Polyfold methods for the study of periodic delay orbits.

We use methods from symplectic geometry to study periodic solutions of differen- tial delay equations (DDEs, also known as retarded functional differential equations, RFDEs). Using polyfold theory, we prove that near a given non-degenerate 1-periodic orbit of a vector field in Rn, there is a 1-dimensional family of 1-periodic delay orbits smoothly parametrized by delay. Then we generalize this result in several ways. More- over, we prove an abstract compactness theorem for perturbed non-local unregularized gradient flow lines in R2n, which is one step towards the construction of Floer theory for Hamiltonian delay equations.