Sheaf Quantization of Lagrangians and Floer cohomology

Abstract: Given an exact Lagrangian submanifold L in the cotangent bundle of N, we want to construct a complex of sheaves F in the derived category of sheaves on N×R, such that its singular support, SS(F), is equal to the cone constructed over L. Its existence was stated in our Eilenberg lectures in 2011, with a sketch of proof, which however contained a gap (fixed here by the rectification). A complete proof was shortly after provided by Guillermou by a completely different method, in particular Guillermou's method does not use Floer theory. The proof provided here is, as originally planned, based on Floer homology. Besides the construction of the complex of sheaves, we prove that the filtered versions of sheaf cohomology of the quantization and of Floer cohomology coincide, and so do their product structures.

Barcodes and area-preserving homeomorphisms (with Frédéric Le Roux and Sobhan Seyfaddini)

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to C0 continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus

The aim of this paper is to use the methods and results of symplectic homogenization (see [V4]) to prove existence of periodic orbits and invariant measures with rotation number depending on the differential of the Homogenized Hamiltonian.

Functors and computations in Floer cohomology. Part II.(postscript)

1996 (2003 revision)

The results in this paper have been announced in a talk at the ICM 94 in Züurich. They concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I, others will be given in forthcoming papers.

1995

Dans ce texte, on consid\`ere une m\'ethode g\'eom\'etrique pour construire des solutions d'\'equations d'Hamilton-Jacobi (cas d'\'evolution du premier ordre) $${\partial \over \partial t}u(t,x) = H(t,x,u,Du) \tag {HJ}$$ $$u(0,x)=u_0(x)$$ o\`u $Du$ d\'esigne la diff\'erentielle de $u$ par rapport \`a la variable $x$. Comme d'habitude, la variable $t$ sera appel\'ee "temps", la variable $x \in N$, "espace". On montre que cette construction, due \`a Sikorav et Chaperon, bas\'ee sur une m\'ethode de fonction g\'en\'eratrice fournit des solutions qui partagent de nombreuses propri\'et\'es des solutions de viscosit\'e de Crandall et Lions, mais peuvent \^etre diff\'erentes.

Solutions généraliséees pour l'équation d'Hamilton-Jacobi dans le cas d'évolution.

1994

This paper studies the variational solutions of Hamilton-Jacobi equations that have been defined by Sikorav and Chaperon. A consequence of "uniqueness of generating functions" from Symplectic topology as the geometry of generating functions is the uniqueness of such varational solutions as well as the possibility to extend their defintion to the case of continuous Hamiltonians, for continuous initial conditions. We prove however, that in contrast to viscosity solutions of Lions and Crandall, the solution operator does not have the Markovian property $T_{s+t}=T_s\Circ T_t$.

Symplectic invariants and symplectic reduction: details of the proof of the camel problem

The aim of this note is to clarify the proof of the camel problem from the paper Symplectic topology as the geometry of generating functions, p.706