## Munich update

Sorry I haven’t had a chance to post any more notes in the last three days. Anyway here is what I talked about in Munich:

In the second lecture, I discussed the ECH index inequality and the definition of ECH, corresponding to installment 7 of the lecture notes, with a bit of background from installment 6. (The Fredholm index, and some of the stuff about the relative adjunction formula, had been previously covered in Sam Lisi’s and Chris Wendl’s lectures.) Vinicius Gripp explained the ellipsoid example (installment 8) in his discussion section.

In the third lecture I finished explaining why the ECH differential is well-defined (installment 9), and then I tried to explain the proof of the writhe bound. The latter proof is from section 4 of the paper “The ECH index inequality revisited”. This was maybe excessively technical, but I thought it fit into the workshop because Chris Wendl had previously introduced relevant material on asymptotic winding numbers, and I thought that the lattice point business in this argument would be an interesting complement to material related to continued fractions that came up in Dusa McDuff’s talks.

Today in the fourth lecture I finished explaining why $\partial^2=0$ when there are no multiply covered elliptic orbits involved (see installment 10 of the lecture notes), and I introduced ECH capacities similarly to the first lecture in the Budapest minicourse. Also, in the afternoon I gave a discussion section on the really technical aspects of the proof that $\partial^2=0$, i.e. an introduction to my two-part 200 page gluing paper with Taubes, which explains how to glue in the presence of multiply covered elliptic orbits, by inserting index zero branched covers of trivial cylinders and counting zeroes of a section of an obstruction bundle. I hadn’t been planning on talking about this at all, but several people requested it, so I tried to explain the basic idea (although I have forgotten a lot of the details from the gluing papers — I write to forget).