# Symplectic Techniques in Dynamical Systems

## ICMAT, Campus de Cantoblanco, Madrid

### November 11-15, 2013

Instituto de Ciencias Matemáticas
C/ Nicolás Cabrera, 13-15
Campus Cantoblanco - UAM
www.icmat.es
Telephone: +34 91 2999704

Alberto Abbondandolo
Title: A local non-squeezing theorem on infinite dimensional phase spaces
Abstract:
I will discuss the proof of a non-squeezing theorem for symplectomorphisms on an infinite dimensional Hilbert space which are close to the identity and some related open questions.

Marta Batoréo
Title: On hyperbolic fixed points and periodic orbits of symplectomorphisms
Abstract:
We will discuss a variant of the Conley conjecture which asserts the existence of infinitely many periodic orbits of a symplectomorphism if it has a fixed point which is "unnecessary" in some sense. More specifically, we discuss a result which claims that, for a certain class of closed monotone symplectic manifolds, any sympectomorphism isotopic to the identity with a hyperbolic fixed point must necessarily have infinitely many periodic orbits as long as the symplectomorphism satisfies some constraints on the flux.

Michael Bialy
Title: Rigidity for totally integrable convex billiards
Abstract:
I am going to report on recent progress on rigidity of totally integrable convex billiards on surfaces of constant curvature. I will explain also that this kind of rigidity holds true also for magnetic billiards. I will discuss Integral geometric inequalities related to this phenomenon.

Paul Biran
Title: Quantum Invariants and Lagrangian Cobordisms
Abstract:
We will discuss calculation of new and older invariants associated to the quantum homology of Lagrangian spheres and other manifolds. We will also explain the relation of these invariants with the ambient quantum homology, as well as how these invariants behave under Lagrangian cobordisms. Joint work with Cedric Membrez.

Lev Buhovski
Title: Unboundedness of the first eigenvalue of the Laplacian in symplectic category
Abstract:
I will discuss a certain symplectic flexibility result concerning the first eigenvalue of the Laplacian. Previous results in this direction were obtained by Leonid Polterovich, and by Dan Mangoubi.

Marc Chaperon
Title: Generating maps and applications to invariant manifolds and conjugacy
Abstract:
For quite a long time, it has been observed that symplectic maps can be defined locally (and sometimes globally) by generating functions having one foot in the source space and the other in the target space. The same remark applies to general differentiable maps and provides a rather pleasant geometric way to get invariant manifolds or conjugacies.

Octav Cornea
Title: Categorification of Seidel's representation
Abstract:
Seidel's representation of the fundamental group of the group of Hamiltonian diffeomorphisms of a symplectic manifold admits a categorification involving the fundamental groupoid of the Hamiltonian diffeomorphisms groups, the derived Fukaya category of the manifold and a cobordism category. In the talk - based on joint work with Francois Charette - I will explain this picture and give some examples.

Başak Gürel
Title: On non-contractible periodic orbits of Hamiltonian diffeomorphisms
Abstract:
In this talk I will discuss a variant of the Conley conjecture for non-contractible periodic orbits of Hamiltonian diffeomorphisms of atoroidal closed symplectic manifolds.

Alberto Ibort
Title: On a symplectic Radon transform
Abstract:
We will describe an extension of the Radon transform to states of classical systems defined on a symplectic manifold $$M, \omega$$ equipped with a momentum map $$J$$.
The theory of quantum tomograms and its relation to group representation theory will reviewed. It will be shown that the symplectic tomograms of a classical state is, as in the case of quantum tomograms, the Fourier transform of the weighted character of the action of the group. The symplectic tomograms of the canonical, or Gibbs state of a Hamiltonian $$H$$ will be computed in the case the $$G$$ is a symmetry of the system by judiciously using the Duistermaat--Heckman formula.

Ely Kerman
Title: All boundaries of contact type can keep secrets
Abstract:
Let $$M, \omega$$ be a symplectic manifold of dimension at least four with a nonempty boundary of contact type. In the early nineties Eliashberg and Hofer posed and investigated the following question: What does the interior of $$M$$ know about its boundary?
Among other conclusions they produced examples of (unseen) contact boundaries which were not fully visible from the interior. In this talk I will show that no contact type boundary is fully visible (rigid). In particular we show that the full action spectrum of the boundary is never observable from the interior. The proof involves the construction of a new dynamical plug for Reeb flows.

David Martínez
Title: Transverse geometry of codimension one foliations calibrated by a closed 2-form
Abstract:
A codimension one foliation (M,F) is calibrated by a closed 2-form w, if this form makes every leaf of F symplectic. Such foliations are natural generalizations of 3-dimensional taut foliations. A version of Donaldson's techniques produces 3-dimensional submanifolds of (M,F,w) in which F induces a taut foliation. In this talk I shall explain how these embedded 3-dimesional taut foliations capture the transverse geometry of the ambient foliated manifold (M,F).

Title: Symplectic methods in c-projective geometry
Abstract:
In my talk will show how effective standard methods of symplectic geometry work in the theory of c-projectively equivalent metrics. I will of course explain what this theory is about, it is one of the classical branches of differential geometry. The results of my talk will include local and global classification of c-projectively equivalent metric and a proof of Yano-Obata conjecture.

Will J. Merry
Title: Orderability and the Weinstein Conjecture
Abstract:
Joint work with Peter Albers and Urs Fuchs.
In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the (non)existence of positive loops of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Klaus Niederkrüger
Title: Understanding subcritical surgeries with holomorphic curves
Abstract:
A common technique of modifying contact manifolds consists in using contact surgery. The aim of our work (in progress) is to show that the belt sphere of a subcritical surgery is contractible in every symplectically aspherical filling. This is a joint work with P. Ghiggini and C. Wendl.

Yaron Ostrover
Title: From Symplectic Measurements to Billiard Dynamics and Back
Abstract:
We shall discuss how symplectic capacities on the classical phase space can be used to obtain bounds on the length of the shortest periodic billiard trajectory in a convex domain. Moreover, going in the other direction, we will explain how billiard dynamics can be used to obtain information on the symplectic size of certain configurations in the phase space. This talk is based on a joint work with Shiri Artstein-Avidan and Roman Karasev.

Leonid Polterovich
Title: Symplectic intersections and invariant measures
Abstract:
We discuss a method which enables one to detect invariant measures with large rotation vectors for a class of Hamiltonian flows. The method, which is based on symplectic topology, is robust with respect to perturbations of the Hamiltonian in the uniform norm and is applicable even in the absence of homologically non-trivial closed orbits of the flow.

Ana Rechtman
Title: Existence of periodic orbits of geodesible and Euler vector fields on 3-manifolds
Abstract:
A non-singular vector field on a closed manifold is geodesible if there is a Riemannian metric making its orbits geodesics. This condition can be interpreted in terms of the curl of the vector field: a vector field is geodesible if its is parallel to its curl. We will study the existence of periodic orbits for such vector fields on closed 3-manifolds.
On 3-manifolds, K. Kuperberg constructed examples of vector fields without periodic orbits. On the other hand, C. H. Taubes proved that the Reeb vector field of a contact form has periodic orbits. Reeb vector fields are geodesible, and also suspensions are geodesible. If we assume that the ambient manifold is either diffeomorphic to the three sphere or has non trivial second homotopy group, we will prove the existence of a periodic orbit for volume preserving geodesible vector fields.
Volume preserving geodesible vector fields form a subset of the vector fields satisfying the time-independent Euler equations, we will explain how the above result extends to the second family of vector fields.

Sheila Sandon
Title: Floer Homology for Translated Points
Abstract:
In 2011 I proposed a contact version of the Arnold conjecture, based on the notion of translated points of contactomorphisms. I will now discuss a work in progress to prove this conjecture for all contact manifolds, by constructing a Floer homology theory for translated points. I will also discuss why I expect this Floer homology theory to provide a perfect tool to study contact rigidity phonomena such as orderability, non-squeezing and existence of bi-invariant metrics on the contactomorphism group.

Iskander Taimanov
Title: Periodic magnetic geodesics on almost all energy levels via variational methods
Abstract:
We shall explain how to establish the existence of periodic magnetic geodesics for exact magnetic fields on almost all energy levels by using variational methods. In particular, we demonstrate how to justify the "throwing out cycles" principle for almost all energy levels.

Chris Wendl
Title: Existence of unknotted Reeb orbits
Abstract:
Hofer's 1993 paper on the Weinstein conjecture proves more than just the existence of a Reeb orbit: in the situations where it applies, it produces orbits that are contractible, thus proving that the contact manifold in question is not hypertight. More recent extensions of these methods, both in dimension 3 and higher, establish the so-called "strong" Weinstein conjecture in various settings, which states the existence of a set of Reeb orbits whose sum is nullhomologous. In general, however, the orbits obtained in this way need not be so geometrically nice, e.g. a contractible orbit could also be multiply covered, and the disk that it bounds need not be embedded. I will describe some work in progress with my student, Alexandru Cioba, using Siefring's intersection theory of punctured holomorphic curves to rule out such possibilities in dimension 3. We can show for instance that for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere (e.g. anything overtwisted or subcritically Stein fillable), generic contact forms admit an embedded and unknotted Reeb orbit.

San Vu Ngoc
Title: Symplectic geometry for classical and quantum magnetic fields
Abstract:
In the late 1970's, there have been many works on the relationships between "the classical Hamitonian has a periodic flow" and "the quantum spectrum has regularly spaced clusters". These operators with "periodic bicharacteristics" were studied using a microlocal refinement of the standard averaging method in classical mechanics. In the case of a magnetic Laplacian, even if the classical Hamiltonian does not have a periodic flow, the existence of the well-known "cyclotron" motion makes it possible to employ a similar method (via symplectic techniques) , and yields very precise asymptotics for the quantum eigenvalues. This is joint work with Nicolas Raymond.

Scientific and
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