Faculties » Faculty of Mathematics » Chairs » Mathematics VII - Analysis

Spring 2021

The talks start at 16.15 .

15.06.2021   Georgios Dimitroglou Rizell (Uppsala) " tba"    via Zoom

04.05.2021   Tobias Weich (Paderborn) "SRB measures for Anosov actions"    via Zoom

Abstract: SRB (Sinai-Ruelle-Bowen) measures for hyperbolic flows (Anosov flows) are known to be among the most significant invariant measures. Using recently established spectral techniques (Ruelle-Taylor resonances) we construct SRB-measures for general higher rank Anosov actions. We examine their properties and provide a Bowen formula which allows to express these measures in terms of closed orbits. This is based on joint work with Y. Guedes-Bonthonneu, C. Guillarmou and J. Hilgert.

27.04.2021   Alexandre Jannaud (Neuchâtel) "Dehn-Seidel twist, C^0 symplectic geometry and barcodes"    via Zoom

Abstract: In this talk I will present my work initiating the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory. Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to $C^0$ settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that no power of the square of the generalized Dehn twist is not in the connected component of the identity in the group of symplectic homeomorphisms. Doing so, we prove the non-triviality of the $C^0$ symplectic mapping class group of some Liouville domains.

WINTER 2020/21

The talks start at 16.15 .

26.01.2021   Klaus Niederkrüger (Lyon 1) "About symplectic fillings of real projective spaces "    via Zoom
                                  THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR  

Abstract: (Joint work with Paolo Ghiggini.) The standard contact structure of a real projective space is always strongly fillable, but it was shown by Eliashberg-Kim-Poltervich, that from dimension 5 on, they are not Weinstein fillable. I will explain a geometric argument that shows that every real projective space of dimension 4k+1 is not Liouville fillable.

19.01.2021   Lucas Dahinden (Heidelberg) "Introducing sub-Riemannian Billiards"    via Zoom

Abstract: In sub-Riemannian geometry we study Riemannian manifolds with movement restricted to a bracket generating sub-bundle of the tangent bundle. We introduce a billiard reflection law in two natural ways: on the one hand by a variational principle and on the other hand by a symplectic description as characteristic flow lines at the boundary of a domain in the cotangent bundle. The two definitions coincide. Finally, we examine the highly symmetric example of the Heisenberg group, which reveals strong connections to magnetic billiards.

12.01.2021   Johanna Bimmermann (Heidelberg) "On the Hofer–Zehnder capacity for twisted tangent bundles over closed surfaces"    via Zoom

Abstract: In this talk I will present the computation of the Hofer–Zehnder capacity for magnetic systems on closed surfaces with constant (weak) magnetic field. While finding a lower bound for the Hofer–Zehnder capacity is relatively easy, as any admissible Hamiltonian function provides one, finding an upper bound is much harder. By a theorem of G. Lu for closed symplectic manifolds an upper bound is given by the symplectic area of a homology class that has a non-vanishing Gromov–Witten invariant. Our strategy is therefore, to find an embedding of the magnetic system into a closed symplectic manifold. We will then use the theorem to find an upper bound of the Hofer–Zehnder capacity. Finally we will see that upper and lower bound agree and therefore determine the Hofer-Zehnder capacity.

15.12.2020   Otto van Koert (Seoul) "A generalization of the Poincare-Birkhoff fixed point theorem
                                                                      and the restricted three-body problem"
    via Zoom
                                  THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR  

Abstract: In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem.

08.12.2020   Benoit Joly (Paris) "Dynamical construction of barcodes of Hamiltonian homeomorphisms of surfaces"    via Zoom

Abstract: The notion of barcodes appears to be a useful tool in C^0 symplectic topology and can be seen as a « path » of all spectral invariants. Nevertheless, the construction relies on Floer Homology which needs a C^2 setting. I will present a new construction of barcodes of Hamiltonian homeomorphisms of oriented compact surfaces which relies on Le Calvez's transverse foliation theory. For a homeomorphism of a surface Le Calvez proved that there exists C^0 foliations associated to some « maximal » isotopies. Moreover, in the case of Hamiltonian homeomorphisms, the foliations are « gradient-like ». This property allows us to construct new barcodes without Floer Homology. I will present the dynamical tools and then I will give the ideas to construct theses barcodes.

01.12.2020   Samuel Lisi (Oxford, USA) "Spinal Open Book Decompositions"    via Zoom
                                  THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR  

Abstract: In work with Jeremy Van Horn-Morris and Chris Wendl, we have introduced the notion of a spinal open book decomposition for contact manifolds in dimension 3. This can be thought of as a generalization of an open book decomposition. The main purpose is to describe a large class of contact manifolds as supported by planar (but spinal) open books. By use of J-holomorphic curve techniques, we are then able to classify symplectic fillings.

17.11.2020   Matthias Meiwes (Aachen) "Hofer's geometry and entropy"    via Zoom

Abstract: A central object in the study of Hamiltonian diffeomorphisms on a symplectic manifold is Hofer's metric, a bi-invariant metric on the group of Hamiltonian diffeomorphisms that displays rigidity features that are special for those diffeomorphisms. The geometry of this metric has been thoroughly studied since its discovery by Hofer and his work in the early 90's and a central theme is to link Hofer's geometry to dynamical properties of the underlying maps. In my talk I will discuss some conditions under which dynamical complexity persists under bounded perturbations in Hofer's geometry. This leads to stable lower bounds on topological entropy and on orbit growth in various situations. This talk is partly based on joint work with Arnon Chor.

10.11.2020   Bernd Stratmann (Bochum) "Nowhere vanishing primitive of a symplectic form and removing parametrized rays symplectically"    via Zoom

Abstract: In the talk, two new results will be explained. The first one is that an exact symplectic form always admits a nowhere vanishing primitive. The second result says that one can extract a ray from an (open) symplectic manifold and the resulting manifold is always symplectomorphic to the inital manifold. The latter can be generalized under a condition to parametrized rays. Both results are proven with classical methods.

03.11.2020   Will Merry (ETH Zürich) "Symplectic cohomology of magnetic cotangent bundles"    via Zoom
                                  THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR  

Abstract: Joint work with Y. Groman (and, time-permitting, Seongchan Kim). We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles.

27.10.2020   Shira Tanny (Tel Aviv) "The Poisson bracket invariant: soft and hard approaches"    via Zoom

Abstract: In 2006 Entov and Polterovich proved that functions forming a partition of unity with displaceable supports cannot commute with respect to the Poisson bracket. In 2012 Polterovich conjectured a quantitative version of this theorem. I will discuss three interconnected topics: a solution of this conjecture in dimension two (with Lev Buhovsky and Alexander Logunov), a link between this problem and Grothendieck's theorem from functional analysis (with Efim Gluskin), and new results related to the Floer-theoretical approach to this conjecture (with Yaniv Ganor).