Lecture notes on ECH 0: An application, and overview of ECH

Today I gave the first two lectures in the minicourse on ECH in Budapest. Instead of telling the story from the beginning, as I have been doing on this blog, I thought it would be better to first state an application of ECH, and then give an overview of the basic structure of ECH, so that the audience would have some motivation for studying the details and some rough idea of where we are going. When I write more formal lecture notes, I plan to start with an introduction along these lines. Tomorrow I will start over and explain the detailed story from the beginning, as I have been doing here (but probably not in as much detail).

I don’t have much time to write right now, so instead of complete notes on the lectures, here is an outline of what I said.

The application I chose to describe was the sequence of obstructions to four-dimensional symplectic embeddings. First, to give some flavor of the subject, I stated McDuff’s theorem giving necessary and sufficient conditions for the existence of symplectic embeddings between four dimensional ellipsoids, and I described McDuff-Schlenk’s computation of the obstructions to embedding an ellipsoid into a ball. I then stated the existence of ECH capacities and described their basic properties, which among other things imply the obstruction half of McDuff’s theorem. I also announced the result of Dan Cristofaro-Gardiner, Vinicius Gripp, and myself, that the “volume conjecture” regarding the asymptotics of the ECH capacities is now a theorem.

I then gave an overview of the structure of ECH. In particular I said what the generators of the ECH chain complex are, but I was very vague about the definition of the differential (which, as you have seen here, is a subtle matter which requires more lecture time). I briefly stated Taubes’s isomorphism of ECH with Seiberg-Witten Floer cohomology. I then described four additional structures on ECH, which I have not yet described here, namely: the canonical class, the U map, filtered ECH, and cobordism maps. Taking all of this for granted, I then showed how one can use this formalism to define the ECH capacities, and the proof of the monotonicity axiom (giving obstructions to embeddings) is then a straightforward exercise.

Most of the above (at roughly the level of detail of my lectures) is contained in the survey article “Recent progress on symplectic embedding problems in four dimensions”. A nearly final version of this article is available here. (The parts of the article about ball packings and the proof of McDuff’s theorem were not covered in the lecture.)

I then had ten minutes left, and I didn’t want to start on the details, so I said a bit about how Taubes’s isomorphism implies the Weinstein conjecture in three dimensions. Another direction of applications of ECH is to refinements of the three-dimensional Weinstein conjecture, but I didn’t have time to discuss this.

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