## Version 2 of “Beyond ECH capacities”

I just posted a revised version of the preprint “Beyond ECH capacities” to the arXiv. It should appear on Monday, but you can view it here first. The new version corrects some embarrassing/horrifying typos, clarifies a few things, and includes the new application to sharpness of symplectic folding from the previous blog post.

There are still tons of calculations to do using the ideas in this preprint, if anyone is interested. Here are some examples of things to try:

• Study symplectic embeddings of the polydisk $P(a,1)$ into a ball when $a>12/5$, improving Theorem 1.3 and/or extending Theorem 1.4 in the preprint.
• Study symplectic embeddings of the polydisk $P(a,1)$ into the ellipsoid $P(bc,c)$ when $b$ is an integer and $a>2$, or when $b$ is not an integer, extending Theorem 1.5 in the preprint.
• Prove Conjecture A.3 in the preprint (regarding the ECH differential on the boundary of a suitably perturbed convex toric domain), which would allow Theorem 1.6 to be improved as explained in Remark 1.8.
• Study symplectic embeddings of the disjoint union of two polydisks into an ellipsoid or polydisk.
• Study symplectic embeddings of concave or convex toric domains into a concave toric domain.

In general, one would like to identify more cases when the obvious inclusion map, or the folding-type constructions in Felix Schlenk’s book, give optimal symplectic embeddings.