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 10.15 am via moodle** and will be announced by email weekly.

Anyone interested is welcome to attend.

Summer term 2021

Date | Speaker | Title |
---|---|---|

05. Mar. 2021 | Scheduling | |

19. Mar. 2021 | Cortez Atance Pablo | optimal transport |

26. Mar. 2021 | Darko Mitrovic | Velocity averaging ## AbstractVelocity averaging is a common name for compactness properties for solutions of transport equations which beside the space and time variables we have also so called velocity variable which plays the role of a parameter (the derivative of the solution to the equation with respect to the velocity variable does not appear in the equation). The equations are nonlinear, but the sink/source term can be very singular (e.g. distribution of finite order). The velocity averaging techniques were used in the proof of existence of weak solutions to the Boltzman equation, regularity properties of entropy solutions to scalar conservation laws and degenerate parabolic equations, existence of strong traces of entropy solutions to scalar conservation laws and degenerate parabolic equations, existence of solutions to some types of evolution equations with discontinuous coefficients... |

16. Apr. 2021 | Matteo Calisti | differential calculus in metric measure spaces ## AbstractTo define a differential calculus in a metric measure space, with the Laplacian as objective, we need to replace the classical fundamental objects like the gradient, its norm and the action of a differential on it with new ones using the duality with measures. Also one can define the Laplacian with the Dirichlet's energy in the Euclidian case, which will be replaced in this setting by the Cheeger's one. In both cases, the calculus' rules are not so different from the ones is the smooth setting. Then an application will be shown to the first Heisenberg group: here the metric Laplacian turns out to behave differently from the usual one. Finally, observations on the Wasserstein space and the tangent space in this setting will be added, showing also some further possible developments like RDC-spaces. |

23. Apr. 2021 | Bachler Phillip Josef | Fock spaces ## AbstractWe will start with infinite direct sum and gain some functional analytic statements that show us the most basic behaviours of the abstract Fock space. After that we take a glimpse on the second quantization operator. |

30. Apr. 2021 | Felix Rott | Curvature comparison in metric spaces: a gluing theorem for CAT($k$) spaces ## AbstractIn this talk we will prove the gluing theorem of Reshetnyak. It states that when gluing CAT($k$) spaces - metric spaces with curvature bounded above by $k$ - along closed and convex isometric subsets the resulting space satisfies the same curvature bound. We start with a brief introduction into the theory of length spaces and describe the gluing process in a bit more detail. We continue with a description of the (Riemannian) model spaces - these are Riemannian manifolds of constant sectional curvature $k$ (sphere, plane and hyperbolic space). Comparing triangles in the model spaces with triangles in more general metric spaces yields the definition of a curvature bound. Then we show Alexandrov's lemma and talk about some basic properties and characterizations of CAT($k$) spaces. Finally, we will prove the gluing theorem and mention generalizations as well as some reasoning why this theorem fails with respect to a lower curvature bound. |

07. May 2021 | Shantanu Dave | cancelled |

14. May 2021 | Zellhofer Paul | Symmetric spaces ## AbstractThis talk will provide an introduction to the theory of Riemannian symmetric spaces. Although they can be defined entirely within the framework of classical Riemannian geometry, it will be apparent in this setting that symmetric spaces are in general best described as homogeneous spaces of Lie groups satisfying certain additional properties. We motivate this shift in perspective and then investigate more closely the Lie algebra structure that is naturally associated with a symmetric space. This description not only demonstrates that symmetric spaces can be completely classified, but it also gives rise to the distinguished class of symmetric spaces of the non-compact type, which are characterized by a particularly simple algebraic and geometric structure. Finally, we aim to give a concrete example on how a symmetric space of the non-compact type can be compactified. |

21. May 2021 | Tobias Beran | Spacetime distances: an exploration (part I) ## AbstractIn general, there is no natural distance function on a spacetime. But assuming a time function, we can introduce several: Riemannization and Null distance. We look at how they behave in generalized Robertson Walker spacetimes (GRW, i.e.\ warped product spacetimes), what happens when one post-composes the time function, and maybe whether they "see" a big bang singularity in a GRW spacetime. |

28. May 2021 | Tobias Beran | Spacetime distances: an exploration (part II) ## AbstractIn general, there is no natural distance function on a spacetime. But assuming a time function, we can introduce several: Riemannization and Null distance. We look at how they behave in generalized Robertson Walker spacetimes (GRW, i.e.\ warped product spacetimes), what happens when one post-composes the time function, and maybe whether they "see" a big bang singularity in a GRW spacetime. |

04. Jun. 2021 | Akbarali Mukhammadiev | Fourier transform for all generalized functions ## AbstractDepending on a fixed infinite number $k\in{^\rho}\tilde{\mathbb{R}}$ in the ring ${^\rho}\tilde{\mathbb{R}}$ of Robinson-Colombeau (here $\rho=(\rho_\varepsilon)$ generalizes the classical case $\rho_\varepsilon=\varepsilon$), we define the $n$−dimensional hyperfinite Fourier transform for any generalized smooth function $f\in {^\rho}\mathcal{GC}^\infty(K,\tilde{\mathbb{C}})$, $K= [−k,k]^n$, as $\mathcal{F}_k(f) (\omega) :=\int_Kf(x)e^{−ix\omega} dx=\int_{-k}^k dx_1\ldots \int_{-k}^k f(x_1\ldots x_n)e^{−ix\cdot\omega}dx_n$. We prove that $\mathcal{F}_k:{^\rho}\mathcal{GC}^\infty(K,\tilde{\mathbb{C}})\to {^\rho}\mathcal{GC}^\infty({^\rho}\tilde{\mathbb{R}}^n,\tilde{\mathbb{C}})$. We also define an appropriate notion of convolution in order to prove the related properties of this transform. We then prove that the hyperfinite Fourier transform satisfies almost all the elementary properties together with the convolutional property and the corresponding inversion formula. We also consider the embedding of all Colombeau generalized functions into the space ${^\rho}\mathcal{GC}^\infty(K,\tilde{\mathbb{C}})$. Finally, we show that this kind of Fourier transform is applicable to a larger class of spaces of generalized functions and differential problems. |

11. Jun. 2021 | Mekonnen Manuel | Introduction to Gromov-Hausdorff convergence ## AbstractThis talk intends to introduce the concept of Gromov-Hausdorff distance which is an often used tool for measuring how far two compact metric spaces are from being isometric. This distance leads to the notion of Gromov-Hausdorff convergence which can be extended to non-compact metric spaces under certain assumptions and allows for a pointed and non-pointed version. With this at hand, a convergence notion of points and maps can be introduced. |

18. Jun. 2021 | Argam Ohanyan | Splitting Theorems and Bartnik's Conjecture ## AbstractThe beginning of the talk will consist of a review of the Riemannian and Lorentzian splitting theorems and how they came about. These theorems (roughly) state that (reasonable) Riemannian manifolds containing a global minimizing (resp. maximizing) geodesic must already be isometric to a product.After that, we turn to the main topic of the talk: Bartnik's conjecture. It is one of the great open problems in mathematical GR/Lorentzian geometry today and it is a splitting conjecture made by Robert Bartnik in 1988 that asserts the rigidity of the Hawking-Penrose singularity theorem. We will discuss one particular approach via Hausdorff closed limits by Galloway and Vega in rather great detail and outline the difficulties that one is confronted with when tackling the conjecture without any extra assumptions. |

25. Jun. 2021 | cancelled |