Tag - Symplectic geometry

Alfonso Sorrentino: On the Rigidity of Integrable Twist Maps of the 2-Dimensional Annulus

27th November, 2023

In the study of Hamiltonian systems, integrable dynamics play a crucial role. Integrability, however, appears to be a delicate property that is not expected to persist under generic small perturbations. Understanding the essence of this fragility presents a compelling task, which turns out to be relevant across different contexts. In this talk, I shall present some results aimed at shedding more light on this issue, within the framework of symplectic twist maps of the 2-dimensional annulus. Specifically, I shall investigate the persistence and the properties of invariant Lagrangian tori that are foliated by periodic points and discuss how their fragility underpins the rigidity of completely integrable twist maps.

Matthias Meiwes: C0 Stability of Topological Entropy for 3-Dimensional Reeb Flows

24th November, 2023

The C0 distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms. In this talk, I will explain some recent progress on the stability properties of the topological entropy with respect to this distance obtained in collaboration with M. Alves, L. Dahinden, and A. Pirnapasov. Our main result states that the topological entropy for closed contact 3-manifolds is lower semi-continuous in the C0 distance for C∞-generic contact froms. Applying our methods to geodesic flows of surfaces, we obtain that the points of lower-semicontinuity of the topological entropy include non-degenerate metrics. In particular, given a geodesic flow of such a metric with positive topological entropy, the topological entropy does not vanish for sufficiently C0-small perturbations of the metric.

Pazit Haim-Kislev: On the Existence of Symplectic Barriers

20th November, 2023

Results concerning rigidity of Lagrangian submanifolds lie at the heart of symplectic topology, and have been intensively studied since the 1990s. An example for this phenomenon is the concept of Lagrangian Barriers, a form of symplectic rigidity introduced by Biran in 2001, which involves obligatory intersections of symplectic embeddings with Lagrangian submanifolds not derived from mere topology. In this joint work with Richard Hind and Yaron Ostrover, we present what appears to be the first illustration of Symplectic Barriers (and in particular not Lagrangian). The key point being that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold does not exhibit flexibility. In our work, we also tackle a question by Sackel–Song–Varolgunes–Zhu and provide bounds on the capacity of the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.

Alberto Abbondandolo: Symplectic Capacities of Domains Close to a Ball and Geodesics in the Space of Contact Forms

13th November, 2023

An old open question in symplectic geometry asks whether all normalized symplectic capacities coincide for convex domains in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex domains which are C2-close to a Euclidean ball. This is related to the question of existence of minimizing geodesics in the space of contact forms on a closed contact manifold equipped with a Banach-Mazur-like metric. On the other hand, there are smooth domains which are arbitrarily C1-close to the ball for which the ball capacity is strictly smaller than the cylindrical capacity.

Marco Mazzucchelli: Locally Maximal Closed Orbits of Reeb Flows

6th November, 2023

A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gurel, I will present a 'forced existence' result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension:

- If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits. 

- If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption.

These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gurel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.

Oliver Edtmair: The Subleading Asymptotics of Symplectic Weyl Laws

3rd November, 2023

Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics (smooth closing lemma) and symplectic geometry (simplicity conjecture). In this talk, I will report on work in progress concerning the subleading asymptotics of symplectic Weyl laws. I will explain the connection to symplectic packing problems and the algebraic structure of groups of Hamiltonian diffeomorphisms and homeomorphisms.

Gabriele Benedetti: Rigidity and Flexibility of Periodic Hamiltonian Flows

30th October, 2023

An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In the last years, interest in this problem has been revitalized since recent research has unveiled a deep relationship between periodic Hamiltonian flows and systolic questions in symplectic and contact geometry. While only trivial examples of periodic flows among magnetic and central systems exist, Zoll and, later, Guillemin have shown that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin's approach, the goal of this talk is to show how the Nash-Moser implicit function theorem can be used to construct magnetic flows on the two-torus which are periodic for a single value of the energy.

Lukas Nakamura: A Metric on the Contactomorphism Group of an Orderable Contact Manifold

27th October, 2023

We discuss some properties of a pseudo-metric on the contactomorphism group of a strict contact manifold M induced by the maximum/minimum of Hamiltonians. We show that it is non-degenerate if and only if M is orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. We also discuss analogous results on isotopy classes of Legendrian submanifolds and on universal covers.

Han Lou: On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds

27th October, 2023

Arnold's conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this talk, I will show that the Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology.