# Seminar – Topics in symplectic geometry

# 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.

# 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

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# 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.