# Seminar – Topics in symplectic dynamics

# SoSe 19

Topic

*Holomorphic Curves in Low Dimensions – From Symplectic Ruled Surfaces to Planar Contact Manifolds*

## Ort und Zeit

Mittwoch 14 - 16 in INF 205 / SR 4

(First meeting is on April 24)

## Abstract

Since the groundbreaking work of Gromov [1] in 1985, holomorphic curves have been a central tool in studying symplectic manifolds, as illustrated in the comprehensive monograph by McDuff and Salamon [3]. When the manifold M has dimension four, the interplay between holomorphic curves and the homological intersection product H2(M; Z) × H2(M; Z) → Z lead to surprising rigidity results.

After giving the basics of the theory of holomorphic curves, the goal of our seminar will be to present some of these rigidity results and, in particular, a theorem of McDuff [2] of 1990, which asserts that a symplectic manifold M of dimension 4 which contains a holomorphic sphere with positive self- intersection is either CP2 or is completely foliated by holomorphic spheres, possibly after blow-down.

In our seminar, we will follow the recent book [4] by Chris Wendl, which gives a modern perspective on the subject.

Literatur:

[1] M. Gromov, Pseudo holomorphic curves in symplectic manifolds

[2] D. McDuff, The structure of rational and ruled symplectic 4-manifolds

[3] D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology

[4] C. Wendl, Holomorphic curves in low dimensions. From symplectic ruled surfaces to planar contact manifolds

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# WiSe 18/19

Topic

*Floer Homology*

## Ort und Zeit

Mittwoch 14 - 16 in INF 205 / SR 4

## Abstract

The purpose of this seminar is to give an introduction to Floer homology. We will study in some detail the construction of Hamiltonian Floer homology and will then, depending on preferences of participants and time, look at other Floer homologies and applications.

Literatur:

Michèle Audin, Mihai Damian - Morse Theory and Floer Homology

Kai Cieliebak, Urs Frauenfelder - A Floer homology for exact contact embeddings

Kai Cieliebak, Alexandru Oancea - Symplectic homology and the Eilenberg-Steenrod axioms

Dietmar A. Salamon - Lectures on Floer homology

Paul Seidel - Fukaya categories and Picard-Lefschetz theory

Joa Weber - Topological methods in the quest for periodic orbits

# SoSe 18

Lecture Series

*Algebraic Methods in the Theory of Integrable Systems*by Michael Gekhtman

## Ort und Zeit

Mittwoch 14 - 16 in INF 205 / SR C

## Abstract

We will start by reviewing classical theory and examples of Liouville completely integrable systems of classical Hamiltonian mechanics. Then the modern theory of integrable models will be discussed including :

- Lax formalism for finite and infinite dimensional systems;

- integrable equations on Lie algebras and Poisson-Lie groups;

- integrable equations and nonlinear waves: the Sato Grassmannian and the KP hierarchy and its reductions;

- discrete integrable systems and connection with the theory of cluster algebras.

Pre-course (by Dr. Fuchs) for the lecture series by Prof. Gekhtman:

1) Lie algebras and their representations, classification of representations of sl_2(C)

2) Solvable, reductive and semisimple Lie algebras, Cartan's criteria for solvable and semisimple Lie algebras

3) Cartan subalgebras and root space decompositions of semisimple Lie algebras

4) Abstract root systems, Weyl group, simple roots, classification of root systems by Dynkin diagrams, classification of complex semisimple Lie algebras

# WiSe 17/18

This seminar can serve as an introduction to contact geometry in dimension 3. Among the topics we will discuss are: existence of contact structures, tight/overtwisted dichotomy, open book decompositions, characteristic foliations, convex surfaces, transverse/Legendrian knots.

## Ort und Zeit

Mittwoch 14 - 16 in INF 205 / SR 3

## Vorträge

Oct 18 Darboux's theorem and Gray stability (Urs)

Oct 25 Normal forms (Urs)

Nov 8 Characteristic foliations on hypersurfaces (Irene)

Nov 14 Morse-Smale foliations (Myeonggi)

Nov 21 Convex surfaces I: Basic properties and dividing sets (Urs)

Nov 28 Convex surfaces II: Realization Lemma and uniqueness of dividing curves (Urs)

Dec 6 Convex surfaces III: Detecting overtwisted tubular neighbourhoods (Urs)

Dec 13 Tomography (Urs)

Dec 20 Uniqueness of tight contact structures on S^3, R^3 and S^2 x S^1 (Urs)

Jan 10 Legendrian knots I: Front projection, C^0 approximation and Seifert surfaces (Ferdinand)

Jan 17 Legendrian knots II: The invariants tb and rot (Falk)

Jan 24 Existence of contact structures via open book decompositions (Anna-Maria)

Jan 31 Detecting overtwisted contact structures (Urs)

Feb 7 Finiteness results for tight contact structures (Urs)

## Literatur

John Etnyre - Introductory Lectures on Contact GeometryHansjörg Geiges - An Introduction to Contact Topology

# SoSe 17

We will discuss generating functions for Lagrangian submanifolds in cotangent bundles (as well as Legendrian submanifolds in 1-jet bundles) and explore their use in establishing rigidity phenomena for such submanifolds.

## Vorträge

April 24 Introduction and generalities (Urs)

May 8 Special generating functions (Anna-Maria)

May 5 Existence of generating functions I (Irene)

May 22 Existence of generating functions II (Irene)

May 29 Uniqueness of generating functions is preserved under Hamiltonian isotopies (Matthias)

June 12 Uniqueness of generating functions for the zero section I (Urs)

June 19 Uniqueness of generating functions for the zero section II (Urs)

June 26 Correspondences and Reductions (Urs)

July 7 Lower bounds on Lagrangian intersection points via generating functions (Myeonggi)

July 10, Invariants of Lagrangian submanifolds via generating functions (Irene)

## Literatur

Yasha Eliashberg and Misha Gromov - Lagrangian intersection theory: finite-dimensional approach.François Laudenbach and Jean-Claude Sikorav - Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent.

Jean-Claude Sikorav - Problèmes d’intersection et de points fixes en géométrie hamiltonienne

David Théret - A complete proof of Viterbo's uniqueness theorem on generating functions.

Claude Viterbo - Generating functions, symplectic geometry, and applications.

Claude Viterbo - Symplectic topology as the geometry of generating functions.