Claude Viterbo

Metric and isoperimetric problems in symplectic geometry

Introduction 411 1. Some basic results in symplectic topology 413 2. Capacity and symplectic reduction 414 3. Volume estimates for Lagrange submanifolds 416 3.1. The case of R. 417 3.2. Deformations of the zero-section in cotangent bundles 419 3.3. Generalization to the case of CP 422 4. An application to billiards 423 5. Geometry of convex sets and periodic orbits 425 6. Compensated compactness and closure of the symplectic group 426 Appendix A. A summary of symplectic geometry through generating functions 427 Appendix B. John’s ellipsoid 428 References 430

Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the

On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows

C^0

Generating Functions, Symplectic Geometry, and Applications

topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.

On the topology of fillings of contact manifolds and applications

We show that if a sequence of Hamiltonian flows has a

Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

C^0

SYMPLECTIC TOPOLOGY AND HAMILTON - JACOBI EQUATIONS

limit, and if the generating Hamiltonians of the sequence have a limit, then this limit is uniquely determned by the limiting

Symplectic Topology: An Introduction

C^0

Properties of Embedded Lagrange Manifolds

flow. This answers a question by Y.G. Oh.

Napoleone, la matematica e l’École Polytechnique

We consider the concepts of rotation number and rotation vector for area preserving diffeomorphisms of surfaces and their applications. In the case that the surface is an annulus A the rotation number for a point x ∈ A represents an average rate at which the iterates of x rotate around the annulus. More generally the rotation vector takes values in the one-dimensional homology of the surface and represents the average “homological motion” of an orbit.A symplectic form on a manifold is a closed two form ω, nondegenerate as a skew-symmetric bilinear form on the tangent space at each point. Integration of the form on a two-dimensional submanifold S with boundary ∂S in M associates to S a real number (positive or negative) the “area of S”, which due to Stoke’s formula only depends on the curves ∂S, and the homology class of S rel ∂S. If moreover the form is exact, that is ω = dλ, the area of S is obtained by integrating λ over ∂S. In this case it is also possible to integrate λ on loops nonhomologous to zero and we get the notion of “area enclosed by a loop”. However this area depends on the choice of λ. If this choice is fixed once for all, we shall then talk about an exact manifold. One should be careful about the fact that this notion is slightly different from that of a symplectic manifold with exact symplectic form (because in the latter case we have not chosen the primitive of ω).