Equivariant KAM Theorem

In integrable Hamiltonian systems solution curves in phase space are confined to invariant tori. These integrable systems are often used as first approximations to other more complicated nonintegrable systems. KAM (Kolmogorov, Arnold, Moser) theory is a set of theorems that detail under what conditions these nonintegrable systems will preserve the invariant tori. If an integrable system is also equivariant with respect to a discrete symmetry group $\Gamma$ then none of the present KAM theorems can be applied. In this talk, I will first introduce KAM theory by taking both a dynamical systems and
historical perspective; justifying the need for an Equivariant KAM theorem. Then I will state the new result, Equivariant KAM theory and give some details of the proof. Throughout the talk, several applications of Equivariant KAM will also be discussed.