# Hauptseminar Symplektische Geometrie

## Time: Wednesdays 4 - 6 pm

## Place: INF 205 / SR 4

## SoSe 18:

April 6: BGHK Seminar in Köln

April 18:

May 17: BGHK Seminar in Bochum

June 12: Alexander Ritter (Oxford) – The cohomological McKay correspondence via Floer theory (HS Geometrie)

June 26: Felix Schmäschke (Berlin) – TBA (HS Geometrie)

July 3: Leonid Polterovich (Tel Aviv) – TBA (RTG Colloquium)

July 9-13: Conference Symplectic Dynamics in Heidelberg

## WiSe 17/18:

Oct 6: BHKM Seminar in Köln

Oct 25: Seongchan Kim (Augsburg) – $J^+$-like invariants and the restricted three body problem

Oct 27/28: Geometric Dynamics Days in Heidelberg

Nov 9: BHKM Seminar in Bochum

Nov 14: Umberto Hryniewicz (Rio de Janeiro) – Using holomorphic curves to construct Conley blocks that obstruct hyperbolicity (HS Geometrie)

Nov 22: Lucas Dahinden (Neuchatel) – Lower complexity bounds for positive contactomorphisms

Nov 28: Marco Mazzucchelli (Lyon) – Minimal Boundaries in Tonelli Lagrangian Systems (HS Geometrie)

Dec 6: Alexandru Doicu (Augsburg) – Compactness Result for $\mathcal{H}-$holomorphic Curves in Symplectizations

Dec 7: BHKM Seminar in Münster

Dec 12: Kai Zehmisch (Münster) – ODD-SYMPLECTIC! (HS Geometrie)

Jan 10: Takahiro Oba (Tokyo Institute of Technology) – Mapping class group techniques for higher-dimensional contact and symplectic manifolds

Jan 17: Marc Kegel (Köln) – The knot complement problem for Legendrian and transverse knots

Jan 24: Tobias Dietz (Leipzig) – Yang-Mills moduli spaces over a surface via Fréchet reduction by stages

Feb 12-14:

**Workshop**

*Holomorphic Methods in Symplectic Geometry***with Hansjörg Geiges (Köln), Will Merry (ETH Zürich), Dietmar Salamon (ETH Zürich), Samuel Trautwein (ETH Zürich) and Kai Zehmisch (Gießen)**

## Abstracts:

**Seongchan Kim**–

**$J^+$-like invariants and the restricted three body problem**

Recently, Cieliebak-Frauenfelder-van Koert observed disasters which can happen in a family of periodic orbits in the planar circular restricted three body problem and defined new invariants of such families, based on Arnold's $J^+$-invariant. In this talk, we recall their results and determine those invariants for a distinguished class of periodic orbits in the restricted three body problem, i.e., periodic orbits of the second kind. This talk is based on a joint work with Joontae Kim.

**Umberto Hryniewicz**–

**Using holomorphic curves to construct Conley blocks that obstruct hyperbolicity**

In this talk we would like to explain how to use finite-energy curves to construct isolated Conley blocks for 3-dimensional Reeb flows. Some of these blocks have trivial Conley index, but non-trivial chain recurrent set. Then we will explain how to use results of Bowen and Franks to prove that there can be no uniform hyperbolicity inside the block. Finally, examples in celestial mechanics will be presented. The construction of the blocks is based on joint work with Salomão.

**Lucas Dahinden – Lower complexity bounds for positive contactomorphisms**

There are various results that connect topological properties of special contact manifolds with the topological entropy of their Reeb flows, see for example Macarini--Schlenk, Frauenfelder--Labrousse--Schlenk, Alves or Alves--Meiwes. The proof of results of this type uses growth (in various senses) of symplectic homology or Rabinowitz--Floer homology.

I will explain how to push one of these results from the realm of Reeb flows to positive contactomorphisms (i.e. time-dependent Reeb flows). I will also explain why positive contactomorphisms seem to be the maximal class of maps for which such a kind of result holds true.

**Marco Mazzucchelli – Minimal Boundaries in Tonelli Lagrangian Systems**

In this talk, which is based on joint work with Luca Asselle and Gabriele Benedetti, I will present a few recent results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an orientable closed surface. I will show that in every level of a suitable low energy range there is a "minimal boundary": a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of the surface. Minimal boundaries turn out to satisfy an analogue of the celebrated graph theorem of Mather: in the tangent bundle, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base. I will also present some corollaries of these statements to the existence of action minimizing simple periodic orbits with low energy on non-orientable closed surfaces, and to the existence of infinitely many closed geodesics on certain Finsler 2-spheres.

**Kai Zehmisch – ODD-SYMPLECTIC!**

Motivated by recent results by Fish-Hofer about non-minimality of Hamiltonian flows on compact energy surfaces in the 4-space I will introduce the odd-symplectic cobordism relation. Using holomorphic curves and local contact Dehn surgery, further, I will explain how to construct an infinite sequence of Hamiltonian flows on the 3-sphere, for which Gottschalk’s conjecture about non-minimality remains an open question. The surgical construction of the corresponding exotic odd-symplectic 3-spheres is joint work with Hansjörg Geiges.

**Alexandru Doicu – Compactness Result for $\mathcal{H}-$holomorphic Curves in Symplectizations**

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. This modification of the pseudoholomorphic curve equation was first suggested by Hofer [H] and used extensively in the program initiated by Abbas et al. [ACH] to prove the strong Weinstein conjecture in dimension three. However, due to the lack of a compactness result of the moduli space of $\mathcal{H}-$holomorphic curves, Abbas and his coworkers were only able to prove the strong Weinstein conjecture in the planar case, i.e. when the leaves of the holomorphic open book decomposition have zero genus. In this talk we describe a compactification of the moduli space of finite energy $\mathcal{H}-$holomorphic curves under certain conditions. This is joint work with Urs Fuchs.

[H] H. Hofer,

*Holomorphic curves and real three-dimensional dynamics*, Geom. Funct. Anal. Special Volume 2000, Part II (2000) 674 – 704

[ACH] C. Abbas, K. Cieliebak, H. Hofer,

*The Weinstein conjecture for planar contact structures in dimension three*, Comment. Math. Helv.

**80**(2005) 771 – 793

**Takahiro Oba – Mapping class group techniques for higher-dimensional contact and symplectic manifolds**

I constructed (4n-1)-dimensional contact manifolds with infinitely many Stein fillings by open books and Lefschetz fibrations. In fact, this construction is a generalization of a low-dimensional example, and combinatorics of mapping class groups play an important role. In this talk, first I will discuss some difficulties of higher-dimensional open books and Lefschetz fibrations. After this, I will focus on mapping class group techniques and present my construction. Also, I will explain why the construction does not work well in the other dimensions.

**Marc Kegel – The knot complement problem for Legendrian and transverse knots**

The famous knot complement theorem by Gordon and Luecke states that two knots in the 3-sphere are equivalent if and only if their complements are homeomorphic. In this talk I want to discuss the same question for Legendrian and transverse knots and links in contact 3-manifolds. The main results are that Legendrian as well as transverse knots in the tight contact 3-sphere are equivalent if and only if their exteriors are contactomorphic.

**Tobias Dietz – Yang-Mills moduli spaces over a surface via Fréchet reduction by stages**

In Yang-Mills theory, many important properties of the physical system are encoded in the structure of the moduli space of connections with respect to the group of gauge transformations. Atiyah and Bott studied this moduli space in their seminal paper by using symplectic reduction adapted to the special case of a Riemann surface as the base manifold. The functional analytic problems were approached using Sobolev space techniques. In this talk, I will show how these results can be extended and reformulated in the framework of Fréchet manifolds. In the spirit of reduction by stages, we first take the quotient by the free action of based gauge transformations and in the second step consider the singular action by the finite-dimensional residual group. In particular, central Yang-Mills connections are realized as a subset of the Fréchet manifold of based gauge equivalence classes of connections and are identified as the inverse image under the Wilson holonomy loop map.

## SoSe 17:

April 21: BHKM Seminar in Köln

May 8: Myeonggi Kwon (Heidelberg) - Morse-Bott spectral sequence in symplectic homology — part I

May 29: Myeonggi Kwon (Heidelberg) - Morse-Bott spectral sequence in symplectic homology — part II

June 2: BHKM Seminar in Heidelberg

June 12: Myeonggi Kwon (Heidelberg) - Morse-Bott spectral sequence in symplectic homology — part III

June 23: BHKM Seminar in Bochum

July 3: Jungsoo Kang (Bochum) - Local systolic inequalities in contact and symplectic geometry