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.

Plakat-Gekhtman



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 Geometry

Hansjörg 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.