# Mini-Workshop on Symplectic Geometry, Universiteit Utrecht, March 12, 2015

## Titles and abstracts for the talks

### Kai Cieliebak:

Knot invariants from contact homology and string topology, parts I and II

In the first part of the talk, a relative version of the Chas-Sullivan string topology operations is used to construct a knot invariant known as Lenny Ng's cord algebra. This invariant distinguishes the unknot from all nontrivial knots, and it contains some classical knot invariants such as the Alexander polynomial.

In the second part of the talk, I will show that the cord algebra agrees with a holomorphic curve invariant, the Legendrian contact homology of the unit conormal bundle.

### Felix Schlenk:

Entropy and slow entropy of Reeb flows on spherizations, and beyond

We consider Reeb flows on spherizations, that is, autonomous Hamiltonian systems on cotangent bundles whose energy levels are fiberwisse starshaped (with respect to the origin). For such flows, positive entropy has been shown by Macarini-Schlenk in the case that the base manifold has rich homology. We now look at the case that the homology is not so rich, ie, the fundamental group and the homology of the contractible based loops grow only polynomially. We then find a lower bound of the ``polynomial entropy'' of Reeb flows in terms of these homological data. In the last ten minutes, we sketch an extension due to Marcelo Alves to Reeb flows on certain closed contact 3-manifolds, that brings into play Legendrian contact homology.

This is joint work with Urs Frauenfelder and Clémence Labrousse, and work by Marcelo Alves.

### Rob van der Vorst:

The Braid classes and the Poincare-Hopf Theorem

Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the standard 2-disc may be regarded as braids, and a such determine relative braid classes. If the Braid Floer homology of a proper relative braid class is non-trivial, then additional closed integral curves of the Hamilton equations are forced via a Morse type theory. In this talk we show that the Euler-Floer characteristic of Braid Floer homology can be used to force closed integral curves of arbitrary vector fields and yields a Poincare-Hopf type Theorem. The Euler-Floer characteristic for any proper relative braid class can be computed via a finite cube complex that serves as a model for the given braid class.

### Frol Zapolsky:

On the Lagrangian Hofer geometry of the Clifford torus in CP^2

The space of Lagrangian submanifolds which are Hamiltonian isotopic to a given one L carries a natural Finsler metric, analogous to the Hofer metric on the Hamiltonian group. It is quite nontrivial to find lower bounds on this metric, and in general it is little studied. In this talk I'll demonstrate that in case L is the Clifford torus in the complex projective space CP^2, this space contains a quasi-isometrically embedded real line and in particular has infinite diameter. This is based on joint work with Misha Khanvevsky and uses results on Lagrangian spectral invariants obtained together with Remi Leclercq.