Mathematics

The first part of the course deals with  local and global  bifurcation theory and applications to dynamical systems and PDEs, like the Lyapunov center theorem, Hopf bifurcation, traveling and Stokes waves for water waves, as well as other bifurcation problems in fluids. At the beginning we shall present the differential calculus and the implicit function theorem in Banach spaces. Later on I will deal with also cases in which the classical implicit function theorem can not be applied since the linearized operator has an unbounded inverse. We shall prove some versions of the Nash-Moser implicit function theorem, providing  applications to the classical Siegel and KAM theorems, as well as local Cauchy existence results of solutions of PDEs.