# 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.

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

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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.