# The DIANA seminar

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.

The the seminar takes place every Friday at 10.15 am in Seminarraum 11 (2nd floor) and will be announced by email weekly.

Anyone interested is welcome to attend.

## Winter Term 2018

Date Speaker Title
05. Oct. 2018 Scheduling
12. Oct. 2018 Clemens Sämann
A very short introduction to optimal transport
AbstractI will introduce the problem of optimal transport and show a basic existence result. If time permits an outlook on more advanced topics will be given. Some knowledge of measure theory is assumed.
19. Oct. 2018 Benedict Schinnerl
The Gannon-Lee theorems of general relativity.
AbstractAs singularity theorems the GL-results describe conditions under which a generic spacetime becomes singular (in the sense of geodesic incompleteness). However, compared to the classical theorems of Hawking and Penrose, they predict geodesic incompleteness under the existence of non-simply connected topology in a (partial) Cauchy surface (instead of e.g. the trapped surface condition in Penrose's theorem). I will discuss the original results obtained independently by Gannon and Lee in the mid-70's as well as refinements by Galloway and Costa e Silva.
09. Nov. 2018 Martin Kirchberger
Triangle Comparison Theorems
AbstractFirst, I will introduce the classic triangle comparison theorem by Toponogov. This theorem is generalized to semi-Riemannian manifolds in a recent paper by Stephanie B. Alexander and Richard L. Bishop titled "Lorentz and Semi-Riemannian Spaces with Alexandrov Curvature Bounds“, which was the starting point for my master’s thesis. The main result will be stated and explained together with an overview of the proof which involves interesting results obtained along the way, for example a unified law of cosines for all semi Riemannian surfaces of constant curvature.
16. Nov. 2018 Paolo Giordano
Lost mathematics: Grassmann's Ausdehnungslehre
AbstractHave you ever dreamt to return back in time of, let's say, 100 years to talk about your present mathematical knowledge? Grassmann essentially had this experience in 1844 talking of sum and product of points, product between point and vector, bivectors, multidimensional spaces, etc. His theory is one of the best way to deal with Euclidean geometry and several other problems. Clearly, no one understood him, and so his discoveries got almost completely lost. Nowadays, someone asked: what's the simplest way to teach geometry to a stupid computer? And someone else answered: well, it's surely Grassmann's Ausdehnungslehre! So, he was right saying: "For I have every confidence that the effort I have applied to the science reported upon here, which has occupied a considerable span of my lifetime and demanded the most intense exertions of my powers, is not to be lost... a time will come when it will be drawn forth from the dust of oblivion and the ideas laid down here will bear fruit... some day these ideas, even if in an altered form, will reappear and with the passage of time will participate in a lively intellectual exchange. For truth is eternal, it is divine; and no phase in the development of truth, however small the domain it embraces, can pass away without a trace. It remains even if the garments in which feeble men clothe it fall into dust".
23. Nov. 2018 Marco Erceg (University of Zagreb)
One-scale H-distributions
AbstractMicrolocal defect functionals (H-measures, H-distributions, semiclassical measures etc.) are objects which determine, in some sense, the lack of strong compactness for weakly convergent $L^p$ sequences. In contrast to the semiclassical measures, H-measures are not suitable to treat problems with a characteristic length (e.g. thickness of a plate), while more recant variants, one-scale H-measures, have property of being extension of both H-measures and semiclassical measures. However, H-measures, as well as one-scale-measures, are adequate only for the $L^2$ framework. As the generalisation of H-measures to the $L^p$ - $L^{p'}$ setting has already been constructed via H-distributions, here we introduce objects which extends the notion of one-scale H-measures, one-scale H-distributions, as a counterpart of H-distributions with a characteristic length.
30. Nov. 2018 Tobias Beran
Causality theory
AbstractI will introduce the notions of causality theory and some basic results, and prove the Penrose singularity theorem.
07. Dec. 2018 Katharina Brazda
Curvature Varifolds
AbstractCurvature varifolds are measure-theoretical generalizations of submanifolds that are equipped with a weak second fundamental form. I will give an elementary introduction to varifolds and briefly indicate how their compactness properties may help to understand the shapes of biological membranes.
14. Dec. 2018 Peter Buder
Stereotype spaces
11. Jan. 2019 Kevin Islami
Global solutions of the Yang-Mills Equations in Minkowski space
AbstractAn analysis of the Yang-Mills equations has been an active area of research in the time where Yang and Mills proposed their theory and it has not lost any interest nowadays due to their importance in physics. In this talk I will describe how the equations are obtained and discuss some global solution theorems of the original Yang-Mills equations and the case where the Yang-Mills field is coupled to a scalar field.
18. Jan. 2019 Manuel Seitz
Distributional Jump Formula
AbstractI will introduce the distributional jump formula and use it to verify the fundamental solution of certain differential operators.
25. Jan. 2019 Zhangwen Guo
Singular distributions on a manifold
AbstractWhen the dimension of a distribution is not assumed to be constant on each point, integrability is a stronger condition than involutivity, unlike that in the Frobenius theorem for distributions of constant rank. We will look at how these two properties and some properties regarding flows on vector fields of the distributions are related, and also we will see that a leaf of an integrable singular distribution is an initial submanifold, same as in the case of a distribution of constant rank.