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. 2011 | Annegret Burtscher | The metric space structure of Riemannian manifolds AbstractThe metric space structure of a Riemannian manifold $(M, g)$ plays a key role in the understanding of its geometry. Already the basic definitions reveal the interrelation of differential and metric geometry. A distance function $d$ on $M$ is constructed by minimizing the integral length of certain curves on $M$. This metric space structure $(M, d)$ in turn induces a length structure on $M$: the length of a curve is computed as its variation (think of polygons approximating a curve in $\mathbb{R}^n$). Do these two notions of length coincide? The aim is to provide minimal criteria on the Riemannian metric $g$ and class of curves for which the answer is "yes", i.e. find conditions for which the integral length and variational length of curves exist and coincide. In the first part of the talk we will focus on smooth Riemannian metrics and extend the standard class of piecewise smooth curves to the class of absolutely continuous curves. Riemannian metrics of low regularity (e.g., continuous metrics) do not provide us with standard tools and have to be treated differently. They will be subject of the second part of the talk. Throughout, techniques from (smooth) Riemannian geometry and metric analysis shake hands and complement one another. Luckily, many questions remain open. Time permitted, some observations regarding the difference of smooth and continuous Riemannian metrics will be shared with the audience. |
| 25. Mar. 2011 | Hideo Deguchi | Reaction diffusion systems |
| 01. Apr. 2011 | Eduard Nigsch | Polynomial distributions and other ideas AbstractPolynomial distributions (i.e., polynomials on the space of test functions) provide a very simple structure for defining multiplication of distributions. One can obtain a Colombeau-type algebra in the usual way. Because the space of polynomial distributions is the smallest algebra (in some sense) containing the distributions, studying it seems to be promising for a better understanding of the foundations of Colombeau-type algebras. Furthermore, the close relation between polynomial and linear distributions allows one to carry over many concepts from linear distribution theory easily. As a main motivation I aim at a better understanding of the ideal of negligible generalized functions, but this still is at an early stage. Depending on time (and interest) I might also touch other topics. |
| 08. Apr. 2011 | Paolo GiordanoProblems approaching infinite dimensional spaces | This Diana seminar aims at introducing the next one about diffeological spaces and the next about Synthetic Differential Geometry (the latter not scheduled yet). The seminar aims at explaining the problems occurring in generalizing the notion of manifold to infinite dimensional spaces so as to obtain a Cartesian closed category. We will firstly motivate mathematically and physically the notion of Cartesian closedness. We will see that Banach manifolds are not Cartesian closed and the incompatibility of locally convex topology with Cartesian closedness. Finally, we will state the Omori theorem about Banach Lie groups. Almost all the theorems are stated, with references, but without proofs. In case of sufficient time, we will also introduce the notion of Froelicher space and of diffeological space. |
| 15. Apr. 2011 | No seminar | |
| 22. Apr. 2011 | No seminar | |
| 29. Apr. 2011 | No seminar | |
| 06. May 2011 | Michael Kunzinger | Wave equations on Lorentzian manifolds |
| 13. May 2011 | Shantanu Dave | Analysis of generalized Duistermaat-Guillemin trace AbstractWe shall give a quick overview of relationship between solutions to hyperbolic (wave) equations on a closed manifold and Hamiltonian dynamics on its cotangent bundle, and provide an outline how singularities of Duistermaat-Guillemin trace relate to length of periodic trajectories of the Hamiltonian flow. We shall also discuss the hypothesis sufficient for a converse, producing singularities in DG trace whenever there are closed trajectories. The talk shall then focus on abstracting this phenomenon in terms of a spectral-triple formulation for wider applicability. Time permited relationnships with spectral zeta functions and heat asymptotics will be covered. |
| 20. May 2011 | Paolo Giordano | Cartesian closure and diffeological spaces AbstractI present a general way to obtain a cartesian closed category embedding a given category $F$ of "smooth spaces". E.g. $F$ can be the category of open subsets of cartesian spaces ($R^n$) and smooth maps (which is the case of diffeological spaces) or manifolds or, why not, generalized functions. The obtained category, called "cartesian closure of $F$", has very good categorical properties and we will see several examples of construction of new spaces starting from given ones. Evaluations, compositions, classical differential and integral operators and inversion of diffeomorphims are all examples of smooth (and continuous) arrows in the cartesian closure. |
| 27. May 2011 | Shantanu Dave | Analysis of generalized Duistermaat-Guillemin trace |
| 03. Jun. 2011 | No seminar | |
| 10. Jun. 2011 | James D.E. Grant | Alexandrov spaces and related topics AbstractThis talk will be largely paedogogical, the aim being to introduce some of the basic modern tools and ideas of synthetic and comparison geometry as developed by Gromov and others. I will introduce the basic ideas of length spaces and corresponding metric space structures. I will then explain how one can give a definition of a (lower) curvature bounds analogous to a lower sectional curvature bound for a Riemannian manifold, building on the Toponogov triangle comparison theorem. If time permits, I will summarise some recent work attempting to relate curvature bounds defined in a synthetic geometric context with more analytical definitions. |
| 17. Jun. 2011 | Nathalie Tassotti | Interpretation of the Riemannian Curvature tensor |
| 24. Jun 2011 | No Seminar |