## 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.