Summer Retreat

CRC/TRR 191

August 21–25, 2023

Ludwig-Windthorst-Haus

Gerhard-Kues-Straße 16

49808 Lingen-Holthausen

Germany

Practical information

Arrival:

August, 21 – before dinner

Departure:

August, 25 – after lunch

(pdf)

Calender

Speakers

Peter Albers, Christian Lang, Leonard Marks

Flows and Frameworks

We start with a live demonstration of our recent advances on WebGL-based visualization of flows on the 3-torus and S3, exemplified with carefully chosen examples. In the second part of this talk, we present a framework that simplifies collaboration by easy and consistent dissemination of custom plugins within the visualization framework ParaView.

Peter Albers, Filip Sadlo, Robin Fleige

Discontinuities and Bifurcations

In the first part of this talk, we report on the progress in subproject C6, with a focus on scientific visualization. We present a generalization of vector field topology to dynamical systems exhibiting discontinuities, and discuss the involved implications regarding uniqueness and reversibility of transport. In the second part, we present an extension of bifurcation analysis to arbitrary number of parameters.

Souheib Allout

On the closed geodesic problem in Lorenz manifolds

It is known that closed geodesics always exist in compact Riemannian manifolds. In this talk we show that this is no longer true in the Lorentzian counterpart. If time permits, we address also some further open questions and work in progress in this direction.

Barney Bramham

Topological entropy and geodesic flow on surfaces

After a general review of the results in project B1, I will focus on two subprojects: 1) Joint with Z. Zhang we use a renormalisation scheme to show that a large class of pseudo-rotations are limits of integrable systems, despite potentially wild dynamics. This relates to a question of Katok, which asks whether (in low dimensions) the set of zero entropy conservative systems is equal to the closure of the set of integrable systems. 2) Recently begun work of M. Stange to construct an infinite energy pseudo-holomorphic plane with interesting asymptotics. More precisely, in the symplectisation of a mapping torus of a monotone twist map, the goal is to construct a plane with infinite lambda energy but finite and non-zero omega-energy, that is asymptotic in the negative direction to (part of) the Aubry-Mather set. The other direction should be asymptotic to another invariant set that intuitively corresponds to the smallest min-max of the action functional. Besides providing an interesting class of examples of infinite energy pseudo-holomorphic curves, it could be a first step towards using more symplectically invariant methods on monotone twist maps.

Stéphanie Cupit-Foutou

On the Gromov-width of spherical compact Kähler manifolds

This talk deals with the part of C3 concerned with computing the Gromov-width of multiplicity-free Hamiltonian manifolds equipped an action of connected compact Lie group; we shall focus of the compact and Kähler ones (aka. spherical varieties). An overview of the results obtained so far in this subproject will be given; a geometric approach, based on these results, to work out the case in general in the 3rd phase of the SFB will be outlined.

Hansjörg Geiges

Magnetic symplectic fillings

It is known that the unit cotangent bundle of a hyperbolic surface, equipped with the Liouville form, admits nondiffeomorphic (strong) symplectic fillings, constructed by attaching nondiffeomorphic symplectic caps to one boundary component of a cylindrical semi-filling. Reporting on joint work with Kai Zehmisch, I present an alternative construction: First modify the standard filling, the disc cotangent bundle with the canonical symplectic form, by adding a magnetic term near the zero section. Then the topology of the filling can be changed by performing Gompf sums. In particular, we show that any finitely presented group can be realised as the fundamental group of a symplectic filling.

Gerhard Knieper

Marked length spectrum rigidity and Thurston's strech metric

In 1984 Burns and Katok conjectured that the marked length spectrum of a closed Riemannian manifold of negative sectional curvature determines the Riemannian metric up to isometry. I will discuss recent results and relations to Thurston’s stretch metric which has been introduced by Thurston on Teichmüller space.

Markus Kunze

Twist maps and minimal geodesics

We explain what B2 is about (mostly for the twist map part) and outline the results that have been obtained so far for our main piece of interest, which are non-periodic twist maps.

Markus Kunze

Symplectic methods in infinite-dimensional systems

We explain what B6 is about and then focus on the particular example of the Vlasov-Poisson system and results related to those equations.

George Marinescu

Geometric quantization

After a short presentation of the activities of the project A3, I will present some recent result about Berezin-Toeplitz quantization.

Stefan Nemirovski

Lorentz and contact geometry

About four decades ago, Penrose initiated the study of the spaces of light rays of spacetimes and noticed that they carry a canonical contact structure. Much later, it was noticed that global contact rigidity plays a role and provides answers to some of Penrose's questions. The project is devoted to the resulting interplay of the two fields - there are direct connections (e.g. one can try to describe certain Lorentz phenomena in contact terms) as well as inspiring (dis)similarities. I will give a brief survey of what's been done already and what we hope to get done in the 3rd period.

Markus Reineke

Floer potentials, cluster algebras and quiver representations

This is a report on progress in project A8, joint with Peter Albers and Maria Bertozzi. We interpret Floer potentials (encoding holomorphic disk counts) of exotic monotone Lagrangian tori in the projective plane as so-called F-polynomials of representations of Markov quivers (encoding Euler characteristic of Grassmannians of subrepresentations) via mutation rules in cluster algebras. All these notions will be explained.

Giulio Sanzeni

Non existence of closed null geodesic in Kerr spacetimes

We define the Kerr-star spacetime and sketch the idea of the proof of the theorem.

Duc Viet Vu

Geometry of singular spaces

After a short presentation of the activities of the project A2, our focus will be on results and problems in the geometry of the space of Kähler potentials and the degeneration of Kähler-Einstein metrics in Kähler spaces.

Kai Zehmisch

Non-spin bLobs?

It seems that in the theory of non-fillable contact manifolds a case was overlooked: Are there closed weakly fillable contact manifolds of dimension greater or equal than 7 that admit a bordered Legendrian open book, which could be orientable but is not spin. Non of the potential fillings can be semi-positive.

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