# Hauptseminar Symplektische Geometrie

## Time: Wednesdays 4 - 6 pm

## Place: INF 205 / SR 4

## SoSe 18:

April 6: BGHK Seminar in Köln

April 25: Luca Asselle (Gießen) – On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

May 17: BGHK Seminar in Bochum

May 23: Gabriele Benedetti (Heidelberg) – Zoll magnetic flows and magnetic curvature, Part I

June 5: Arnaud Maret (ETH Zürich) – Forcing relations for periodic orbits

June 6: Gabriele Benedetti (Heidelberg) – Zoll magnetic flows and magnetic curvature, Part II

June 11: Alexander Ritter (Oxford) – The cohomological McKay correspondence via Floer theory — the full truth (4 - 6 pm in room ???)

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

June 20: Andy Sanders (Heidelberg) – The symplectic geometry of the $SL(2,\mathbb{C})$ character variety

June 21: Paul Seidel (MIT) – Mirror Symmetry and Lagrangian Tori (Emil Artin Lecture)

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

July 3: Leonid Polterovich (Tel Aviv) – Persistence barcodes in analysis and geometry (RTG Colloquium)

July 9-13: Conference Symplectic Dynamics in Heidelberg

July 24: Stephan Mescher (Leipzig) – Topological complexity of symplectic manifolds (HS Geometrie)

## Abstracts:

**Gabriele Benedetti – Zoll magnetic flows and magnetic curvature**

Magnetic geodesics are curves on a closed Riemannian manifold with a prescribed acceleration, and can be thought as a generalization of standard geodesics. In the first part of the talk, we present a notion of curvature for magnetic flows and discuss its implication to the study of periodic magnetic geodesics. In the second part, we focus on Zoll magnetic flows, for which all magnetic geodesics are closed, and present a few open questions and some recent work in progress revolving around this topic.

**Periodic orbits play a leading role in the study of the dynamics of iterated transformations of a given space. This talk outlines an abstract approach to the analysis of dynamical forcing relations between periodic orbits of a fixed transformation. In other words, we present a strategy to answer the following question: "Under what conditions on a space, does the existence of some periodic orbit imply, in a general way, the existence of other orbits ?". Every periodic orbit is specified by topological properties that will serve as a base for the comparison. The expected forcing relations define a preorder on the set of such specifications. Understanding the preorder is of main interest in this context. The above framework is illustrated at its best by the Sharkovskii Theorem on interval dynamics. In dimension two, achievements have been made towards the establishment of an analogue for surface homeomorphisms.**

Arnaud Maret – Forcing relations for periodic orbits

Arnaud Maret – Forcing relations for periodic orbits

**Alexander Ritter – The cohomological McKay correspondence via Floer theory**

The goal of my talk is to present joint work with Mark McLean (Stony Brook, NY), which proves the cohomological McKay correspondence using symplectic topology techniques. This correspondence states that given a crepant resolution Y of the singularity Cn/G, where G is a finite subgroup of SL(n, C), the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y . This statement was proved by Batyrev (1999) and Denef-Loeser (2002) using algebraic geometry techniques. We instead construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

**Andy Sanders – The symplectic geometry of the SL(2,C) character variety**

Given a closed, connected, oriented surface S of genus at least two, the SL(2,C) character variety is the space of conjugacy classes of reductive homomorphisms of the fundamental group of S into SL(2,C). In this talk, I will try to give a historical account of the (symplectic) geometry of this space, with the starting point being the discovery of Atiyah-Bott (and later Goldman) that the SL(2,C) character variety has a natural holomorphic symplectic structure. In the course of this discussion, we will meet Teichmuller space, quasi-Fuchsian space, and the space of complex projective structures on the surface S. The ultimate goal of this talk is to explain the remarkable interaction between geometrically defined coordinate systems on these spaces, and the behavior of the Atiyah-Bott-Goldman symplectic structure with respect to these coordinate systems.

**While originated in topological data analysis, persistence modules and their barcodes provide an efficient way to book-keep homological information contained in Morse and Floer theories. I shall describe applications of persistence barcodes to symplectic topology and geometry of Laplace eigenfunctions. Based on joint works with Iosif Polterovich, Egor Shelukhin and Vukasin Stojisavljevic.**

Leonid Polterovich – Persistence barcodes in analysis and geometry

Leonid Polterovich – Persistence barcodes in analysis and geometry

Stephan Mescher – Topological complexity of symplectic manifolds

Stephan Mescher – Topological complexity of symplectic manifolds

Topological complexity (TC) was introduced by M. Farber as a numerical homotopy invariant motivated by the motion planning problem from robotics. It bears similarity with the Lusternik-Schnirelmann category. In this talk, I will present a result from joint work with Mark Grant in which we identify a topological condition on a symplectic manifold that ensures TC to coincide with a standard dimensional upper bound. This result is the TC analogue of a theorem by Rudyak-Oprea on the Lusternik-Schnirelmann category of symplectically aspherical manifolds. After an introduction to TC and the presentation of some basic results, I will explain how the cohomology groups of a space may be used to derive lower bounds on TC. I will then outline how these bounds are combined with infinite-dimensional de Rham theory to provide the abovementioned result for symplectic manifolds.

## 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