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 - 11:00Sergei Tabachnikov – Cross-ratio dynamics on ideal polygons
Coffee break
12:00 - 13:00Kai Zehmisch
Lunch break
15:00 - 15:30Jagna Wiśniewska – Rabinowitz Floer Homology for Tentacular Hamiltonians
Coffee break
16:00 - 16:30Lei Zhao – Consecutive collision orbits in the planar circular restricted three-body problem
17:00 - $\infty$Wine and Cheese Reception

Tuesday
10:00 - 11:00Anna Wienhard – Symplectic geometry of representation varieties
Coffee break
12:00 - 13:00Joel Fish
Lunch break
15:00 - 16:00Ed Belbruno – Capture in Celestial Mechanics Using Dynamical Systems and Stochastic Methods
Coffee break
17:00 - $\infty$Discussions

Wednesday
9:30 - 10:00Milena Pabiniak
Coffee break
11:00 - 12:00 Yael Karshon
Coffee break
12:30 - 13:00Gabriele 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 - 13:00Jungsoo Kang
Lunch break
15:00 - 16:00Michael Gekhtman
Coffee break
17:00 - 18:00Public Lecture by Richard Schwartz Geometry, Computer, and Art

Friday
10:00 - 11:00Alexandru Oancea
Coffee break
12:00 - 13:00Leonid 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.