theory of distributions. The occasion given by my present lecture course on this matter, we shall consider the case of convolutions of two distributions with at least one of them having compact support. I will demonstrate how the cut-off functions occurring abundantly in the respective definitions, theorems and proofs can be dispensed with by a little additional input from functional analysis. This input, in turn, is related to the forthcoming talk of Eduard Nigsch on June 15 on regularizations of distributions. |
| 23. Mar. 2012 | Paolo Giordano | Generalized functions as smooth set-theoretical maps AbstractAfter a (hopefully conceptual and not only formal) introduction to Colombeau's generalized function (CGF), we will see a work in progress (with M. Kunzinger) concerning the idea to see generalized functions as suitable set-theoretical maps defined (and with values) in Colombeau's generalized numbers (CGN). Some motivating questions are: - Is it possible to define a generalized function as a set-theoretical map generated by nets of ordinary smooth functions u_\eps \in \cinfty(\Omega_\eps, \R^t) (note that the domain depends on \eps)? - Is it possible to define them so as to include both CGF and tempered CGF? - Do they form a category without limitation about composition? - How can we perform a differential calculus with them? - Do they avoid functions like i(x)=1 if x is infinitesimal and i(x)=0 otherwise? Our attempt (which is work in progress!) see a strong interplay between the sharp topology and the Fermat topology, i.e. that generated by balls (w.r.t. the generalized absolute value in CGN) with standard real radius. |
| 30. Mar. 2012 | Katharina Brazda | Continuum mechanics deals with the motion, the deformation, and the interaction of continuous bodies which possess certain material properties. I will discuss various approaches towards a "correct" mathematical formulation of these physical concepts in the sense of "including all that can possibly be imagined by an engineer but excluding all that can be dreamt up only by an ingenious mathematician". |
| 06. Apr. 2012 | No seminar | |
| 13. Apr. 2012 | No seminar | |
| 27. Apr. 2012 | Prof. Jiri Podolsky | Interpreting spacetimes of any dimension using geodesic deviation AbstractWe present a general method that can be used for geometrical and physical interpretation of an arbitrary spacetime in four or any higher number of dimensions. It is based on the systematic analysis of relative motion of free test particles. We demonstrate that the local effect of the gravitational field on particles, as described by the equation of geodesic deviation with respect to a natural orthonormal frame, can always be decomposed into a canonical set of transverse, longitudinal and Newton–Coulomb-type components, isotropic influence of a cosmological constant, and contributions arising from specific matter content of the Universe. In particular, exact gravitational waves in Einstein’s theory always exhibit themselves via purely transverse effects with D(D-3)/2 independent polarization states. To illustrate the utility of this approach, we study the family of pp-wave spacetimes in higher dimensions and discuss specific measurable effects on a detector located in four spacetime dimensions. For example, the corresponding deformations caused by generic higher-dimensional gravitational waves observed in such physical subspace need not be trace-free. |
| 04. May 2012 | Gudrun Szewieczek | Solvability of PDEs with generalized complex coefficients AbstractThe aim of this talk is to discuss the solvability of PDEs with generalized complex coefficients. I will present an adapted version of the Malgrange-Ehrenpreis-Theorem for fundamental solutions of these PDEs [C.Garetto] and based on this result an equivalent assertion to the solvability for compactly supported right-hand sides. [G. Hörmann/M. Oberguggenberger] Since it turned out that in the Colombeau setting dual spaces are an appropriate framework for fundamental solutions, I will give a short introduction to them before. |
| 11. May 2012 | Katharina Kienecker | The Peter-Weyl theorem AbstractThe Peter-Weyl theorem treats representations of compact groups. To obtain integration on locally compact groups, I will introduce the Haar measure, and then deal with some results from representation theory. We will use this and the spectral theorem for compact operators to prove the Peter-Weyl theorem. Then I will discuss its consequences in Lie group theory. |
| 18. May 2012 | Milena Stojković (maybe) | Causality Theory |
| 25. May 2012 | Christoph Karner | The Mountain Pass Theorem AbstractThe Mountain Pass Theorem is a result that deals with existence of critical points of a functional on a Hilbert space. Beyond its utility in the calculus of variations, it can also be used to prove the existence of non-trivial (weak) solutions of certain PDE. I will present a proof of the MPT using the Deformation Lemma and then show how one can use this result in PDE theory. |
| 01. Jun. 2012 | Ronald Quirchmayr | Introduction to the Calculus of Variations AbstractI will discuss the following topics: (a) (free) minima of functionals - existence/uniqueness (b) G-differentiable functionals (c) connections between convex functionals F and monotone operators F' (d) criteria for potential operators. |
| 08. Jun. 2012 | No seminar | |
| 15. Jun. 2012 | Eduard Nigsch | Regularization of distributions AbstractApproximating distributions by sequences of smooth functions is classically done by convolving them with smooth mollifiers. In Colombeau theory more general regularizations are needed, which gives the motivation for studying the space of all continuous linear mappings from the space of distributions into the space of smooth functions. In order to understand this space and give a nice representation of it, I will essentially give a résumé of L. Schwartz' 1954 paper 'Espaces de fonctions différentiables à valeurs vectorielles', which contains all the functional analytic background and results we need. |
| 22. Jun. 2012 | Eduard Nigsch | Regularization of distributions AbstractApproximating distributions by sequences of smooth functions is classically done by convolving them with smooth mollifiers. In Colombeau theory more general regularizations are needed, which gives the motivation for studying the space of all continuous linear mappings from the space of distributions into the space of smooth functions. In order to understand this space and give a nice representation of it, I will essentially give a résumé of L. Schwartz' 1954 paper 'Espaces de fonctions différentiables à valeurs vectorielles', which contains all the functional analytic background and results we need. |
| 29. Jun. 2012 | Martina Glogowatz | T.B.A. |