Fundamental and Statistical Symplectic Topology

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From 2004 to 2010, my student Martin Pinsonnault and myself (Silvia Anjos joined us later) discovered what could be considered as the first phase transition in symplectic topology: indeed, we found in some ruled 4-dimensional symplectic manifolds $(M, \omega)$ a critical value $c_{crit}$ for the capacity of balls such that below that value, the infinite dimensional space of symplectic embeddings of the standard 4-ball of given capacity $c < c_{crit}$ inside $(M, \omega)$ retracts on the finite dimensional manifold of symplectic frames on $M$, while beyond that critical value, that space does not retract on any finite dimensional CW complex (it has homology in dimensions as high as we wish). What is spectacular is that the change in homotopy only occurs starting at the $\pi_3$ level, so that this critical value could not have been detected by physicists. This raises three questions: the first one is to understand this phenomenon physically as the expression of uncertainty, which is naturally quantized in our theory. We are looking for a general framework to interpret this phenomenon rigorously as a phase transition. The second question is to try to generalise this to other toric manifolds by using very different techniques. The third and most interesting question is to relate these critical values, which express uncertainty, to other values that express the level of Poisson anti-commutativity of the manifolds.