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Dustin Connery-Grigg (IMJ-PRG):
Spectral invariants and dynamics of low-dimensional Hamiltonian systems
Since their introduction by Schwarz in 2000 (following an earlier idea of Viterbo), spectral invariants have become a central tool in the modern Floer-theoretic study of Hamiltonian isotopies and diffeomorphisms. Unfortunately, given a particular Hamiltonian isotopy, it is often very difficult to compute its associated spectral invariants, and the relationship of these invariants to the underlying dynamics remains opaque. In this talk I will discuss a novel class of spectral invariants for Hamiltonian systems which share the main properties that make the classical spectral invariants useful, but which have the advantage of admitting a completely dynamical interpretation for generic Hamiltonian systems on surfaces.
(Abstract ein-/ausblenden)
Emilia Alves (Universidade Federal Fluminense, Brazil):
Intersection of real Bruhat cells
We examine arbitrary intersections of real Bruhat cells. Arising in various contexts across several disciplines -- such as in the Kazhdan-Lusztig theory and the study of the spaces of locally convex curves -- these objects have attracted the attention of many authors. We present a stratification of an arbitrary pairwise intersection of real Bruhat cells. Also we show that the CW-complex which is roughly speaking a dual cell structure to such stratification, is homotopically equivalent to the intersection under analysis. Finally, both classical and new topological results about such intersections stem from our methods. We include many examples and perform explicit computations to illustrate our methods. This includes joint work with G. Leal (PUC-Rio), N. Saldanha (PUC-Rio), B. Shapiro (Stockholm University).
(Abstract ein-/ausblenden)
Leonardo Masci (RWTH Aachen):
A Poincaré-Birkhoff theorem for asymptotically linear Hamiltonian systems
The celebrated Poincaré-Birkhoff theorem on area-preserving maps of the annulus is of fundamental importance in the fields of Hamiltonian dynamics and symplectic topology. In this talk I will formulate a generalization of the Poincaré-Birkhoff theorem, which applies to asymptotically linear Hamiltonian systems on linear phase space. In order to explain some elements of the proof, I will explore the main difficulties in setting up a Floer homology for this class of Hamiltonian systems, and develop two techniques which can be combined to relate the filtered Floer homologies of different iterates of the same asymptotically linear Hamiltonian system.
(Abstract ein-/ausblenden)
Marcelo Alves (University of Antwerp):
C0-stability of topological entropy for 3-dimensional Reeb flows
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, and a generalisation of the C0-distance on the space of Riemannian metrics. I will explain the following recent result, obtained in collaboration with Lucas Dahinden, Matthias Meiwes and Abror Pirnapasov: the topological entropy of Reeb flows on contact 3-manifolds is lower semicontinuous with respect to the C0 metric on a C-infinity open dense set on the space of Reeb flows.
Applied to geodesic flows of Riemannian metrics on surfaces, this says that for "most" Riemannian metrics on closed surfaces, one cannot destroy positivity of topological entropy
by C0-small perturbations of the metric. This is in some sense unexpected, as the geodesic flow depends on the derivatives of the Riemannian metric.
(Abstract ein-/ausblenden)
Urs Fuchs (RWTH Aachen):
A primer on circle packings
Circle packings are well known to yield an approach to discretize conformal geometry on surfaces.
Projectively circle packed surfaces also naturally occur as boundaries at infinity of certain hyperbolic three manifolds.
I will review the relevant notions together with basic existence, uniqueness and density results. Along the way
we will see how circle packings arise from (and give rise to) interesting geometric constructions in two and three dimensions.
This talk is based on joint work with Jessica Purcell and John Stewart.
(Abstract ein-/ausblenden)
Michael Hutchings (University of California, Berkeley):
Unknotted Reeb orbits and the first ECH capacity
Patrice Le Calvez (Sorbonne Université):
Non-contractible periodic orbits for area preserving surface homeomorphisms
Jacob Rasmussen (University of Cambridge):
Floer homology for knots in the solid torus (joint seminar with EDDy)
In the first part of the talk, I'll discuss Floer homology for
3-manifolds.
The Seiberg-Witten invariants of a smooth 4-manifold M are defined by
counting solutions to a certain PDE on M. The theory of Floer homology
for 3-manifolds was developed to understand these invariants using cut
and paste topology: we split a given 4-manifold M in half along a
3-manifold Y and express the invariants of M in terms of relative
invariants of the two halves. The relative invariants live in a vector
space (the Floer homology) associated to Y. There's a related invariant
(knot Floer homology) which assigns a vector space to a knot inside a
3-manifold.
A similar thing happens when we split Y along a surface S, but now the
relative invariant is an object in a category. The second half of the
talk will focus on what happens when one of the two pieces is a solid
torus containing a knot. This is an extension of previous joint work
with Hanselman and Watson.
(Abstract ein-/ausblenden)
Murat Sağlam (Universität zu Köln):
Contact 3-manifolds with integrable Reeb flows
In this talk we discuss the existence of Reeb flows that admit a Morse-Bott type integral of motion on 3-dimensional closed contact manifolds. After addressing the obstruction on the topology of such manifolds, we present some methods to construct integrable Reeb flows and show that all contact structures on S^3, T^3 and S^1×S^2 admit such Reeb flows.
(Abstract ein-/ausblenden)
David Bechara Senior (RWTH Aachen):
The asymptotic action of area preserving disk maps and some of its properties
Given a compactly supported diffeomorphism of the disk that preserves the standard symplectic form, I will introduce the asymptotic action associated to this map. I will then show a pointwise formula relating the asymptotic action to the asymptotic winding number of pairs of points. As a corollary one obtains a new proof for a well known result by A. Fathi which gives a formula for the Calabi invariant of a disk map in terms of its mean winding numbers.
(Abstract ein-/ausblenden)
Rohil Prasad (Princeton University):
Volume-perserving right-handed vector fields are conformally Reeb
A non-singular vector field on rational homology 3-sphere is right-handed if, roughly speaking, any pair of long flowlines have positive linking number after closing them with small segments into loops. They exhibit a fascinating dynamical and topological property. Ghys (and later Florio-Hryniewicz) showed that any collection of periodic orbits bounds a global surface of section and is therefore a fibered link. With this in mind, it is natural to ask whether every right-handed vector field has a periodic orbit. In this talk, I will explain a proof that if a right-handed vector field preserves a smooth volume form, then it is equal to a Reeb vector field after multiplication by a positive smooth function, and therefore has at least two periodic orbits.
(Abstract ein-/ausblenden)
Abror Pirnapasov (ENS Lyon):
The mean action and the Calabi invariant
Hutchings used Embedded Contact Homology to show the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: if the asymptotic mean action on the boundary is greater than the Calabi invariant, then the infimum of the mean action of the periodic points is less than or equal to the Calabi invariant. In this talk, I explain how to extend this result to all orientation and area-preserving disc diffeomorphisms. This is joint work with Barney Bramham
(Abstract ein-/ausblenden)
Valerio Assenza (Universität Heidelberg):
Magnetic Curvature and Existence of Closed Magnetic Geodesics
A Magnetic System is the toy model for the motion of a charged particle moving on a Riemannian Manifold endowed with a magnetic field. Solutions for such systems are called Magnetic Geodesics and preserve the Kinetic Energy. One of the most relevant investigative interest in the theory is to understand the existence and in case the topological nature of Closed Magnetic Geodesics (periodic solutions) in a given level of the energy. I will introduce the Magnetic Curvature, an object which encodes the geometrical properties coming from the Riemannian Curvature structure together with terms of perturbation due to the magnetic interaction. We will see how a positive curved Magnetic System carries a Contractible Closed Magnetic Geodesic for small energies.
(Abstract ein-/ausblenden)
Urs Frauenfelder (Universität Augsburg):
GIT quotients and Symmetric Periodic Orbits
This is joint work with Agustin Moreno and Dayung Koh. The restricted three-body problem is invariant under various antisymplectic involutions. These real structures give rise to the notion of symmetric periodic orbits which simultaneously have a closed string interpretation namely as a periodic orbit as well as an open string interpretation as Hamiltonian chords. This makes the bifurcation analysis of symmetric periodic orbits very intriguing since under bifurcations two local Floer homologies are invariant, the periodic one as well as the Lagrangian one. In this talk we explain how methods from symmetric space theory can help to extract efficiently datas from reduced monodromy matrices of periodic orbits helping to analyse the possible bifurcation patterns.
(Abstract ein-/ausblenden)
Benoit Joly (Ruhr-Universität Bochum):
Barcodes for Hamiltonian homeomorphisms of surfaces
In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.
(Abstract ein-/ausblenden)
Alberto Abbondandolo (Ruhr-Universität Bochum):
Bi-invariant Lorentz-Finsler structures on the linear symplectic group and on the contactomorphism group
It is well known that the linear symplectic group and the contactomorphism group admit no bi-invariant metrics which are continuous with respect to the Lie group topology. I will show that they admit natural bi-invariant Lorentz-Finsler structures, which can be seen as generalizations of the anti-de-Sitter space time. I will discuss to what extent these structures can be used in order to define bi-invariant measurements and present several open questions. This talk is based on some recent joint work with Gabriele Benedetti and Leonid Polterovich.
(Abstract ein-/ausblenden)
Tobias Soethe (RWTH Aachen):
Sharp systolic inequalities for rotationally symmetric 2-orbifolds - Part II
In this talk I will show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed. In this context, suitable means that we consider systoles that are defined in terms of closed geodesics that lift to contractible loops on certain covers of the unit sphere bundle. This result generalizes an earlier result of Abbondandolo, Bramham, Hryniewicz and Salomão for spheres of revolution. This is joint work with Christian Lange.
(Abstract ein-/ausblenden)
Tobias Soethe (RWTH Aachen):
Sharp systolic inequalities for rotationally symmetric 2-orbifolds
In this talk I will show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed. In this context, suitable means that we consider systoles that are defined in terms of closed geodesics that lift to contractible loops on certain covers of the unit sphere bundle. This result generalizes an earlier result of Abbondandolo, Bramham, Hryniewicz and Salomão for spheres of revolution. This is joint work with Christian Lange.
(Abstract ein-/ausblenden)
Alberto Abbondandolo (Ruhr Universität Bochum):
CANCELED
Erman Çineli (IMJ-PRG):
Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective
In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.
(Abstract ein-/ausblenden)
Matthias Meiwes (RWTH Aachen):
Braid stability and Hofer's metric
A central object in the study of Hamiltonian diffeomorphisms on a symplectic manifold is Hofer's metric dH, a bi-invariant metric on the group of Hamiltonian diffeomorphisms that displays rigidity features that are special for those diffeomorphisms.
In my talk I will discuss a result stating that, under certain conditions, the braid type of a set of periodic orbits of Hamiltonian diffeomorphisms is stable under perturbations that are sufficiently small with respect to Hofer's metric.
This can be applied to obtain stability properties of topological entropy. For example one obtains that the topological entropy is lower semi-continuous on the group of Hamiltonian diffeomorphisms on a closed surface, as well on the group of compactly supported Hamiltonian diffeomorphisms on the 2-disc. This is joint work with Marcelo Alves.
(Abstract ein-/ausblenden)
Lucas Dahinden (Universität Heidelberg):
The Bott-Samelson Theorem for positive Legendrian Isotopies
The classical Bott-Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space, i.e., a quotient of a sphere. I will explain how this theorem is proved and generalized by weakening the assumption of the flow being geodesic (a positive Legendrian isotopy is enough), and being a flow (the isotopy needs not be autonomous).
(Abstract ein-/ausblenden)
Marco Mazzucchelli (ENS Lyon):
Existence of global surfaces of section for Kupka-Smale 3D Reeb flows
In this talk, which is based on joint work with Gonzalo Contreras, I will sketch a proof of the fact that a Kupka-Smale Reeb flow on a closed 3-manifold admits a Birkhoff section: a compact surface whose interior is embedded and transverse to the Reeb vector field, and whose boundary is immersed and tangent to the Reeb vector field. The proof employs the broken book decompositions of closed contact 3-manifolds, recently introduced in a seminal work of Colin-Dehornoy-Rechtman.
(Abstract ein-/ausblenden)
Yuri Lima (Universidade Federal do Ceará):
Symbolic Dynamics for Maps with Singularities in High Dimension
We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, as well as includes all the recent results of the literature. As an application, we prove exponential growth rate on the number of periodic orbits. Joint work with Ermerson Araujo and Mauricio Poletti
(Abstract ein-/ausblenden)
Pedro Salomão (Universidade de São Paulo):
Genus zero global surfaces of section for Reeb flows and a result of Birkhoff
An important result due to G. Birkhoff states that a simple closed geodesic of a positively curved Riemannian 2-sphere lifts to an annulus-like global surface of section for the geodesic flow on the unit sphere bundle. This talk will present a symplectic dynamics proof of Birkhoff's theorem and a generalization of this result for Reeb flows in dimension three. More precisely, let (M,ξ) be a closed contact 3-manifold that admits an open book decomposition with binding L and genus zero pages. If a Reeb flow on (M,ξ) has L as a set of periodic orbits and every periodic orbit in the complement of L is linked with the pages of the open book, then L is the boundary of a global surface of section for the Reeb flow. If the contact structure is trivial, it is possible to restrict the linking hypothesis to periodic orbits whose indices lie in a specific interval depending only on (M,ξ). This is joint work with U. Hryniewicz and K. Wysocki.
(Abstract ein-/ausblenden)
Gabriele Benedetti (VU Amsterdam):
The dynamics of strong magnetic fields on surfaces: periodic orbits and trapping regions
How does a magnetic field influence the motion of a charged particle on a surface? Are there periodic orbits or trapping regions for the particle? How difficult is to construct a magnetic field for which all orbits are periodic? In this talk we will see that, if the magnetic field is strong, a normal form going back to the Russian school allows us to use Birkhoff theorem and KAM theory to tackle these questions. This is joint work with Luca Asselle.
(Abstract ein-/ausblenden)
Jungsoo Kang (Seoul National University):
Symplectic Homology of Convex Domains
In the category of convex domains (as opposed to starshaped domains), the minimal period of the Reeb flow on the boundary is a symplectic capacity. One way to see this is to use Clarke's dual action functional which only exists under the convexity condition. I will describe the Morse homology of Clarke's dual action functional and its isomorphism with the symplectic homology. An application of the isomorphism will also be presented. This is joint work with Alberto Abbondandolo.
(Abstract ein-/ausblenden)
Stefan Suhr (Ruhr-Universität Bochum):
New developments in the theory of Lyapunov functions for cone fields
Lyapunov functions are a well known tool in the theory of dynamical systems to distinguish different regimes of the dynamics. A similar problem appeared in the theory of spacetimes for General Relativity, in the form of time/temporal functions. The theory of Lyapunov functions for cone fields joins both points of view into a single one. In my talk I will introduce Lyapunov functions for cone fields and the main results of the theory. At the end I will comment on new developments in the field. This is joint work with Patrick Bernard (ENS Paris).
(Abstract ein-/ausblenden)
Barney Bramham (Ruhr-Universität Bochum):
Symbolic Dynamics for Reeb Flows
I will explain how to construct symbolic dynamics for 3D Reeb flows with non-degenerate periodic orbits when a transversal foliation exists, e.g. via the Hofer-Wysocki-Zehnder holomorphic curve theory. The following dichotomy is a consequence: Either there is a global section or a horse-shoe. This is joint work with Umberto Hryniewicz and Gerhard Knieper.
(Abstract ein-/ausblenden)
Anna Florio (IMJ-PRG):
Torsion of Conservative Twist Maps on the Annulus
For a C1 diffeomorphism f on the annulus isotopic to the identity, the torsion is the limit of the average rotational velocity of the images of tangent vectors through the differential of f. We consider torsion of conservative twist map on the annulus and we discuss conditions to assure the existence of points with non-zero torsion. As an outcome, we prove that the set of points with non-zero torsion within bounded instability regions has positive Lebesgue measure.
(Abstract ein-/ausblenden)
Marcelo Alves (Université Libre de Bruxelles):
Symplectic invariants and topological entropy of Reeb flows
Reeb flows are an important class of dynamical systems which
appear in geometry, symplectic topology and mathematical physics. The
topological entropy is a non-negative number associated to a dynamical
system which quantifies the exponential instability of the system.
Positivity of the topological entropy means that the dynamical system
presents some type of chaotic behavior. In this talk I will explain
how one can extract information about the topological entropy of Reeb
flows from the behavior of invariants from symplectic topology. These
invariants are constructed, following a recipe discovered by Floer, by
studying the spaces of solutions of certain partial differential
equations on a symplectic manifold. The talk will be aimed also at
non-specialists. Our hope is to present to viewers some of the
techniques used to study the dynamics Reeb flows by looking at their
applications to the entropy problem.
(Abstract ein-/ausblenden)
Gabriele Benedetti (Universität Heidelberg):
Periodic motions of a charged particle in a stationary magnetic field
In this talk, I will present some recent developments about
periodic motions of a charged particle in a magnetic field on a closed
manifold. Using the topology of the manifold and tackling some
interesting analytic issues, one can show that periodic motions exists
for almost every value of the kinetic energy. When the manifold is
two-dimensional and the magnetic field is large with respect to the
kinetic energy much stronger results are known. As a sample, we show
that there exists a countable family of magnetic fields on the
two-torus such that all motions with unit kinetic energy are periodic.
(Abstract ein-/ausblenden)
Louis Merlin (RWTH Aachen University):
Global invariants of symmetric spaces
After I describe a Riemannian geometric structure of
somewhat algebraic nature (symmetric spaces), I'll discuss the minimal
entropy conjecture and its relation to the study of other global
Riemannian invariants, including the minimal volume of Gromov, the
length spectrum and the bounded cohomology.
(Abstract ein-/ausblenden)
