Abstract: Embedded contact homology and periodic Floer homology are Floer homology theories of similar flavour defined for Reeb flows and surface diffeomorphisms respectively. They were originally proposed by Hutchings and rigorously defined by Hutchings and Taubes. The difference with the usual Floer homology theories is threefold:
• their differential counts embedded holomorphic curves
• the topological type of those curves is not fixed a priori, and
• the resulting homologies are topological, rather than symplectic, invariants; in fact it has been proved by various groups of people that they are isomorphic to Heegaard Floer homology and monopole Floer homology.
In the first talk I will introduce and motivate the ECH index, while in the second talk I will define the chain complexes for closed manifolds and for manifolds with sutured boundary.