SEMINAR ON SYMPLECTIC AND CONTACT GEOMETRY

Université Libre de Bruxelles

First Semester 2009-2010

Monday 14:00-15:00

Room 2.NO.906

Thursday, October 8th

Brussels-Cologne joint seminar, Universität zu Köln
Math building (Weyertal 86-90), 2nd floor (Großer Hörsall).
Directions

  
14:00 Sheila Sandon (Instituto Superior Técnico, Lisboa)
Contact non-squeezing via generating functions
15:30 Sinem Çelik Onaran (METU Ankara and Oberwolfach)
Invariants of Legendrian knots from open book decompositions

Monday, October 12th

Clément Hyvrier (ULB)
Sur les invariants de Gromov-Witten des fibrations Hamiltoniennes

Apres avoir introduit la définition des invariants de Gromov-Witten (invariants GW) d'une variété symplectique, nous verrons comment, sous certaines conditions, certains invariants GW d'une fibration Hamiltonienne, de base symplectique, sont donnés par un produit d'invariants GW de la base et d'une fibration Hamiltonienne au dessus d'une surface de Riemann. Nous montrerons comment la formule produit obtenue permet de montrer qu'une fibration Hamiltonienne au dessus d'une base (symplectiquement) rationnelle est nécessairement (symplectiquement) uniréglée.

Monday, October 19th

Evgeny Volkov (ULB)
Stable Hamiltonian structures and open books

Tuesday, October 27th

Frédéric Bourgeois (ULB)
ContactMath : Legendrian contact homology and generating families

This is a presentation of a research project funded by the European Research Council, starting in a few days for the next 5 years. The main goal of this project is to show that the generating family homology and the linearized Legendrian contact homology can be defined for the same class of Legendrian submanifolds, and are isomorphic. This result can then be used to obtain more general structural results on linearized Legendrian contact homology, to extend recent results on existence of Reeb chords, and to gain a much better understanding of the geography of Legendrian submanifolds.

Friday, November 6th

Brussels-Cologne joint seminar, Université Libre de Bruxelles
NO building (Campus Plaine), 5th floor (Salle Solvay).
Directions

  
14:15 Wilhelm Klingenberg (Durham)
Mean Curvature Flow, Holomorphic Discs, and the Caratheodory Conjecture
We outline joint work with Brendan Guilfoyle, which establishes a proof of the Caratheodory Conjecture. This claims that every C3 - differentiable sphere in Euclidean space admits at least two umbilic points. (These are locally spherical points; at such points both principal curvatures are equal, and every tangent vector is a principal direction).
Remark: This is one more umbilic than needs to appear for topological reasons, namely the nonvanishing of the Euler number of the sphere (thereby presents an instance of rigidity).
Our proof is inspired by Gromov's symplectic rigidity-flexibility dichotomy (specifically by his approach to the rigidity of convex surfaces which lead him to the development of his theory of pseudoholomorphic curves). It uses new a - priory gradient estimates for Mean Curvature Flow in manifolds of split signature (building on work of Bartnik and Ecker-Huisken). The latter allows us to construct a holomorphic disc with boundary encircling an isolated umbilic point (in a symplectic model space). This results in sufficient rigidity to prove CC in the spirit of said dichotomy.

15:45 Alexander Ritter (Cambridge)
Novikov-symplectic cohomology and exact Lagrangian embeddings
We are interested in finding topological obstructions to the existence of exact Lagrangian submanifolds L inside a cotangent bundle T*N. Under mild homotopy assumptions on N, I proved that the image of π2(L) inside π2(N) has finite index. This result makes no assumption about the Maslov class of L, and the manifolds need not be orientable. My approach builds on Viterbo's work: by using symplectic cohomology we construct a transfer map on the Novikov homologies of the free loop spaces of N and L. The result then follows from a vanishing result for the Novikov homology of loop spaces.

Saturday, November 28th

Brussels-Cologne joint seminar, Universität zu Köln
Math building (Weyertal 86-90), 2nd floor (Großer Hörsall).
Directions

  
14:15 Hiraku Nozawa (ENS Lyon)
Five-dimensional K-contact geometry via Morse theory on moment maps
15:45 Ferit Öztürk (Boğaziçi Üniversitesi İstanbul)
First steps in real contact topology

Tuesday, December 15th

Benoit Tonnelier (Polytechnique)
Exemples de sous-variétés coisotropes de S2 × S2

Cet exposé concerne le problème d'intersections coisotropes posé par Bolle et Ginzburg. Le produit direct de deux sphères est un exemple d'une variété torique M de dimension 4. L'image de son application moment est le carré X=[-1,1]2 (modulo translations). Soient A et B deux points du bord de X, différents des sommets. La préimage du segment [AB] est une sous-variété coisotrope C de M, les préimages de A et de B étant des feuilles isotropes. Durant l'exposé, on décrira comment C est déplacée par isotopies hamiltoniennes. Selon les positions de A et de B, des estimations seront données sur un nouvel invariant : l'énergie d'intersection coisotrope.

Wednesday, December 16th

Dmitri Panov (Imperial College)
Construction of symplectic manifolds of "Fano" and "Calabi-Yau" type

The notions of Fano and Calabi-Yau manifolds in complex algebraic geometry can be extended to symplectic geometry. It is natural to ask if symplectic "Fanos" and "Calabi-Yaus" resemble the algebraic ones. I will discuss possible ways to show the divergence of symplectic and algebraic examples, using in particular 4 and 6 dimensional hyperbolic geometries. This is joint work with Joel Fine.
Previous Semesters :
First Semester 2005-2006
Second Semester 2005-2006
First Semester 2006-2007
Second Semester 2006-2007
First Semester 2007-2008
Second Semester 2007-2008
First Semester 2008-2009
Second Semester 2008-2009
Next Semester :
Second Semester 2009-2010

Maintained by Frédéric Bourgeois.