Abstract. Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [Vit92], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a "classical" Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

Abstract. We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C 0 -dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C 0 -Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.

Let (M, ω) be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold L such that ω| π 2 (M,L) = 0 and µ| π 2 (M,L) = 0, where µ is the Maslov index. Given any Lagrangian submanifold L ′ , Hamiltonian isotopic to L, we define Lagrangian spectral invariants associated to the non zero homology classes of L, depending on L and L ′ . We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case and carry strictly more information than the classical ones. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.