SYMPLECTIC DYNAMICS

9 - 13 July 2018

Ruprecht-Karls-Universität Heidelberg

This conference is part of the *Focus Semester **Symplectic Dynamics*

at

MAThematics Center Heidelberg (MATCH)

All talks will take place at the University of Heidelberg, in the

Maths building (Mathematikon) on the 5-th floor.**Directions**

**Speakers and Titles:**(Abstracts are below)

- Ed Belbruno (Princeton) – Capture in Celestial Mechanics Using Dynamical Systems and Stochastic Methods
- Joel Fish (Boston) – Feral curves and minimal sets
- Michael Gekhtman (Notre Dame) – Cluster structures on Poisson-Lie groups
- Sonja Hohloch (Antwerpen)
- Jungsoo Kang (Seoul) – Systoles in contact geometry
- Yael Karshon (Toronto)
- Alexandru Oancea (Paris)
- Leonid Polterovich (Tel Aviv)
- Richard Schwartz (Brown) – Geometry, Computer, and Art (Public Lecture)
- Sergei Tabachnikov (PennState) – Cross-ratio dynamics on ideal polygons
- Anna Wienhard (Heidelberg) – Symplectic geometry of representation varieties
- Kai Zehmisch (Gießen)

**Short Talks**:

- Gabriele Benedetti (Heidelberg) – On the curvature of magnetic flows
- Milena Pabiniak (Köln)
- Lei Zhao (Augsburg) – Consecutive collision orbits in the planar circular restricted three-body problem
- Jagna Wiśniewska (ETH Zürich) – Rabinowitz Floer Homology for Tentacular Hamiltonians

Tentative schedule | |
---|---|

Monday | |

9:30 - 10:00 | Registration and Coffee |

10:00 - 11:00 | Sergei Tabachnikov – Cross-ratio dynamics on ideal polygons |

Coffee break | |

12:00 - 13:00 | Kai Zehmisch |

Lunch break | |

15:00 - 15:30 | Jagna Wiśniewska – Rabinowitz Floer Homology for Tentacular Hamiltonians |

Coffee break | |

16:00 - 16:30 | Lei Zhao – Consecutive collision orbits in the planar circular restricted three-body problem |

17:00 - $\infty$ | Wine and Cheese Reception |

Tuesday | |

10:00 - 11:00 | Anna Wienhard – Symplectic geometry of representation varieties |

Coffee break | |

12:00 - 13:00 | Joel Fish |

Lunch break | |

15:00 - 16:00 | Ed Belbruno – Capture in Celestial Mechanics Using Dynamical Systems and Stochastic Methods |

Coffee break | |

17:00 - $\infty$ | Discussions |

Wednesday | |

9:30 - 10:00 | Milena Pabiniak |

Coffee break | |

11:00 - 12:00 | Yael Karshon |

Coffee break | |

12:30 - 13:00 | Gabriele Benedetti |

Free afternoon | or hike of the Philosophenweg |

19:00 - $\infty$ | Conference dinner at Die Burgfreiheit |

Thursday | |

10:00 - 11:00 | Sonja Hohloch |

Coffee break | |

12:00 - 13:00 | Jungsoo Kang |

Lunch break | |

15:00 - 16:00 | Michael Gekhtman |

Coffee break | |

17:00 - 18:00 | Public Lecture by Richard Schwartz – Geometry, Computer, and Art |

Friday | |

10:00 - 11:00 | Alexandru Oancea |

Coffee break | |

12:00 - 13:00 | Leonid Polterovich |

Lunch break | |

Departure | |

Unfortunately, we have limited space available. If you want to participate you are kindly requested to send an e-mail to Mrs Nicole Umlas (numlas "at" mathi.uni-heidelberg.de) not later than Monday, May 21, 2018. Please indicate whether you wish to participate in the conference dinner on Wednesday. If you want us to book a hotel room, please mention this in your e-mail to Mrs Umlas. We may have some funds to support the hotel costs.

This meeting is supported by

**MAThematics Center Heidelberg (MATCH)**and

**SFB/TRR 191 - Symplectic Structures in Geometry, Algebra and Dynamics**.

Organizing board: Alberto Abbondandolo (Bochum), Hansjörg Geiges (Köln), Peter Albers (Heidelberg), Gabriele Benedetti (Heidelberg)

**Abstracts:**

**Ed Belbruno – Capture in Celestial Mechanics Using Dynamical Systems and Stochastic Methods**

The problem of capture about the small body in the Newtonian restricted three and four body problems can be solved using weak stability boundaries, offering an understanding of the capture dynamics. It also offers important applications to aerospace engineering where new routes to the Moon can be found using much less fuel. One of these was demonstrated in 1991 to rescue a Japanese lunar spacecraft. Recently, work on a different topic in cosmology involves regularization of the big bang singularity using dynamical systems and stochastic methods. These methods have an interesting bearing on the capture problem.

**Gabriele Benedetti – On the curvature of magnetic flows**

In this talk, which report on joint work with Jungsoo Kang and Luca Asselle, we discuss the role of curvature in the study of magnetic flows on surfaces. In particular, we analyse its relation with integrable flows, Zoll flows and systolic inequalities in this category.

**Joel Fish – Feral curves and minimal sets**

I will discuss some current joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral curves) to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian in $\mathbb{R}^4$, then no regular energy level of H is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

**Michael Gekhtman – Cluster structures on Poisson-Lie groups**

The connection between cluster algebras and Poisson structures is by now well-documented. Among the most important examples in which this connection has been utilized are coordinate rings of double Bruhat cells in semisimple Lie groups equipped with (the restriction of) the standard Poisson–Lie structure. In this talk, based on the joint work with M. Shapiro and A. Vainshtein, I will describe a construction of a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson–Lie group GL(n), a generalized cluster structure on GL(n) compatible with the push-forward of the dual Poisson–Lie bracket and exotic cluster structures on GL(n) compatible with Poisson–Lie brackets arising from the Belavin-Drinfeld classification

**Jungsoo Kang – Systoles in contact geometry**

The systole of a contact manifold is the minimal period of periodic Reeb orbits. Two miracles of systoles happen for convex hypersurfaces. First, systoles measure the symplectic size of convex hypersurfaces. Second, there is a bound on the systole of a convex hypersurface in terms of the volume (namely, a contact systolic inequality). After reviewing such facts, I will talk about some extensions of these phenomena to general contact manifolds. This is joint work with Gabriele Benedetti.

**Sergei Tabachnikov – Cross-ratio dynamics on ideal polygons**

Define a relation between labeled ideal polygons in the hyperbolic space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics. This is work in progress, joint with M. Arnold, D. Fuchs, and I. Izmestiev.

**Lei Zhao – Consecutive collision orbits in the planar circular restricted three-body problem**

In the restricted three-body problem, consecutive collision orbits are those orbits which start and end at collisions with one of the primaries. Interests for such orbits arise not only from mathematics but also from various engineering problems. In this talk we shall discuss the existence of such orbits via the use of Floer theory. This is a joint work with Urs Frauenfelder.