Today was the end of the Budapest minicourse. Basically it covered installments 1-10 of the lecture notes (with a lot of details omitted), plus the application to symplectic embeddings which was outlined in installment 0.

The Munich course is supposed to cover some more material, so hopefully I will post more notes here. Here are some things I think belong in an introduction to ECH and which I might post about (even though they will not all fit in the minicourse):

- Outline of the proof of the writhe bound and the partition conditions.
- Details of why defining cobordism maps without Seiberg-Witten is difficult.
- The U map. (This was briefly mentioned in installment 0.)
- The ECH contact invariant. (This was briefly mentioned in installment 0.)
- Filtered ECH. (This was briefly mentioned in installment 0.)
- Topological complexity of holomorphic curves in ECH.
- Forced transversality and finite energy foliations.
- Application to contact three-manifolds with all Reeb orbits elliptic.
- Exact symplectic cobordisms and application to the chord conjecture.
- ECH capacities.
- Introduction to the calculation of the ECH of T^3 and related examples.
- The symmetric product picture, how this helps motivate the definition of ECH, and how it might lead to a simpler proof that .

Please feel free to let me know what you would most like to hear about, or to request other things.

I have also figured out (or partially figured out) some new things which I am eager to write about when I can. I am currently trying to finish writing a paper explaining the details of my blog post from May about how to make ECH into a field theory for strong symplectic cobordisms. (It’s pretty straightforward once you have the setup from that blog post, but to explain everything takes longer than I expected.) As I like to say, in any research project that I do, figuring out the basic idea takes 3 percent of the time, and writing up all the details takes 97 percent of the time. A blog is a nice place to communicate the first 3 percent, before the remaining 97 percent is done.

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I really appreciate your series of blog posts about ECH. I am looking forward to the workshop in Munich.

I am very interested in the U map. You have also piqued my curiosity with the bullet point “topological complexity of holomorphic curves”. If you end up having time, I’d be interested in the T^3 computation too.