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 |
|---|---|---|
| 18. Mar. 2016 | Clemens Sämann | Causal Relationship AbstractWe will discuss a novel notion to relate two Lorentzian manifolds introduced by García-Parrado and Senovilla (in CQG 20 (2003)) and its applications to causality theory. |
| 08. Apr. 2016 | Robert Švarc | General Relativity and Gravitational Waves AbstractEinstein's general relativity is the most successful theory of gravity so far. It passed many sophisticated experimental tests of which the recent gravitational waves direct detection is the physical milestone. Basic overview of the GR main ideas together with its application in current astrophysics and cosmology will be briefly presented. |
| 15. Apr. 2016 | Alexei Cheviakov | Local Conservation Laws for Nonlinear Differential Equations: Theory, Systematic Construction, Computation, and Examples AbstractA local conservation law is an expression of the form divergence(flux vector)=0, holding on solutions of a given system of differential equations. Local conservation laws of ordinary differential equations (ODE) yield their first integrals. For partial differential equations (PDE), local conservation laws reveal globally conserved quantities, such as energy and momentum, and provide important analytical information for existence, uniqueness, stability, linearization, integrability analysis, and so on. Conserved forms of the equations are also of high importance for numerical simulations.In this talk, we will discuss some theory, including that of variational systems, the first Noether's theorem, construction of Lagrangians, and the relationship of local symmetries and conservation laws. For the majority of nonlinear equations, a physical variational formulation does not exist, and the Noether's theorem is not applicable. An systematic direct construction method, applicable to any ODE/PDE system, will be presented. We will discuss the underlying theory and the algorithm of the method. Examples of conservation law computation for some classical nonlinear DEs, as well as the equations of nonlinear mechanics, will be presented. Examples of symbolic computations using Maple-based symbolic software will be given. Time permitting, we will discuss fundamental results and recent developments pertaining to the conservation law analysis of Euler and Navier-Stokes fluid dynamics models, as well as some related models: vorticity-type equations, shallow water models, and surfactant dynamics equations. It will be shown how sets of local conservation laws may be significantly extended for models admitting geometrical symmetries, such as rotationally and helically symmetric flows and plane flows. |
| 22. Apr. 2016 | Jiří Podolský | Algebraic structure of Kundt and Robinson–Trautman geometries in four and higher dimensions AbstractWe study a general class of D-dimensional spacetimes that admit a non-twisting and shear-free null vector field. This includes the famous non-expanding Kundt family and the expanding Robinson-Trautman family of spacetimes. In particular, we show that the algebraic structure of the Weyl tensor is I(b) or more special, and derive surprisingly simple conditions under which the optically privileged null direction is a multiple WAND (PND). Some of these conditions are identically satisfied in four dimensions. All possible algebraically special types, including the refinement to subtypes, are thus identified. No field equations are applied, so that the results are valid not only in Einstein's theory but also in its generalizations. Applying vacuum Einstein's field equations, we reveal fundamental algebraic differences between the D=4 and D>4 cases. We also give a short discussion of some interesting particular subcases (exact gravitational waves, gyratons, non-rotating p-form black holes etc.). |
| 29. Apr. 2016 | Eduard Nigsch | The wavefront set - products and pullbacks of distributions and nonlinear generalized functions AbstractIn classical distribution theory, the product of distributions (in the sense of Hörmander) can only be defined under a certain condition on their wavefront sets, while in Colombeau algebras one can always form their product. Similarly, the pullback of distributions along a given smooth function can be defined only for those distributions whose wavefront set is in a favorable position. The question of how to restrict elements of full Colombeau algebras to submanifolds so far has only been dealt with in the case of linear subspaces of Euclidean space.The talk will consist of three parts: In the first part, I will thoroughly motivate and introduce the notion of wavefront set of distributions. In the second part, I will discuss two recent papers [1,2] about the space $\mathcal{D}'_{\Gamma}$ of distributions having their wavefront set contained in a fixed cone. In the third part, I will outline how a diffeomorphism invariant full Colombeau algebra on a manifold, possessing a restriction mapping to a given submanifold, could possibly be constructed. References: [1] Functional properties of Hörmander’s space of distributions having a specified wavefront set.Commun. Math. Phys. 332, No. 3, 1345-1380 (2014). [2] Boundedness and continuity of the fundamental operations on distributions having a specified wave front set. (with a counter example by Semyon Alesker). |
| 06. May. 2016 | Eberhard Mayerhofer | Option Portfolios on Multiple Assets AbstractWe derive optimal portfolios for an investor who trades several assets and European options on each of them, available in multiple strikes, with the goal of maximizing the Sharpe ratio. Optimal weights are characterized by a system of integral equations which admits a unique solution, and reduces to a linear system in a discrete setting. The solution separates in the cross section only if assets are independent. Joint work with Paolo Guasoni. |
| 13. May. 2016 | Melanie Graf | Lorentzian splitting theorems and the Busemann function AbstractIn this talk I will give a brief introduction to the splitting problem in Lorentzian geometry, i.e., the question under which conditions a spacetime satisfying a lower bound on the timelike Ricci curvature is isometric to a warped poduct. As there are many different versions of such splitting theorems the goal is to give an overview of the main ideas their proofs have in common without getting lost in too many technicalities. A particular focus will be the (Lorentzian) Busemann function and its level sets, for which we will show some basic inequalities and estimates. Together with a (low-regularity) variant of the maximum principle this will lead to a local splitting result (which may then be extended globally). |
| 20. May. 2016 | Lorenzo Luperi Baglini | ODEs in GSF spaces: some results and some ideas AbstractOur goal is to try to recover, in GSF spaces, as many of the "classical" properties of ODEs as we can. We will start by (re)showing a Banach fixed point theorem for GSF and the related Picard-Lindelöf theorem, that allows to deduce the existence and uniqueness of solutions of generalized ODEs for (in general) infinitesimal intervals of time. Then we will discuss certain problems and certain ideas that arise if we want to extend this intervals of existence from infinitesimal to finite radius. This will lead to discuss a different Banach fixed point theorem that talks about infinite iterations of contractions, and its related Picard-Lindelöf theorem. Finally, we will apply these results to show some properties of linear ODEs in GSF spaces and, if time allows, to discuss, by means of examples, the relationships between solutions in GSF spaces and embedded distributions. This is a joint work with Paolo Giordano. |
| 03. Jun. 2016 | Michael Kunzinger | Pseudodifferential Operators AbstractThis talk is designed for an audience with some prior knowledge of distribution theory wanting to find out what pseudodifferential operators are about: what they are good for, how they are constructed, how they fit into the general framework of distribution theory, and how to use them to solve PDEs. The emphasis will be on ideas rather than on technical details. |
| 10. Jun. 2016 | Paolo Giordano | Thoughts in the searching for a third system AbstractThe first system is called capitalism and it's a social and economical system which irrationally idolizes the false efficiency of a free market. The second system is called communism: by pursuing an economic egalitarianism, it necessarily deletes private property; it is usually followed by those who don't live it directly and still have their own reach private properties. We will describe a mathematical model for Zipf's law about the frequency of words in a given natural language. Then, we will use the same ideas to hint at the mathematical model of von Thünen's law about patterns of land uses in a spatial economic system. These models inspire a possible informal definition of complex adaptive system, which will be the source of our (very) free (and possibly crappy) thoughts in the searching for an efficient, egalitarian and quantitative third system. |
| 17. Jun. 2016 | Alexander Lecke | Introduction to the Calculus of Variations in Generalized Smooth Functions AbstractThe aim of this talk is to introduce the calculus of variations into the theory of generalized smooth functions (GSF) [1, 2, 3]. GSF are smooth set-theoretical functions defined on a non-Archimedean extension of the real field. They embed Schwartz distributions but are freely close with respect to composition. This feature facilitates the transposition of classical results into this generalized setting. In order to do this, we begin with a brief introduction to the theory of generalized smooth functions. After this, we give some interesting results like the fundamental lemma of calculus of variations or the Legendre - Hadamard condition in the GSF setting. We conclude the talk with examples from low regular Riemannian geometry such as that (with some assumptions) the standard part of the minimal length in GSF exists and is equal to the minimal lenght in the "standard world." This is a joint work with Lorenzo Luperi Baglini and Paolo Giordano (University of Vienna).[1] Giordano P. and Kunzinger M., A convenient notion of compact set for generalized functions, Accepted in Proceedings of the Edinburgh Mathematical Society. [2] Giordano P. and Kunzinger M., Inverse Function Theorems for Generalized Smooth Functions, arxiv 1411.7292. [3] Giordano P. and Kunzinger M. and Vernaeve H., Strongly internal sets and generalized smooth functions, Journal of Mathematical Analysis and Applications 422 (2015). |