## Summer 2024

The talks start at 14.15 in Seminar Room IA 1/177 .

**18.07.2024** Stefan Matijevic "Positive (S^1-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality"**Abstract:**

We show that the positive (S^1-equivariant) symplectic homology of a convex do-

main is naturally isomorphic to the singular (S^1-equivariant) homology of Clarke’s dual associ-

ated with the convex domain. Consequently, we show that Gutt–Hutchings capacities coincide

with the spectral invariants introduced by Ekeland-Hofer on the set of convex domains. Also,

as a corollary, we get that barcode entropy associated with the singular homology of Clarke’s

dual is a lower bound for the topological entropy of the Reeb flow on the boundary of a con-

vex domain in R^2n. In particular, the above-mentioned barcode entropy coincides with the

topological entropy for convex domains in R^4.

**11.07.2024** Jacobus Sander de Pooter "Minimality for non-convex Reeb flows in certain cotangent bundles"

**02.07.2024** Michael Vogel *"*Local higher systolic inequalities in higher dimensions*"* EXCEPTIONALLY AT 16.15 IN ROOM IA 1/181**Abstract:**

In 2018 ABHS proved a local systolic inequality for Reeb flows on S3. I.e. for all contact forms in a neighbourhood of a Zoll one they gave an upper bound on the length of the shortest Reeb orbit in terms of the contact volume. Since then this has been generalised multiple times. For instance, the same statement has been extended to contact manifolds in any dimension. On the other hand, in dimension three a similar statement involving higher systolic inequalities has been proven. In this talk we will present some progress on unifying these two directions.

**16.05.2024** Paul Anton Wilke *"Morse homology for Dirac equations"* **Abstract:** tba **25.04.2024** Johanna Bimmermann *"Magnetic billiards and the Hofer-Zehnder capacity of disk tangent bundles of lens spaces"* CHANGE: **Friday, May 3** at **12:15** p.m. in the seminar room **IA 01/473!!!****Abstract:** We compute the Hofer–Zehnder capacity of disk tangent bundles of certain lens spaces with respect to the round metric. This gives a first example, where Gromov width and Hofer–Zehnder capacity of a disk tangent bundle disagree. Techniques we use include for the lower bound magnetic billiards and for the upper bound pseudoholomorphic curves.**18.04.2024** Lucas Dahinden *"On the Causal Discontinuity of Morse Spacetimes"* **Abstract:** Morse spacetimes are generalisations of globally hyperbolic Lorentz spacetimes: they feature singularities where time-level sets undergo a topology change. Borde and Sorkin conjectured that such Morse spacetimes are causally continuous if and only if neither the index nor the coindex of the critical points is 1. This has been recently confirmed by Garcia Heveling for the case of small anisotropy and Euclidean background metric. Here, we provide a complementary counterexample: a four dimensional Morse spacetime whose critical point has index 2 and large enough anisotropy is causally discontinuous. Thus, the Borde-Sorkin conjecture does not hold. The proof features a low regularity causal structure and causal bubbling.

## Winter 2023/24

The talks start at 14.15 in Seminar Room IA 1/177 .**01.02.2024** Jan Eyll *"Systolic Inequalities in Riemannian Surfaces"* **Abstract:** The systole of a nonsimply connected Riemannian manifold is the length of the shortest noncontractible closed curve. In the 1940s, it was discovered that for any Riemannian metric on the two-torus, its area is bounded from below by a constant multiple of the square of the systole. Inequalities of this kind are known as systolic inequalities, and in this talk we will present and prove sharp systolic inequalities for some explicit surfaces. **25.01.2024** Lucas Dahinden *"C^0-stability of topological entropy for geodesic flows in dimension 3"* **Abstract:** Topological entropy is a numerical measure of how chaotic a map is. We are interested in how topological entropy depends on the map, especially how stable it is under perturbation. For example for diffeomorphisms homotopic to the identity on three dimensional manifolds it is well known that topological entropy can collapse to 0 after smooth perturbations, so it can be very unstable. We have managed to prove that for Reeb flows on three dimensional co-oriented compact contact manifolds topological entropy is surprisingly stable: a generic Reeb flow (with respect to \(C^\infty-topology)\) is a point of lower semi-continuity (with respect to C^0-topology) of the topological entropy. The tool that allows us to conclude this is a mechanism discovered by Matthias Meiwes, by which one can approximate topological entropy using links of Reeb chords extracted from a horseshoe. In our work we show stability of these links under perturbation. This is joint work with Marcelo Alves, Matthias Meiwes and Abror Pirnapasov.**18.01.2024** Stefan Nemirovski *"Sheaves, generating functions, and positive Legendrian isotopies"* **Abstract:** I will try to show in a simple example how the first two objects in the title are related and how they can be used to study the third.**11.01.2024** Lars Kelling *"On the geometric construction of the Chas-Sullivan product"* **Abstract:** Given a closed oriented Riemannian manifold M, Chas and Sullivan introduced a product on the loop homology, i.e. the ordinary homology of the space of (free) loops in M. For two chains its product is given by the intersection of marked points in M followed by the concatenation of appropriate loops. I will discuss its geometric construction as a lift of the classical intersection product with focus on a direct description of tubular neighbourhoods to avoid the delicate setting of normal bundles on infinite dimensional Hilbert manifolds.**14.12.2023** Niklas Tuschinski *"Der Satz von Arnold-Liouville"* **23.11.2023** Patrick Henning *"Computing ground states of Bose-Einstein condensates"* **Abstract:** In this talk we discuss the ground states of Bose-Einstein condensates which can be characterised as global minimizers of an energy functional on a Riemannian manifold. By formulating a suitable Riemannian gradient flow and by studying its decay properties, we infer a corresponding discrete gradient flow that can be used for practical computations. We will further discuss its global convergence and the asymptotic decay rates. For that we will show and exploit various links to a certain class of nonlinear eigenvalue problems.**16.11.2023** Pierre-Alexandre Arlove *"Contact orderability and spectral selectors"* **Abstract:** Some contactomorphisms groups and isotopy classes of Legendrians are known to admit a natural partial order. In this talk I will use this partial order to define functions on the latter spaces analogous to the spectral invariants in Symplectic Geometry coming from Lagrangian Floer homology. For Legendrians, I will show that these functions are spectral selectors and, from there, derive dynamical applications such as the existence of translated points and Reeb chords. After discussing the non-degeneracy of the selectors, I will present applications to the geometric study of these spaces, namely : construction of time-functions and the study of various (bi-invariant) metrics. This is a joint work with Simon Allais.**09.11.2023** Souheib Allout *"Compact Lorentz manifolds without closed geodesics"* **Abstract:** In the previous Retreat I gave a short talk about constructing some examples of Lorentz manifolds without closed geodesics. In this talk, we give more details, examples and questions in this direction.**26.10.2023** Alberto Abbondandolo *"On Hamiltonian diffeomorphisms and domains"* **Abstract:** Suitably compactly supported Hamiltonian diffeomorphisms of the (2n-2)-dimensional ball can be lifted to smooth domains in the 2n-dimensional vector space.I will discuss this construction, its properties and how it can be used to construct counterexamples and symplectic embeddings. **19.10.2023** Alberto Abbondandolo *"Symplectic capacities and geodesics in the space of contact forms"* **Abstract:** An old open question in symplectic geometry is whether all normalized symplectic capacities coincide on convex bodies. I will discuss this question, together with its relationship with Viterbo‘s conjecture and systolic inequalities, and the argument for giving it a positive answer in the case of smooth convex bodies which are C^2-close to a Euclidean ball. This argument involves understanding minimizing geodesics in the space of contact forms equipped with a Banach-Mazur pseudo-metric.

## Summer 2023

The talks start at 14.15 in Seminar Room IA 1/177 .

**22.06.2023** Simon Vialaret *"Systolic inequalities for invariant contact forms on S^1-principal bundles"* **Abstract:** In Riemannian geometry, a systolic inequality aims to give a uniform bound on the length of the shortest closed geodesic for metrics with fixed volume on a given manifold. This notion generalizes to contact geometry, replacing the geodesic flow by the Reeb flow, and the length by the period. As opposed to the Riemannian case, it is known that there is no systolic inequality for general contact forms on a given contact manifold. In this talk, I will state and prove a systolic inequality for invariant contact forms on S^1-principal bundles over S^2.**15.06.2023** Liang Jin *"Weak KAM theory for contact vector fields"* **Abstract:** Contact vector fields are natural generalizations of Hamiltonian vector fields on contact manifolds. Motivated by the classical weak KAM theory established for Tonelli Hamiltonian vector fields, a similar theory for certain "Tonelli" contact vector fields on the manifold of 1-jets was partially built. In this talk, we shall give a very brief introduction of such a theory without going to the details, then offer simplest dynamical feedback from it.**04.05.2023** Johanna Bimmermann *"Capacities of Hermitian Symmetric Spaces"* **Abstract:** We will introduce Hermitian symmetric spaces and focus on their description as (co-)adjoint orbits. From this description it will be easy to give a Hamiltonian that generates a circle action. Finally this Hamiltonian circle action can be used to determine the Gromov width and the Hofer-Zehnder capacity of any Hermitian symmetric space of compact type.**27.04.2022** Michael Vogel *"b-contact structures on tentacular hyperboloids"* **Abstract:** b-symplectic and b-contact structures were introduced as a model for symplectic and contact manifolds with boundary. However, in recent years they have been studied in their own right. I will give an introduction to these singular structures and some of the known theory about them. Then I will present a construction relating them to tentacular hyperboloids.**20.04.2023** Jonas Fritsch *"The Arnold conjecture in the case of the symplectic torus"* **Abstract:** The Arnold conjecture states that any Hamiltonian diffeomorphism on a compact symplectic manifold, possesses at least as many fixed points as a smooth function on that manifold must possess critical points. This conjecture was first proven in the special case of the symplectic Torus by E. Zehnder and C. C. Conley. In this talk I will discuss a very similar proof of this case, by first reframing the question into a variational problem on the Sobolev space of loops on the torus, and then performing a saddle point reduction to restrict the problem to a finite dimensional subspace of the loop space to apply principles similar to Morse-Theory.

## Winter 2022/23

The talks start at 14.15 in Seminar Room IA 1/177 .**26.01.2023** Stefan Suhr *"Gromov-Hausdorff convergence and Lorentzian geometry, Part II"* **Abstract:** Following the discussion for the metric case I will propose a definition for Lorentzian-metric spaces and a Gromov-Hausdorff metric for these spaces. I will further discuss first results, especially stability of curvature and compactness. This is joint work with Ettore Minguzzi (Florence). **19.01.2023** Stefan Suhr *"Gromov-Hausdorff convergence and Lorentzian geometry, Part I"* **Abstract:** I will introduce the Gromov-Hausdorff metric for compact metric spaces and discuss first properties which will lead to a proof of the Gromov precompactness Theorem for spaces of bounded Ricci curvature. **12.01.2023** Barney Bramham *"Higher dimensional versions of Fathi’s formula for the Calabi invariant"* **Abstract:** For a Hamiltonian system on the 2-disc Fathi found a dynamical interpretation of the Calabi invariant as the average linking of pairs of trajectories. In this talk I will present formulae that extend this to arbitrary dimensions. **08.12.2022** Xian Dai *"Teichmüller theory and Thurston's asymmetric metric, Part III"* **01.12.2022** Xian Dai *"Teichmüller theory and Thurston's asymmetric metric, Part II"* **24.11.2022** Xian Dai *"Teichmüller theory and Thurston's asymmetric metric, Part I"* **Abstract:** In these talks, we will give an overview of the classical Teichmüller theory which involves the interplay of many mathematical subjects, including geometry, dynamics and topology. After the exposition, we will focus on an asymmetric Finsler metric on the Teichmüller space proposed by Thurston which has natural relation to study of extreme Lipschitz maps and length functions on hyperbolic surfaces.**17.11.2022** Luca Asselle *"Exotic Zoll magnetic systems on the two-torus via the Nash-Moser IFT, Part II "* **10.11.2022** Luca Asselle *"Exotic Zoll magnetic systems on the two-torus via the Nash-Moser IFT, Part I"* **Abstract:** In this series of talks I will show how to construct examples of magnetic systems on the two-torus (different from the trivial ones) with the property that every orbit with speed 1 is periodic. In the first talk we will reformulate the question as a problem of determining the pre-image of zero under a suitable „action“ map, and explain why the standard IFT fails in this setting. In the second talk we will instead show how such a map can be dealt with by mean of a Nash-Moser type IFT. **03.11.2022** Lucas Dahinden *"Robustness of Topological Entropy for Geodesic Flows"* **Abstract:** Topological entropy (=h) is a numerical invariant of maps that measures chaotic behaviour. It is well known, that h is not lower semi-continuous: There are examples where smooth perturbations lead to total collapse of h. However, sometimes a geometric feature prevents this collapse. In this talk we explore some of these features and discuss their consequences: On the 2-torus we investigate contractible closed geodesics, an intersection pattern that we call “ribbon” and on general manifolds we investigate a feature that we call “retractable neck”.

This is joint work with Marcelo Alves, Matthias Meiwes and Louis Merlin.**27.10.2022** Manuel Stange *"Construction of geodesic foliations"* **Abstract:** In my talk i will discuss a theorem by H.Gluck & D.Singer adressing the following problem: Given a Riemannian manifold N, which is seperated into two components by a compact submanifold M of codimension 1, and two geodesic fields G and G' transversal to M - When does there exist a new metric on N, which conincides with the original metric outside an arbitrary small neighbourhood U of M, and a field of geodesics (wrt. this new metric), which connects G to G' within U (according to some given map on M, which will describe exactly, what curves of G and G' will be pieced together).