Moduli Space of Symplectic Balls in 4-manifolds and Packing (1578)
Dates : 22 June – 3 July at CIRM (Marseille Luminy, France)
We investigated one of the most elusive questions in symplectic topology: is there a notion of « statistical symplectic topology »? What does happen when we take various limits, say on the dimension of the configuration space or on the radius of embedded symplectic balls in a given fixed symplectic manifold? In terms of physics, this boils down to understanding if there is any reasonable notion of probabilistic String theory. Note that it might perfectly be that this notion does not make sense from the String approach, but does make sense for the point of view of Symplectic Topology.
There are evidences that support both points of view. Indeed, the inexistence of a statistical Symplectic Topology is supported by the fact that the moduli space of k symplectically embedded balls of equal radius seems to loose all of its hard characteristics as k goes to infinity and the radius goes to zero. That is to say, it behaves like volume-perserving balls. This was probably best exemplified in a series of papers by Anjos-Lalonde-Pinsonnault that showed, for the first time, the existence of a critical value in moduli spaces in Symplectic Topology: below that critical value (of the radius of a symplectic ball in a rational 4-manifold M), the full moduli space retracts to the space of frames, that is to say to a compact finite dimensional CW complex. But above that value, the space is genuinely infinite dimensional and contains homology and homotopy in dimensions as high as one wishes. That proof uses heavy machinery, relying on hard elliptic methods, on a (cofinite) stratification of the infinite dimensional space of almost complex structures on M tamed by the symplectic form on which various groups act, and on a comparison between the stratification for the J-structures on the blow-up of M at the given ball and the stratification for the J-structures on M itself. In any case, one needs hard methods to prove that dichotomy between hard and soft moduli spaces.
More recently, an isoperimetric conjecture of Viterbo that states that the round ball achieves the minimum for some symplectic capacities, has been proved by Ostrover and Artstein-Avidan. Even if the conjecture is stated in finite dimensions, their proof relies on the asymptotic behaviour of these capacities. This supports the idea that some limits, when carefully taken, can carry more information than what one would find in the soft world of volume-preserving maps.
Note that the space of embeddings of balls and ellipsoids lead in general, if no limit is taken, to an extraordinarily rich corpus, where Number theory intervenes in an essential way. This was best illustrated by the recent work of McDuff and Schlenk. Olguta Buse has also recently done substantial work in proving that limits lead to soft objects.