Lyon
IX Workshop on
Symplectic Geometry, Contact Geometry,
and Interactions



École normale supérieure de Lyon, France
January 29 - 31, 2015



The talks will take place at the Amphithéâtre Descartes of the École normale supérieure de Lyon, see the practical information page for directions.


Schedule

Abstracts
  • Alberto Abbondandolo, On the length of the shortest closed geodesic on a Riemannian two-sphere
    In 1988 Croke studied geodesic flows on a Riemannian two-sphere and proved a uniform upper bound bound for the ratio between the square of the length of the shortest closed geodesic and the area of the two-sphere. The question of determining the best upper bound is still open, but it is known that the round metric is not a global maximiser. Babenko and Balacheff have conjectured that the round metric is a local maximiser for the above ratio. In this talk I will discuss the proof of a theorem which confirms this conjecture and complements it with a statement about the length of the longest simple closed geodesic. This theorem was proved in collaboration with B. Bramham, U. Hryniewicz and P. Salomão.


  • Denis Auroux, Two new constructions of monotone Lagrangian tori
    We will discuss some recent constructions of "exotic" Lagrangian tori in simple symplectic manifolds such as CP2 (work of Renato Vianna) and R6 that are not Hamiltonian isotopic to previously known examples, inspired by wall-crossing phenomena and mirror symmetry.


  • Roger Casals, Geometric criteria for overtwistedness
    In this talk, we focus on characterizing overtwisted contact  structures. First  we introduce the overtwisted--tight dichotomy in contact topology and detail the definition of the overtwisted disk. Then we explain a criterion to detect whether a contact manifold is overtwisted. In particular, we relate different notions of flexibility and provide explicit examples of overtwisted contact manifolds. This is joint work with E. Murphy and F. Presas.


  • Georgios Dimitroglou Rizell, Lagrangian unknottedness in the cotangent bundle of a torus and applications
    The homotopically non-trivial Lagrangian tori in the cotangent bundle of a two-torus are classified up to Hamiltonian isotopy. This is done by following ideas and techniques due to Ivrii, who treated the case when the torus is homologous to the zero-section. We apply this result to obtain answers concerning the classification of monotone Lagrangian tori in a symplectic vector space. Both questions are studied by considering the behaviour of pseudo-holomorphic foliations when stretching the neck around the Lagrangian submanifold.


  • Urs Fuchs, Gromov compactness revisited
    I will outline a proof of Gromov compactness for closed pseudoholomorphic curves by following Gromov's original article and emphasizing the study of the metrics induced on pseudoholomorphic curves. Then I will indicate, how doubling constructions allow to carry over this proof to situations, where the pseudoholomorphic curves have totally real or otherwise controlled boundary. This talk relies on joint work with Lizhen Qin.


  • Stéphane Guillermou, Sheaves associated with exact Lagrangians in cotangent bundles
    The microlocal theory of sheaves associates with a sheaf its microsupport which is a conic coisotropic subset of the cotangent bundle. We will see how we can use this theory to recover recent results on exact Lagrangians in cotangent bundles, namely that they have Maslov class zero and that they are homotopy equivalent to the zero section.


  • Ailsa Keating, Symplectic properties of non-ADE Milnor fibres
    Given any hypersurface singularity, Picard-Lefschetz theory readily provides distinguished exact Lagrangian spheres (vanishing cycles) and symplectomorphisms (Dehn twists about them). In the ADE (i.e., modality zero) case, all exact Lagrangians are known to be spheres, and Dehn twists generate the symplectic mapping class group in the A case. We will explain how to construct exact Lagrangian S1xSn-1's in the Milnor fibres of all non-ADE (i.e., positive modality) singularities. The focus will be on modality one singularities in three variables. Time allowing, we give applications to their symplectic mapping class group.


  • Thomas Kragh, Simple homotopy type of exact Lagrangians in cotangent bundles
    I will start by recalling how the simple homotopy type of manifolds can distinguish some smooth structures on manifolds with the same homotopy type. I will then describe some recent progress, joint with M. Abouzaid, where we prove that exact Lagrangians in cotangent bundles are simple homotopy equivalent to the zero section, which thus provides new restrictions on the smooth structure of exact Lagrangians.


  • Vinicius Gripp Ramos, Billiards, toric domains and ECH capacities
    ECH capacities have been shown to give sharp obstructions to many symplectic embedding problems. In this talk, I will explain how they are defined and give some examples of the problems for which they are sharp. In particular, I will explain how to compute them for concave toric domains and what that tells us about the space of billiard trajectories on a round disk.


  • Sobhan Seyfaddini, Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants
    We construct a dynamical invariant, denoted by N, for Hamiltonians of R2 and closed surfaces with positive genus.  We prove that, on the set of autonomous Hamiltonians,  this invariant coincides with the spectral invariants constructed by Viterbo on R and Schwarz on closed surfaces of positive genus.  This is joint work with Vincent Humilière and Frédéric Le Roux.


  • Anne Vaugon, Periodic Reeb orbits and bypass attachments
    Starting from a contact 3-manifold with boundary, a way to change the contact structure is to choose a suitable Legendrian arc on the boundary and glue to it half of a thickened overtwisted disk. This elementary operation, called a bypass attachment, may yield a Reeb vector field with additional periodic orbits. In this talk, I will describe the new orbits created by a bypass attachment  (for a suitable choice of contact form) in terms of words on the Reeb chords of the attaching arc. Following Giroux and Ozbagci, I will also explain how a bypass attachment can be expressed as the combination on two consecutive contact handle attachments, a point of view that may lead to generalizations of the previous result.


  • Naïm Zénaïdi, Lagrangian Floer Theory for translated points of contactomorphisms
    I will describe a Bott Framework of Lagrangian Floer  theory suitable to study pairs of cleanly intersecting Lagrangian submanifolds with one positive end and explain how to use this set-up to obtain new insights on the contact version of Arnold conjecture formulated in 2012 by S. Sandon.


UMPA
ENS Lyon
ICJ
UCLB
Université de Lyon
MiLyon
ESF
ERC

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