The workshop is aimed at graduate students that are interested in the topic. The first three days we will work in small groups on some foundations of the field. On Thursday we will again in small groups look at some more recent papers (or advanced topics) and have a conference dinner in the evening. Finally on Friday we shall have a small conference.

Every participant will play an active role in the seminar - either by preparing a talk in the first part of the workshop or by presenting their favourite paper on pseudoholomorphic curves in Hamiltonian dynamics.

**We distributed the topics acording to the preferences of the participants. Here is the list of topics and the timetable!**

The workshop will take place from **February 27th to March 3rd, 2023 ** at the Mathematical Institute Heidelberg 5th floor.

**Conference Speakers:**

**Alexandru Oancea:**Frobenius algebra structure for Rabinowitz Floer homology**Barney Bramham:**Higher dimensional versions of Fathi’s formula for the Calabi invariant*(at 11 am)*For a Hamiltonian system on the 2-disc A. Fathi found a dynamical interpretation of the Calabi invariant as the average linking of pairs of trajectories. In this talk I will present formulae that extend this to arbitrary dimensions.

**Chris Wendl:**How to view multiple covers as a feature instead of a bug*(at 2 pm)*As every symplectic topologist learns in Kindergarten, the moduli space of J-holomorphic curves is generically smooth only if one restricts to the open subset of simple curves; the multiple covers are a problem, and they cannot in general be perturbed away. This is a fundamental difficulty in every equivariant Fredholm problem, resulting from the essential incompatibility between transversality and symmetry. Sometimes, however, the symmetry is important and you really don't want to perturb it away -- and it turns out that sometimes, it is possible to achieve transversality anyway, or something almost as nice! In this talk I will present a general method for understanding such issues and illustrate it via two popular examples of equivariant Fredholm problems: (1) closed orbits for a 1-parameter family of contact forms or Hamiltonian structures, and (2) unbranched covers of J-holomorphic curves in a symplectic cobordism with only hyperbolic asymptotic orbits.

**Urs Frauenfelder:**The Symplectic Moon

*(at 9 am)*

Rabinowitz Floer homology can be seen as a version of symplectic homology which is adapted to the study of contact manifolds. I will explain the construction of the Frobenius algebra structure and discuss some implications. Joint work with Kai Cieliebak and Nancy Hingston.

*(at 3:45 pm)*

Newton once said, that if he thinks of the moon, he always gets a headache. In fact the orbit of the moon is much more involved than the orbits of the planets since the attractive forces of the earth and the sun on the moon are of comparable size. The idea of Hill was to approximate the orbit of the moon by a periodic orbit in a coordinate system where both the earth and the sun are at rest. This is the restricted three-body problem. It was Hill's work which inspired Poincare to his famous dictum that periodic orbits are the only breach to enter an otherwise impenetrable stronghold. His lifelong quest for periodic orbits brought him as well to formulate his concept of a global surface of section. In my talk I plan to discuss how questions which started with the description of the orbit of the moon finally led to advanced topics in Symplectic geometry like holomorphic curves in a symplectization and its connection with systolic questions for the restricted three-body problem.

**Organizers:**

- Johanna Bimmermann
- Levin Maier
- Richard Siefring