Frobenius manifolds

In these lectures, some of the geometrical themes in the work of Boris Dubrovin on Frobenius manifolds are discussed. We focus principally on those aspects which have a symplectic flavour, including Hamiltonian flows on coadjoint orbits, Poisson structures on loop spaces, and the symplectic geometry of flat connections on a punctured sphere. A major theme is to study the problem of solving the differential equations for a Frobenius manifold, These are nonlinear equations which appear in disguise in many other branches of mathematics. We show how to reformulate the equations in terms of the problem of determining flat connections on surfaces with given holonomy, the classical subject of isomonodromic deformations.