# Oberseminar Geometrie und Analysis

Das Oberseminar gewöhnlich findet im Raum 008 SeMath, Pontdriesch 14-16 (1950|008) statt.

## Bevorstehende Vorträge im Oberseminar

## Vergangene Vorträge

**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 *C*^{1} 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)*

#
*Rabinowitz Floer Homology*

## RWTH Aachen 2019, 15-16 November

## Vortragende:

Prof. Dr. Kai Cieliebak

Prof. Dr. Urs Frauenfelder

**Abstract.** Rabinowitz Floer homology is the semi-infinite dimensional Morse homology in the sense of Floer associated to a Lagrange multiplier action functional used by Rabinowitz in his pioneering work on applying global methods to Hamiltonian dynamics. This action functional detects periodic orbits of fixed energy but arbitrary period. Since the period is allowed to be negative as well one can think of Rabinowitz Floer homology as a kind of Tate version of Symplectic homology. Rabinowitz Floer homology has a broad range of applications to various fields, including the theory of contact embeddings, magnetic fields and Mañé's critical values, translated points, the global perturbation theory of Hamiltonian systems, the contactomorphism group, symplectic homology, and string topology. While the critical point equation is local the gradient flow equation is not local anymore so that one can think of Rabinowitz Floer homology as an example of a nonlocal Floer homology.

In the talks we plan to explain the construction of Rabinowitz Floer homology and discuss some of its applications.

Hier klicken für weitere Informationen.