Algebraic methods in random matrices and enumerative geometry

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...Comment: review article, Latex, 139 pages, many figure

Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds

We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. We also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper40.abs.htm