Workshop on Symplectic Geometry, Contact Geometry and Interactions, Les Diablerets 2013
Workshop on Symplectic Geometry, Contact Geometry and Interactions, Les Diablerets 2013
| 10 Jan. | 11 Jan. | 12 Jan. | |||
|---|---|---|---|---|---|
| 06:00-12:00 | First discussions... | 08:30-09:30 | Yaron Ostrover | 08:30-09:30 | Nicolas Vichery |
| 09:30-10:00 | Coffee break | 09:30-10:00 | Coffee break | ||
| 10:00-11:00 | Marco Mazzucchelli | 10:00-11:00 | Patrick Massot | ||
| 11:00-11:15 | Break | 11:00-11:15 | Break | ||
| 11:15-12:15 | Barney Bramham | 11:15-12:15 | Will Merry | ||
| 12:30 | Lunch | 12:30 | Lunch | 12:30 | Lunch |
| 14:00-15:00 | Reception | ||||
| 15:00-16:00 | Urs Frauenfelder | ||||
| 16:00-16:30 | Coffee break | 16:00-16:30 | Coffee break | ||
| 16:30-17:30 | Yochay Jerby | 16:30-17:30 | Lev Buhovsky | ||
| 17:30-18:00 | Break | 17:30-18:00 | Break | ||
| 18:00-19:00 | Alexander Ritter | 18:00-19:00 | Emmanuel Opshtein | ||
| 19:30 | Dinner | 19:30 | Dinner | ||
All lectures will take place at the Maison des Congrès which is 5 minutes walk from the Hotel Les Sources and the Hotel Victoria.
Barney Bramham : Mixing for Hamiltonian disk maps
It is an open question whether there exists a Hamiltonian diffeomorphism of the 2-disk that is mixing and has zero entropy. (For surfaces of genus at least one such maps are known to exist.) In this talk we will discuss how finite energy foliations can be used to rule out mixing for a substantial portion of the possible candidates.
Lev Buhovsky : Unboundedness of the first eigenvalue of the Laplacian in symplectic category
The main subject of this talk is the question of Leonid Polterovich, about the symplectic flexibility of the first eigenvalue of the Laplacian. First I will describe the history of this problem, and the results of Leonid Polterovich and of Dan Mangoubi, which provided its partial solutions. In the second half of the talk, I will give a sketch of my solution of the question.
Urs Frauenfelder : A Gamma-structure for the Lagrangian Grassmannian
The notion of Gamma manifold goes back to Hopf who proved that a Gamma structure gives rise to a Hopf algebra structure on the cohomology ring. In this talk I explain how to construct a Gamma structure on the Lagrangian Grassmannian by interpreting Lagrangians as fixed point set of orthogonal antisymplectic involutions. This is joint work with Peter Albers.
Yochay Jerby : The symplectic topology of projective manifolds with small dual
The class of projective manifolds with small dual is classically studied in algebraic geometry. One of their properties is that they admit a stronger version of the Lefschetz hyperplane theorem. In this talk, we present applications of methods from symplectic topology to projective manifolds with small dual. Our main tool is the Seidel representation associated to Hamiltonian fibrations. We note that to a projective manifold with small dual X one can associate a Hamiltionian fibration over S^2, whose fiber is a hyperplane section Y of X. We show that at certain cases one can calculate the Seidel elment of this fibration, and show that it is non-trivial. In turn, the non-triviality of the Seidel element gives rise to new restrictions on the homologies of Y and X, generalizing those obtained by standard Lefschetz theory. This is joint work with Paul Biran.
Patrick Massot : Fillability of higher dimensional contact manifolds
I will explain my work with Klaus Niederkrüger and Chris Wendl on obstructions to fillability of contact manifolds. In particular we generalize the example of tight contact structures on T^3 to get, in any dimension, non-fillable contact structures which are not flexible and have Reeb fields without contractible closed orbits, hence cannot be called overtwisted in any reasonable sense. Along the way I'll explain how to construct compact Liouville manifolds with disconnected boundary in any dimension using number theory.
Marco Mazzucchelli : Symplectically degenerate maxima via generating functions
In this talk, we will discuss a new approach to the study of symplectically degenerate maxima by means of generating functions techniques. In particular, we will provide a simple proof of a theorem due to Nancy Hingston, asserting that isolated symplectically degenerate maxima of a Hamiltonian diffeomorphism of a standard symplectic torus are non-isolated points of its average-action spectrum.
Will Merry : Orderability, non-squeezing and Rabinowitz Floer homology
We study Liouville fillable contact manifolds with non-zero Rabinowitz Floer homology and assign spectral invariants to paths of contactomorphisms. We prove all such contact manifolds are orderable in the sense of Eliashberg and Polterovich. If the contact manifold is in addition periodic or a prequantization space M × S1 for M a Liouville manifold, then we construct a contact capacity, define a bi-invariant metric on the group of contactomorphisms, and prove a general non-squeezing result. This is joint work with Peter Albers.
Emmanuel Opshtein : Symplectic embeddings in dimension 4
The equidimensional symplectic embeddings of open objects (balls or ellipsoids for instance), as well as the problem of symplectic embeddings of closed codimension 2 objects lie at the core of rigidity in symplectic geometry. In dimension 4, the relation between these two problems is well-known, and has been a source of many results. I will discuss this relation in this talk, with a special focus on curve singularities. I will also explain a less-known relation between embeddings of ellipsoids and Lagrangian intersections, that supports Arnold's famous conjecture : "Everything is Lagrangian".
Yaron Ostrover : Billiard Dynamics in the Eye of Symplectic Geometry
In this talk we shall discuss how symplectic capacities on the classical phase space can be used to obtain bounds and inequalities on the length of the shortest periodic billiard trajectory in a smooth convex domain. Moreover, going in the other direction, we will explain how billiard dynamics can be used to obtain information on the symplectic size of certain configurations in the phase space. This talk is based on a joint work with Shiri Artstein-Avidan from Tel Aviv University.
Alexander Ritter : Floer theory for negative line bundles
Let E be a negative line bundle over a symplectic manifold. The first half of the talk is about how the Gromov-Witten theory determines the symplectic cohomology of E. The second half discusses recent joint work with Ivan Smith on how the symplectic cohomology determines the wrapped Fukaya category when E is a negative line bundle over projective space. More concretely, this involves the existence of a non-displaceable monotone Lagragian torus in E.
Nicolas Vichery : Symplectic homegenization, toward a non-convex Aubry-Mather theory
We will review applications of the homogenization of lagrangian spectral invariants in cotangent bundle as studied in joint work with Monzner and Zapolsky. This can be seen as an extension of Viterbo's "symplectic homogenization" process in the cotangent bundle of tori. We will focus on new links with Aubry-Mather theory, especially on rotation vectors of some invariant measures and on subdifferentials of the homogenized hamiltonian.