Exchange Seminar

Welcome to the exchange seminar! It is organized by Amanda Hirschi and myself. The first semester will be an online reading seminar on ECH. The next semester will be a (presumably in person) seminar on virtual fundamental classes and related fun.

I should note this is meant to be a reading seminar - I will not give lectures. The hope is to read through some basic texts on ECH (e.g. Michael Hutchings lecture notes) and some recent papers, depending on interest.

Time: TBA.

Schedule:

• Week 1: Section 1 of "Lecture notes on ECH"

• Week 2: Sections 3.1-3.3 of "Lecture notes on ECH"

  • Supplementary reading: Siefring's paper on asymptotics of holomorphic curves. You can just read the statement. These kind of exponential decay type results are really quite general, they apply for Reeb chords also for example.  https://www.mathi.uni-heidelberg.de/~siefring/publications/asymptotics.pdf

  • The proof of the adjunction inequality. https://arxiv.org/pdf/math/0112165.pdf
    Sections 2 and 3 of this paper flesh out the outline in "Lecture notes". The adjunction inequality is proved in Section 3.1. The proof really isn't bad.

  • There is a newer definition of relative intersection pairing found in Section  2.7 of https://arxiv.org/pdf/0805.1240.pdf. It is equivalent to the old one.

• Week 3: Section 3.4-3.7 of "Lecture notes on ECH"

  • For relation between Conley-Zehnder index see the exposition in  Section 3.4 of https://www.mathematik.hu-berlin.de/~wendl/pub/SFTlectures_book3.pdf
    The orginal proof (and a more detailed exposition) is found in Section 3 of  https://link.springer.com/article/10.1007/BF01895669

  • The key to the ECH index inequality is the writhe bound. Section 5.1 of "lecture notes on ECH" gives an outline of the argument. See Section 6 of "An index inequality for embedded pseudoholomorphic curves in symplectizations" for the full argument. Note the combinatorial inequality ("workhorse inequality") has a simpler proof now, given in Section 4.9 of https://arxiv.org/pdf/0805.1240.pdf. The outline in section 5.1 of "lecture notes on ECH" also follows this simpler argument.

  • That is most of the work of the ECH index inequality. There is a tiny amount more work to include the case of trivial cylinders, in Section 7 of "An index inequality for embedded pseudoholomorphic curves in symplectizations", but I say this is optional.

  • Be sure you understand the classification of low ECH index curves as proved in Proposition 3.7 of lecture notes on ECH. Try to figure out why the same argument does not work for symplectic cobordisms.

  • For details on grading on ECH see this paper: https://msp.org/agt/2013/13-4/agt-v13-n4-p13-s.pdf  (I haven't read this one, I just know it's here).
    I haven't needed to use something this general yet. However it's often useful to know the absolute grading for ECH on S^3 (as illustrated in the "lecture notes on ECH"

  • That's probably enough for one week, we can take a look at the items below for next week.

• Week 4: 

  • Section 5.3 of lecture notes to see the differential is well defined. Note: Gromov compactness for currents works in higher dimensions also.

  • Section 3.8 on the U map. Section 3.9 on partition conditions.

  • Section 5.5 on Cobordism maps on ECH. For details (stronly recommended) see Section 1.5 of this paper https://arxiv.org/pdf/1111.3324.pdf. People usually just use these cobordism maps axiomatically. Pay particular attention to the "J-holomorphic curves axiom" - this is what we mean when we say cobordism maps in ECH counts "ECH index 0 buildings". Now is also a good time to review the definition of ECH capacities in the very beginning of "lecture notes" and convince yourself they make sense.

  • Optional: There is also now a "chain level" cobordism map on ECH. This is described in Section 6 in https://arxiv.org/abs/1509.02183. It's useful for studying the knot filtration on ECH.

  • With the writhe bound and adjunction inequality in hand, there's more useful things they can do. They can rule out certain breakings of J-holomorphic curves in low dimensions. See this blogpost https://floerhomology.wordpress.com/2013/11/05/hidden-branched-covers-of-trivial-cylinders/. These techniques can be used to show cylindrical contact homology is well defined without abstract perturbations in certain settings, see for example Section 3 of this paper (used to rule out bad breakings):https://arxiv.org/pdf/1407.2898.pdf  (Section 3 of this paper and the blog post actually describe the same problem, so just pick whichever you want to read).

• Week 5: Obstruction bundle gluing.

  • Notes by Chris Gerig on the subject: https://scholar.harvard.edu/files/gerig/files/note.pdf

  • A 3 hour lectures series by Michael Hutchings. Starts with a finite dimensional model, circle valued Morse theory, then proceeds to explain the ECH case: https://www.youtube.com/watch?v=_4rJnq9Fn1Y

  • Video by Ipsita explaining also the circle valued Morse theory case: https://www.jacobrooney.com/obg-seminar-2020/ (scroll)

  • Video by me explaining the obstruction bundle gluing in the case of a pair of pants: https://www.jacobrooney.com/obg-seminar-2020/  (scroll)

  • Blogpost by Michael on circle valued Morse theory:https://floerhomology.wordpress.com/2014/07/14/gluing-a-flow-line-to-itself/

  • This is a lot of videos. I would recommend focus on understanding the Circled valued Morse theory case. Start by watching Michael's lectures (until he finishes the circled value Morse case). Supplement this with Chris Gerig's notes and Michael Blog post (and see Ipsita's video if you are still confused). 

  • Then the convex span of rest of Michael's lectures, my talk, and Chris Gerig's notes should explain how to do the pair of pants case. You can just pick one of these or all three, depending on how much time you have.

• Week 6: We start on the computation of ECH of T^3, following the paper https://arxiv.org/abs/math/0410061

  • Section 1-4 for the definition of the combinatorial chain complex

  • Optional Sections 5-8 for the computation of the combinatorial chain complex.

  • No holomorphic curves this week, hurrah!

• Week 7: We continue the paper https://arxiv.org/abs/math/0410061, please read sections 9 and 10 of this paper!