Quantum topology of lagrangian submanifolds

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The goal of this proposal is to investigate the topology of Lagrangian submanifolds. We are going to establish a new research program that develops a Floer theory for Lagrangian intersections without obstructions. The main new object is the cluster moduli space associated to any Lagrange submanifold L in M and a generic Morse function f from L to M. It is obtained by attaching to each homotopy class lambda and set x, x_1,..., x_k in Crit(f) the space of all tree-like configurations starting at x and ending at x_1,..., x_k where each edge is sent to a negative flowline of f and each interior vertex is replaced by a holomorphic disc D mapped to M with boundary mapped to L so that the sum of the homotopy classes of all discs be equal to lambda. This is made from more elementary moduli spaces by gluing them along their common boundary strata (the point being that the dimension loss in the bubbling off of a disc A from a disc B is compensated by the additional real parameter that specifies the time of the flowline between two discs). We consider coefficients which reflect the loop structure of the boundary of such configurations. Some applications are presented - for instance the proof in some instances of the well-known conjecture that for any almost complex structure J and any Lagrangian submanifold L that can be disjoined from itself, there is a J-holomorphic disc with boundary passing through any point of L, or the proof of inequalities on the Maslov indices of Lagrangian submanifolds, and applications to the existence of periodic orbits of the ambient space obtained by stretching the neck of specific almost complex structures near the Lagrangian submanifold. We will also study the Homological Lagrangian Monodromy using the tools of absolute and relative Seidel morphisms. This monodromy, for Hamiltonian loops taking a given Lagrangian submanifold L back to itself, has been computed in some special cases by Mei-Lin Yau. Recently, we established the fundamental result on this monodromy stating that the rational homological monodromy of a weakly exact Lagrangian submanifold is always trivial. The next goal is to compute this monodromy in many cases, starting from the Clifford and Chekanov tori.