## Budapest minicourse conclusion

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 $\partial^2=0$.

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.