Skip to main content

Legendrian contact homology and generating families

Objective

A contact structure on an odd dimensional manifold in a maximally non integrable hyperplane field. It is the odd dimensional counterpart of a symplectic structure. Contact and symplectic topology is a recent and very active area that studies intrinsic questions about existence, (non) uniqueness and rigidity of contact and symplectic structures. It is intimately related to many other important disciplines, such as dynamical systems, singularity theory, knot theory, Morse theory, complex analysis, ... Legendrian submanifolds are a distinguished class of submanifolds in a contact manifold, which are tangent to the contact distribution. These manifolds are of a particular interest in contact topology. Important classes of Legendrian submanifolds can be described using generating families, and this description can be used to define Legendrian invariants via Morse theory. Other the other hand, Legendrian contact homology is an invariant for Legendrian submanifolds, based on holomorphic curves. The goal of this research proposal is to study the relationship between these two approaches. More precisely, we plan 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 correspondence should be established using a parametrized version of symplectic homology, being developed by the Principal Investigator in collaboration with Oancea. Such a result would give an entirely new type of information about holomorphic curves invariants. Moreover, it can 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.

Call for proposal

ERC-2009-StG
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

UNIVERSITE PARIS-SUD
Address
Rue Georges Clemenceau 15
91405 Orsay Cedex
France
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 223 762,25
Principal investigator
Frédéric Bourgeois (Prof.)
Administrative Contact
Nicolas Lecompte (Mr.)

Beneficiaries (2)

UNIVERSITE PARIS-SUD
France
EU contribution
€ 223 762,25
Address
Rue Georges Clemenceau 15
91405 Orsay Cedex
Activity type
Higher or Secondary Education Establishments
Principal investigator
Frédéric Bourgeois (Prof.)
Administrative Contact
Nicolas Lecompte (Mr.)
UNIVERSITE LIBRE DE BRUXELLES

Participation ended

Belgium
EU contribution
€ 486 237,75
Address
Avenue Franklin Roosevelt 50
1050 Bruxelles
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Christine Courillon (Dr.)