Abstract I: We will see how positive S^1-equivariant symplectic homology allows one to tackle questions about the number of non-diffeomorphic contact structures on the sphere, or the minimal number of periodic Reeb orbits on some contact manifolds. This will be done by establishing properties of positive S^1-equivariant symplectic homology, namely a computational property, a functoriality property and an invariance property. Abstract II: In this talk, I will discuss how to define invariant local Morse homology of an isolated critical point of an invariant function on a manifold with a certain Z/kZ action. The definition relies on the construction of an invariant perturbation near the critical point. Moreover, I will discuss potential applications in contact geometry. For example, it might be possible to use this construction to define local contact homology via invariant local Floer homology using finite dimensional reduction and generating functions. This is joint work with Umberto Hryniewicz and Leonardo Macarini.