HOLOMORPHIC METHODS IN SYMPLECTIC GEOMETRY

12 - 14 February 2018

Ruprecht-Karls-Universität Heidelberg

Lecture series by

Dietmar Salamon (ETH Zürich) — *Geometric invariant theory in finite and infinite dimensions*

Kai Zehmisch (Gießen) —*The filling-by-holomorphic-discs method*

Additional talks by

Hansjörg Geiges (Köln), Samuel Trautwein (ETH Zürich) and Will Merry (ETH Zürich)

All talks will take place at the University of Heidelberg,

in the Maths building (Mathematikon) on the ground floor,

in the **Hörsaal**.**Directions**

Schedule | |

Monday | |

9:00 - 10:30 | Kai Zehmisch — $\Gamma_4=0$ |

Coffee break | |

11:30 - 13:00 | Dietmar Salamon — GIT and the moment map I (The moment-weight inequality) |

Lunch break | |

15:00 - 16:00 | Hansjörg Geiges — $\Gamma_3=0$ |

Coffee break | |

17:00 - | Visit Heidelberg Palace and workshop dinner at the Restaurant Zum güldenen Schaf |

Tuesday | |

9:00 - 10:30 | Dietmar Salamon — GIT and the moment map II (The Hilbert-Mumford criterion) |

Coffee break | |

11:30 - 12:30 | Samuel Trautwein — Gauge groups, moment maps, and the symplectic vortex equation |

Lunch break | |

14:00 - 15:30 | Kai Zehmisch — $S^3$ holomorphic |

Coffee break | |

16:30 - 17:30 | Will Merry — Floer homology and contact Hamiltonians |

Wednesday | |

9:00 - 10:30 | Kai Zehmisch — Extending over the ball |

Coffee break | |

11:30 - 13:00 | Dietmar Salamon — Diffeomorphism groups, complex structures, and the Ricci form |

Lunch / departure |

Participation is open. Participants are kindly requested to send an e-mail to Mrs Nicole Umlas (numlas "at" mathi.uni-heidelberg.de) not later than Monday, January 29, 2018. Please indicate whether you wish to participate in the conference dinner on Monday. 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 the

**DFG**through

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

**RTG 2229 - Asymptotic Invariants and Limits of Groups and Spaces**

**.**

**Lecture Notes:**

Dietmar Salamon - Lecture 1 , Lecture 2

Hansjörg Geiges - $\Gamma_3=0$

Kai Zehmisch - The filling-by-holomorphic-discs method

**Abstracts:**

**Kai Zehmisch — The filling-by-holomorphic-discs method**

In 1968 Cerf proved his celebrated result stating that all diffeomorphisms of the 3-sphere extend to a diffeomorphism of the 4-ball. Based on his classification of contact structures on the 3-sphere Eliashberg proposed in 1992 to use Gromovʼs holomorphic curves technique to give a contact theoretic proof of Cerfʼs theorem. In this lecture series I will present the analytic details of Eliashbergʼs proof of Cerfʼs theorem based on a joint paper with Geiges from 2010.

In lecture 1 I will show how to reduce Cerfʼs theorem to symplectic geometry via Eliashbergʼs classification theorem. In lecture 2 main concepts from contact geometry will be introduced in order to understand the standard holomorphic disc filling of the 3-sphere. In lecture 3 I will demonstrate how analytic tools from elliptic PDEs can be used for an actual extension of a given diffeomorphism of the 3-sphere.

**Hansjörg Geiges — $\Gamma_3=0$**

In this talk I will provide some differential topological background for the lectures by Zehmisch. In particular, I will outline Smale's proof that the diffeomorphism group of the 2-sphere retracts to the group of isometries.