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.****** This goes in the following way, according to Leonid Polterovich and al., given a symplectic manifold $(M, \omega)$, consider a covering by a finite number of open sets, and a partition of unity $f_1,..., f_k$, and take the sup over $a_1,...,a_k, b_1,...,b_k$ of the norm of the Poisson bracket of $\sum_i a_if_i$ with $\sum_i b_i f_i$ with the only constraint that $a_i, b_i \in [-1,1]$. Then take the inf over all partitions of unity. This gives a number attached to the open cover. The main result is that this number is bounded below by a constant that depends only on the number of open sets. Polterovich conjectured that there should be a positive constant that does not even depend on the number of open subsets. Our goal here is to extend this theory to coverings given by a continuum of open sets, like the one given by a representative of a homotopy class in the $\pi_3$ of the space of symplectic embeddings of balls of given capacity. We have indeed developed a theory of partitions of unity for coverings made of continuous families of open subsets endowed with corresponding functions where sums are replaced by integrals. We can indeed acheive this if Polterovich conjecture is true. There remains to compare critical values in non-commutativity and uncertainty. This project includes also three other substantial problems related to the cluster complex in the Atiyah-Floer conjecture, the Viterbo conjecture and the problem of determining how hard are the foundations of Symplectic Topology.