Munich conclusion

In my last lecture today I explained how to compute the ECH capacities of X_\Omega, where \Omega is a convex domain in the first quadrant of the plane, and X_\Omega\subset{\mathbb C}^2 is its inverse image under the moment map (at least assuming that \Omega does not touch the axes, which is probably not necessary). The answer is that c_k is the shortest “\Omega-length” of a convex polygon \Lambda enclosing at k+1 lattice points, where “the \Omega-length of \Lambda” means the sums of the norms of the edge vectors of \Lambda with respect to the dual of a Finsler norm whose unit ball is a translate of \Omega. (Whew!) I also explained why if \Omega is a polygon, then in this calcuation you can assume that the edges of \Lambda are ninety degree rotations of the edges of \Omega, up to scaling. See e.g. my blog post “Fun with ECH capacities” from last October. In particular we get the right answer for an ellipsoid or a polydisk , where \Omega is a right triangle or a rectangle. The proof of the formula for c_k(X_\Omega) involves understanding the U map on the ECH of T^3, as in my paper with Michael Sullivan. That was fun.

I’m not going to attempt to summarize the other lectures and discussions here, but they were great. It was nice to have a focused conference in which there were multiple series of lectures, all on related topics.

I want to post more notes here, but I expect the pace of blogging to slow down now, as I am no longer trying to keep up with my lecture series, and I have some other things to do.

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