In the fourth and penultimate lecture of the Budapest minicourse, I stated the formulas of ECH index theory and defined all the ingredients that go into them. The contents of the lecture roughly correspond to installments 5 and 6 of these notes and part of installment 7. I stopped just short of officially defining ECH, although the definition should be clear anyway: the differential counts I=1 curves. Anyway, in the last lecture I will spell this out and describe why it works.
I had to charge through a lot of definitions and formulas today, but an example (the ellipsoid) should help make sense of a lot of this. The plan is to cover that example in the discussion section. If you want to think about this example before the discussion, it is discussed in installment 8, where some of the calculations are left as exercises.
There is a lot more material (beyond the 10 installments of notes posted here so far) which I think belongs in an introduction to ECH. I hope to write notes on this at some point. The minicourse in Munich should be able to go further than the one in Budapest.