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. Oct. 2017 | Scheduling | |
| 13. Oct. 2017 | Martina Glogowatz | PhD Defense |
| 20. Oct. 2017 | Eduard Nigsch | Factorization in Banach modules and applications AbstractThe Cohen-Hewitt factorization theorem allows one to factorize certain elements of Banach modules over Banach algebras having an approximate unit. I will present a concise proof of this theorem due to Koosis. Extending this result to simultaneous factorization of compact sets, this leads to a characterization of compact subsets of $L^p$. As an application, we are able to describe the $\kappa$-topology on $L^q$, i.e., the topology of uniform convergence on absolutely convex compact sets, in terms of 'nice' seminorms. |
| 27. Oct. 2017 | No seminar. | |
| 03. Nov. 2017 | No seminar. | |
| 10. Nov. 2017 | Artur Sergyeyev | New integrable systems in (3+1) dimensions and contact geometry. AbstractWe introduce a novel systematic construction for integrable (3+1)-dimensional systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter. Further details can be found at arXiv:1401.2122 (to appear in Letters in Mathematical Physics). |
| 17. Nov. 2017 | James Grant | An update on a low-regularity positive mass theorem AbstractIn the paper "A positive mass theorem for low-regularity metrics" (https://arxiv.org/abs/1205.1302), Tassotti and I developed a version of the positive mass theorem for continuous Riemannian metrics on an $n$-dimensional manifold $M$ that lie in the Sobolev space $W^{2, n/2}_{\mathrm{loc}}(M)$. For manifolds with spin structure, Lee and LeFloch later developed a version of the positive mass theorem for continuous metrics with $W^{1, n}_{\mathrm{loc}}(M)$ regularity, a significant improvement (on spin manifolds) on the result of Tassotti and myself. It seems unlikely that the techniques used in the Grant--Tassotti paper can be adapted to give a proof of the positive mass theorem on non-spin manifolds at the level of regularity of the Lee--LeFloch paper. Nevertheless, on revisiting our paper it is clear that our results can be strengthened in certain ways. (In fact, we had proved a stronger result than we claimed.) In particular, they yield an approximation result for rough non-negative scalar curvature metrics reminiscent of recent results of Huneau and Luk on limits of Ricci flat Lorentzian manifolds. Some of these "reinterpretations" of our results may also be of relevance to recent work on low-regularity versions of the singularity theorems. |
| 24. Nov. 2017 | Klaus Kröncke | Stability of ALE Ricci-flat manifolds under Ricci flow AbstractWe prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to $g$ exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable. This is joint work with Alix Deruelle. |
| 01. Dec. 2017 | Vladimir Jacimovic | Collective motion of coupled particles on compact Lie groups: Physics and applications |
| 15. Dec. 2017 | Benedict Schinnerl | Length Spaces and Geodesics in Metric Spaces AbstractI will talk about a possible way to generalize the notion of geodesic to metric spaces as locally shortest paths. Starting out with a basic framework about the variational length and shortest paths, we will be able to state analogs of the theorem of Hopf-Rinow. Further I will provide some existence results of geodesics. Lastly I will compare the introduced notion of geodesics to the one from Riemannian Geometry for metrics of low regularity. |
| 12. Jan. 2018 | Leopold Veselka | Schaefer's Theorem and its applications AbstractIn tomorrows talk the main theorem will be the fixed point theorem of Schaefer. To be able to prove the theorem we need some additional information, which I will present first. After that the theory of Schaefer together with two applications will follow. |
| 19. Jan. 2018 | Michael Koch | The Cauchy problem in General Relativity AbstractAfter some motivations to the topic, we describe how to solve the Cauchy problem in general relativity. We introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated. The initial value point of view illustrates the strong connection between Einsteins general theory of relativity and the theory of hyperbolic PDEs. |
| 26. Jan. 2018 | Paolo Giordano | From subpoints to the Grothendieck topos of generalized functions. AbstractAn old guy once said "A point is that which has no part". But Colombeau's generalized points are so rich that we can define the notion of subpoint. This reveals to be very useful since it allows to satisfy properties of points by considering them only on well-behaving subpoints (near-standard or infinite), to prove a general sheaf properties and to construct the Grothendieck topos of generalized functions. The framework is that of generalized smooth functions (GSF). GSF are set-theoretical maps defined on, and taking values in the non-Archimedean ring of Robinson-Colombeau, and form a concrete category which unifies and extends Schwartz distributions and Colombeau generalized functions. The calculus of these generalized functions is closely related to classical analysis, with point values, free composition and several classical theorems such as intermediate value theorem, mean value theorems, extreme value theorem, local and global inverse and implicit function theorems, multidimensional integration, a theory of singular nonlinear ODE and PDE, calculus of variations, etc. The Grothendieck topos of generalized functions is a universe that embeds classical smooth manifolds and is closed with respect to products, sums, pull-backs, push-outs, equalizers, infinite-dimensional function spaces, arbitrary subspaces, etc. It is hence a potential good framework for the study of spaces and functions with singularities. In this seminar we introduce GSF, define the concept of subpoint and hint at the intuitive meaning of the other results. This is a research seminar, so a certain knowledge of Colombeau's theory could be surely useful. Moreover, is some of its part it is a work in progress, so mistakes have a greater probability to appear than, e.g., in other math papers. |