I’ve been struggling through Hutchings’ lecture notes on ECH capacities for bit of time now and so I thought that I may as well post my progress for some katharsis. Expect these notes to be a under development as a result. I’m principally interested in the case of ECH capacities of toric domains (as a starting point at least), and in the appearance of lattice point counts therein. There are several aspects of the calculations that intrigue me – quaint applications of Pick’s theorem, a different sort of duality on the level of norms instead of (well, maybe comparably to) polytopes, combinatorial descriptions of the ECH differential and relevant chain maps, a sumptuous formula for the action functional,… – that I attempted to capture and communicate. You can be the judge of that.
Thanks are due to Julian for spending an afternoon computing integrals with me.