Symplectic embeddings I

I’ve been participating in the Berkeley-Tokyo winter school on geometry, representation theory, and topology for the last week – you can find the notes from my talk last Thursday on my webpage – and have found it unexpectedly interesting being around many people interested in 4-dimensional topology.

A highlight was a killer talk by Michael Hutchings (whose blog I recently discovered thanks to Alex) on symplectic embeddings. I tend not to take notes at such events, but jotted down what I could remember and scoured the immediate references to start to fill in some of the gaps. The problem is essentially as follows: given two symplectic manifolds of the same dimension, can you tell when there is a symplectic embedding between the two? As usual, ‘symplectic’ means that it respects the symplectic forms on each space. Hutchings’ talk addressed a result that follows from an aggregation of his work and that of Dusa McDuff describing when two 4-dimensional ellipsoids can be embedded into one another. Such an ellipsoid is, in complex coordinates, described by two positive real numbers a,b giving the axial radii. Denote the corresponding ellipsoid by E(a,b). The result is a beautiful fusion of symplectic geometry and the geometry of numbers based on the sequence of numbers of the form ma+nb for m,n\in\mathbb{N} listed in ascending order and including repeats. These turn out to be precisely a stream of invariants from contact homology called ECH capacities defined by Hutchings.

As someone who works in mirror symmetry, I’ve often felt guilty about my lack of exposure to ‘authentic’ symplectic geometry. I intend to use the next few blog posts – intermingled with some talk prep for upcoming talks on intersection theory and ribbon graphs (separately…) – to begin to amend that. This compelling story also has an algebraic and a combinatorial twist: these invariants very clearly have something to do with lattice point counts and hence Ehrhart theory (which was made rigorous in some cases by some students at Berkeley a few years ago), and, less obviously, continued fractions and homology of blowups of \mathbb{P}^2 past the Fano-threshold. As my posts to date (and general interests) suggest, these latter two topics are very close to my heart and so might be good anchors to use when approaching the more unfamiliar symplectic side of the story. I hope your interest is as piqued as mine is.

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