Dusa McDuff recently proved that the interior of the 4-dimensional ellipsoid symplectically embeds into if and only if . Here, if and are positive real numbers, then denotes the sequence obtained by taking all nonnegative linear combinations of and and sorting them in nondecreasing order.
What seems strange to me about this theorem is that Seiberg-Witten theory is used in the proof of both directions! I attempted to outline this story in my survey article “Recent progress on symplectic embedding problems in four dimensions”, but let me briefly indicate here where Seiberg-Witten theory comes in:
The “only if” part of the theorem follows immediately from the monotonicity property of ECH capacities. But the proof of the monotonicity property of ECH capacities uses Seiberg-Witten theory to construct ECH cobordism maps.
The “if” part of the theorem uses another result of McDuff that the existence of an ellipsoid embedding as above (when and are rational) is equivalent to the existence of a certain symplectic ball packing. The existence of this ball packing is equivalent to the existence of a symplectic form on a blowup of in a certain cohomology class. This symplectic form is constructed using “inflation” along a certain holomorphic curve. And the existence of this holomorphic curve is proved using Taubes’s “Seiberg-Witten=Gromov” theorem together with the wall crossing formula for Seiberg-Witten invariants on a blowup of .
McDuff gave an alternate proof of the “only if” part of her theorem without using ECH capacities, but it still uses the ball packing story and with it the above business involving Seiberg-Witten theory on blown up .
Anyway, it might be an interesting project to try to prove the “only if” part of the theorem without Seiberg-Witten theory, by directly proving the relevant special case of the monotonicity property of ECH capacities. One would want to prove that if there is a symplectic embedding of into the interior of , then there is a map from to which respects the symplectic action filtrations, the contact invariants, and the maps. The obvious approach is to regard minus the image of the interior of as an exact symplectic cobordism from to , to “complete” this cobordism by attaching “symplectization ends”, and to count holomorphic curves in the completed cobordism with ECH index . The difficulty is that multiply covered curves with negative ECH index will arise. So the cobordism map will probably have to count some broken configurations involving a negative ECH index multiple cover in the cobordism level together with a positive ECH index curve in one or both ends. It might be quite complicated to figure out how to count these and prove the gluing theorem needed to show that the cobordism map respects the map.
In any case, if you want to try to define ECH cobordism maps using holomorphic curves, then this is a special case of the problem which, while it might be quite difficult, appears a lot less hard than the full problem and might be doable.