# The DIANA seminar -- Archive of Talks

(incomplete)

## Winter term 2015

Date Speaker
October 2 First meeting for scheduling.
October 16 Sebastian Fischer: $\mathscr{D}'$
October 23 Markus Sobotnik: $\mathscr{D}'$
October 30 Denise Schmutz: $\mathscr{D}'$
November 6 Paolo Giordano: t.b.a.
November 13 Eduard Nigsch: whatever
November 20 Lorenzo Luperi Baglini: t.b.a.
November 27 Roland Steinbauer: t.b.a.
December 4 Melanie Graf: t.b.a.
December 11 Alexander Lecke: t.b.a.
December 18 Clemens Sämann: t.b.a.
January 8 Zorana Matić: t.b.a.
January 15 Michael Koch: GR
January 22 Olga Borodina: t.b.a.
January 29 Michael Kunzinger / Eduard Nigsch: regularity structures

## Summer term 2015

Date Speaker
March 6 First meeting for scheduling.
March 27 Roman Popovych: Noether theorems.
Abstract: After recalling classical Noether's first and second theorems on conservation laws of differential equations, we enhance and generalize Noether's second theorem. More specifically, we prove that a system of differential equations is abnormal, i.e., it has an identically vanishing differential consequence, if and only if it possesses a trivial conserved vector corresponding to a nontrivial characteristic. Moreover, the above properties are also equivalent to that the system admits a family of characteristics that is parameterized by an arbitrary function of all independent variables in a local way and whose Fréchet derivative with respect to the parameter-function does not vanish on solutions of the system for a value of the parameter-function. The theorem is illustrated by physically relevant examples.
April 17 Melanie Graf: The Hawking and Penrose Singularity Theorem.
Abstract: As all singularity theorems, the singularity theorem of Hawking and Penrose proves that under certain (mainly curvature and causality) assumptions a spacetime has to be singular, i.e., there exists an incomplete causal geodesic. In this talk I will present this theorem and briefly discuss some of its assumptions and its relation to the other singularity theorems. The second part of the talk will be dedicated to sketch some parts of the proof that highlight the difficulties one faces when trying to show an analogous theorem for C^(1,1) metrics.
April 24 Eduard Nigsch: The theory of vector valued distributions and its applications.
May 8 Enxin Wu: An overview of the technique of diffeology.
May 15 Lorenzo Luperi Baglini: A fixed-point iteration method for arbitrary generalized ODE.
May 22 Paolo Giordano: Universal properties of Colombeau algebras.
May 29 Roland Steinbauer: Geodesics in non-expanding impulsive gravitational waves with $\Lambda$.
June 12 Walter Simon: Initial data for rotating cosmologies.
June 19 Norbert Ortner (University of Innsbruck): On the space $\dot{\mathcal{B}}$ of Laurent Schwartz.
June 26 Clemens Sämann: Geodesic completeness and global hyperbolicity for non-smooth spacetimes.

## Winter term 2014

Date Speaker
October 3 First meeting for scheduling
October 10 No seminar
October 17 No seminar
October 24 Lorenzo Luperi Baglini: An introduction to Nonstandard Analysis with some applications.
Abstract: We assume that the audience of this seminar does not know NSA. We will try to explain its basic ideas both from an hystorical point of view and by mean of some simple examples. To avoid the heavy logical apparatus usually needed to construct nonstandard extensions, we will base our approach on some "topological" considerations. In the final part of the talk we will introduce a particular family of generalized functions, called ultrafunctions. We will finally show how ultrafunctions can be used to solve certain differential problems that have no classical solution.
October 31 Paolo Giordano: It would be beautiful if... Generalized Smooth Functions and their future steps.
Abstract: Cauchy, Heaviside and Dirac invented the Dirac delta by using an intuitive point of view, which is still in use among several physicists and engineers. After the work of L. Schwartz, they can all end up this "definition" by citing the fairy tail: "Of course, this is not an ordinary function... and the important property is that it works as a functional". However, they still get into (temporary) trouble by making some calculations with the Heaviside function... This happens because Schwartz substituted the effect (they are functionals) with the cause (they are not ordinary functions). Generalized smooth functions (GSF) are set theoretical functions defined and with values in the ring $\tilde{\mathbb{R}}$ of Colombeau generalized numbers that share several properties with ordinary smooth functions, but include all Colombeau generalized functions (and hence all Schwartz distributions). E.g. GSF are closed w.r.t. composition, can be differentiated using (generalized) incremental ratios, have chain rule, primitives, integration by substitution, intermediate value theorem, mean value theorems, any form of Taylor theorem, extreme value theorem and a useful sheaf property. We can also define spaces of compactly supported GSF and the corresponding strict inductive limit, which have very good properties and can be easily studied using $\tilde{\mathbb{R}}$ valued seminorms. Using the words above, we can say that GSF are a possible formalization of the cause, not of the effect. If time allows, we will explore some future steps, with the idea that in case I'll say bullshits, we can all have a good laugh: Banach-like fixed point theorem and Picard-Lindeloef theorem, very weak solution of any differential equation, Hahn-Banach theorem for generalized smooth functionals... This is a joint work with M. Kunzinger and H. Vernaeve. A minimal knowledge of Colombeau theory is assumed.
November 7 Eduard Nigsch (Wolfgang Pauli Institute): Reduction to trivial bundles.
Abstract: Well-known isomorphisms involving spaces of sections of vector bundles are usually proved by reduction to the case of the trivial line bundle. I will explain the conceptual background of this reduction precedure in easy category theoretical terms using the notion of additive functors and categories. Moreover, I will point out how in principle the respective isomorphisms can be seen to hold also topologically.
November 14 Shantanu Dave: Controlling geometry and analysis at infinity.
Abstract: This talk will provide an introduction to large scale geometry of metric spaces specially of Riemannian manifolds. In particular we recall some properties of rigidity of “quasi-regular” mappings and some implication to low regularity geometry. We shall consider examples of Gromov hyperbolic spaces for illustration. Some aspects like conformal stability of asymptotic dimensions will be covered. The main aim to argue that these ideas are also relevant in semi-Riemannian settings and a plan of some ongoing research.
November 21 Shantanu Dave: Controlling geometry and analysis at infinity, part 2.
November 28 Clemens Sämann: Global hyperbolicity for spacetimes with continuous metrics.
December 12 David Rottensteiner (Imperial College London): Time-Frequency Analysis on the Heisenberg Group.
Abstract: I'll give a summary of my PhD thesis, in which I studied the Weyl quantization and modulation spaces on the Heisenberg group H_n. The connection between these two concepts is very strong in the Euclidean, i.e., \R^n-setting, but it can also be recovered for arbitrary nilpotent Lie groups such as the H_n. Of particular importance in this context is the work of N.V. Pedersen (1949-1996).
January 9 Milena Stojković: Strong differentiability of the exponential map for $C^{1,1}$ metrics.
Abstract: This talk is based on the paper "Convex neighborhoods for Lipschitz connections and sprays" by Ettore Minguzzi, in which he proves that the exponential map of a $C^{1,1}$ metric is locally a bi-Lipschitz homeomorphism and strongly differentiable at the origin. We will recall Peano's definition of strong differential and its basic properties, as well as Leach's inverse function theorem. The proof of the strong differentiability of the exponential map will pass through a local analysis based on the Picard-Lindelöf approximation method. It will then follow that the inverse is Lipschitz by means of Leach's inverse function theorem.
January 16 Alexander Lecke: Geodesics in low regularity.
January 23 Alexander Lecke: Geodesics in low regularity, part 2.

## Summer term 2014

Date Speaker
March 7 First meeting for scheduling.
March 14 No seminar
March 21 Alexander Lecke: An Introduction to Direct Methods in Variational Calculus.
Abstract: I will give a short introduction to direct methods in the calculus of variations, answering questions like: 'What is a direct method?', 'Do solutions exist?', and 'Where are problems?'
March 28 Michael Oberguggenberger (University of Innsbruck): Stochastic Fourier integral operators.
April 4 Milena Stojković: Causality Theory for $C^{1,1}$ Lorentzian metrics.
Abstract: In the standard references to causality theory smoothness of the metric is usually assumed. However, it was of interest for some time to determine the minimal degree of regularity of the metric for which standard results of causality theory remain valid. A reasonable candidate is given by metrics of class $C^{1,1}$ since it represents the threshold where one still has unique solvability of the geodesic equation. The first part of the talk is going to be about the main results of causality theory with smooth metrics and the second part is about the key ingredients and techniques used to develop it for $C^{1,1}$ metrics and show that fundamental results hold true in this case.
April 11 Paolo Giordano: How many big-Os do you know? I.e. how to get rid of quantifiers in the full Colombeau algebraHow many big-Os do you know? I.e. how to get rid of quantifiers in the full Colombeau algebra
Abstract: The full Colombeau algebra (CA) has been introduced to have an intrinsic embedding of Schwartz distributions. The resulting definition is usually perceived as more complicated w.r.t. the special CA, first of all for the greater number of quantifiers involved. We'll see how a suitable definition of "set of indices" permits to introduce new notions of big-O formally behaving as the usual one and allowing a unification of several CA. This is a typical work of foundational nature, i.e. rising from the collision of a lacking of beauty and the searching for a better understanding in topics usually taken for granted. I hope this will also give us the opportunity to discuss about the comparative difficulty in creating definitions, statements, and proofs in Mathematics.
May 2 No seminar
May 9 Lorenzo Luperi Baglini: Asymptotic Gauges
Abstract: In some sense, two of the basic ingredients in the Colombeau constructions are the set of indeces $(0,1]$ and of choice of the "growth condition" given by the net $1/\varepsilon$ with $\varepsilon \in (0,1]$. The idea behind the notion of asymptotic gauges is that the usual Colombeau constructions can be generalized by taking (almost) any set of indeces and (almost) any growth condition defined starting from that set of indeces. This generalization can be done by preserving in a quite natural way many of the constructions and results of the usual Colombeau case. In particular, we will show how to generalize the notions of Colombeau special and full algebras and how they can be applied to solve some very easy differential equations with generalized coefficients. If time permits, we will also show that, in some sense, from this generalized point of view the distinction between special and full algebras vanishes.
May 16 Melanie Graf: Distributions with support in a semi-Riemannian hypersurface.
Abstract: It is well-known that any distribution on $\mathbb{R}^n$ with support in a single point can be expressed as a sum of derivatives of Dirac's delta distribution. The goal of this talk is to present a similar result for distributions with support in a hypersurface. In the beginning we will quickly review some basic facts about distributions on manifolds (taking into account the added structure given by a semi-Riemannian metric). Then we are going to define single- and multilayer distributions and use them to give alternate expressions for both the pullback of delta on a submanifold and the exterior derivative of a function with a jump discontinuity across a hypersurface. Last but not least we will show that any distribution with support in a (closed, oriented, semi-Riemannian) hypersurface can be written as a locally finite sum of such multilayers (sketching the proof if time permits).
May 23 Albert Huber: Allgemeine Relativitätstheorie, niedrige Regularität
June 6 Michael Kunzinger: Geodesics in low regularity.
Abstract: In this talk I will mainly elaborate a very instructive example, due to Hartman and Wintner, showing how standard properties of geodesics may fail if the regularity of the Riemannian metric is below $C^2$. If time permits I will also make some general remarks on the existence of shortest paths as well as on the relation between shortest paths and geodesics for $C^1$-metrics.