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) – On focus-focus singularities in semitoric systems
- Jungsoo Kang (Seoul) – Systoles in contact geometry
- Yael Karshon (Toronto) – Non-linear Maslov index for lens spaces
- Alexandru Oancea (Paris) – Algebraic consequences of the convex/concave duality
- Leonid Polterovich (Tel Aviv) – Quantum footprints of symplectic rigidity
- Richard Schwartz (Brown) – Inscribing rectangles in polygons and Jordan loops
- (Public Lecture) Richard Schwartz – Geometry, Computer, and Art
- Sergei Tabachnikov (PennState) – Cross-ratio dynamics on ideal polygons
- Anna Wienhard (Heidelberg) – Symplectic geometry of representation varieties
- Kai Zehmisch (Gießen) – Symplectic dynamics of $S^3$

**Short Talks**:

- Gabriele Benedetti (Heidelberg) – On the curvature of magnetic flows
- Milena Pabiniak (Köln) – Sandon's conjecture about minimal number of translated points of a contactomorphism for lens spaces
- 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 – Symplectic dynamics of $S^3$ |

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 – Feral curves and minimal sets |

Lunch break | |

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

Coffee break | |

17:00 - 18:00 | Richard Schwartz – Inscribing rectangles in polygons and Jordan loops |

Wednesday | |

9:30 - 10:15 | Milena Pabiniak – Sandon's conjecture about minimal number of translated points of a contactomorphism for lens spaces |

Coffee break | |

11:15 - 12:00 | Yael Karshon – Non-linear Maslov index for lens spaces |

Coffee break | |

12:30 - 13:00 | Gabriele Benedetti – On the curvature of magnetic flows |

Lunch break | |

Free afternoon | or organized visit of the Schloß, Molkenkur and the Königstuhl |

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

Thursday | |

10:00 - 11:00 | Sonja Hohloch – On focus-focus singularities in semitoric systems |

Coffee break | |

12:00 - 13:00 | Jungsoo Kang – Systoles in contact geometry |

Lunch break | |

15:00 - 16:00 | Michael Gekhtman – Cluster structures on Poisson-Lie groups |

Coffee break | |

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

Friday | |

10:00 - 11:00 | Alexandru Oancea – Algebraic consequences of the convex/concave duality |

Coffee break | |

12:00 - 13:00 | Leonid Polterovich – Quantum footprints of symplectic rigidity |

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

**Sonja Hohloch – On focus-focus singularities in semitoric systems**

A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poisson- commute and induce an $(S^1 × \mathbb{R})$-action that has only nondegenerate, nonhyperbolic sin- gularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo & Vu Ngoc by means of five invariants.

Two of these five invariants are the so-called Taylor series invariant and the twisting index. The first one describes the behaviour near the focus-focus singular fibre and the second one compares the ‘distinguished’ torus action given near each focus-focus singular fiber to the global toric ‘background action’.

Recently there has be made some progress in computing these two invariants and, in this talk, we present the (results of the) finished and ongoing project with J. Alonso (Antwerp), H. Dullin (Sydney), and J. Palmer (Rutgers):

- Taylor series and twisting index for coupled spin oscillator and coupled angular momenta.

- Putting the twisting index in relation with wellknown notions from classical dynamical systems like rotation number, winding number, intersection number etc.

- Change of the Taylor series and twisting index when varying the parameters of the systems.

**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.

**Yael Karshon – Non-linear Maslov index for lens spaces**

Let G be the universal cover of the identity component of the contactomorphism group of a lens space. I will discuss the construction of a quasimorphism on G that can be used to detect discriminant points of contactomorphisms.

This work is joint with Gustavo Granja, Milena Pabiniak, and Sheila Sandon. Some of its applications for contact rigidity properties of lens spaces are discussed in the lecture by Milena Pabiniak; however, our lectures are independent of each other.

**Alexandru Oancea – Algebraic consequences of the convex/concave duality**

The Goresky-Hingston product on the cohomology of free loop spaces is the algebraic structure that detects geodesics with maximal index growth. It is defined topologically as a secondary operation, and I will explain how to interpret it as a primary operation in Floer homology. This algebraic fact is a consequence of the interplay between convexity and concavity regions of suitable Hamiltonians. Based on this, I will discuss the duality between symplectically degenerate maxima and symplectically degenerate minima for Reeb flows.

**Milena Pabiniak – Sandon's conjecture about minimal number of translated points of a contactomorphism for lens spaces**

Diffeomorphisms preserving a symplectic form enjoy many rigidity properties. One of the most striking, called the Arnold conjecture, is that the number of fixed points of those generated by Hamiltonian functions is bounded from below by the topology of the manifold. This question was translated by Sheila Sandon to contact geometry in terms of translated points of contactomorphisms.

Together with G. Granja, Y. Karshon and S. Sandon we prove Sandon's Conjecture for lens spaces equipped with the standard contact form. For that purpose we follow ideas of Givental and construct a quasimorphism G -> (R,+), i.e. a homomorphism up to bounded error, for G the universal cover of the identity component of the contactomorphism group of lens spaces.

This quasimorphism, called a non-linear Maslov index, helps to understand the contactomorphisms of lens spaces. Apart from Sandon's Conjecture we prove:

- that G is orderable,

- that G can be equipped with unbounded bi-invariant metrics (which is important because any bounded bi-invariant metric must be trivial),

- existence of non-displaceable pre-Lagrangian submanifolds of lens spaces,

- that any contact form defining the standard contact structure on lens spaces has closed Reeb orbits (Weinstein Conjecture).

In this talk I will present Sandon's Conjecture and outline its proof via non-linear Maslov index.

**Leonid Polterovich – Quantum footprints of symplectic rigidity**

We discuss interactions between quantum mechanics and symplectic topology including a link (joint work with Laurent Charles) between symplectic displacement energy, a fundamental notion of symplectic dynamics, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes.

**Say that a rectangle R graces a Jordan curve J if all 4 vertices of R lie in J and the cyclic ordering of these vertices is the same if they are considered as a subset of R or as a subset of J. I'll sketch a proof of the following theorem: Every Jordan curve has the property that all but at most 4 points are vertices of rectangles which grace the curve. When the Jordan curve is a generic polygon, I'll show that the set of these gracing rectangles forms a 1-manifold whose ends are critical points for the distance function, and that the components obey a conservation law. The picture makes me wonder if there is something related to symplectic geometry going on.**

Richard Schwartz – Inscribing rectangles in polygons and Jordan loops

Richard Schwartz – Inscribing rectangles in polygons and Jordan loops

**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.

**Jagna Wiśniewska – Rabinowitz Floer Homology for Tentacular Hamiltonians**

Rabinowitz Floer homology relates the question of existence of periodic orbits of a Hamiltonian system on a fixed energy hypersurface to the geometry of this hypersurface. I will present how to extend the definition of Rabinowitz Floer homology to non-compact hypersurfaces for a class of tentacular Hamiltonians. This is joint work with Federica Pasquotto.

**Kai Zehmisch – Symplectic dynamics of**$S^3$

Aperiodic symplectic flows on open manifolds strongly influence the underlying smooth structure. Equipped with natural boundary conditions such flows defined on knot exteriors lead to a characterization of the 3-sphere in terms of symplectic dynamics. In the first part of my talk I will explain how the diffeomorphism type is related to Eliashberg-Hofer’s filling by holomorphic discs argument. In the second I will indicate a construction of exotic symplectic flows whose defining odd-symplectic form (Hamiltonian structure) extends to a symplectic form on the 4-ball. Minimality properties for these flows are not known as well as properties of Fish-Hofer’s feral holomorphic curves.

**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.