Fundamental and Statistical Symplectic Topology

从2004年到2010年，我的学生Martin Pinsonnault和我（Silvia Anjos后来加入了我们）发现了辛拓扑中的第一相变：事实上，我们在一些规则四维辛流形中发现了这种相变 $(M, \omega)$ 临界值 $c_{crit}$ 对于小于该值的球的容量，给定容量的标准四球的辛嵌入的无限维空间 $c < c_{crit}$ 内部 $(M, \omega)$ 上辛坐标系的有限维流形上的缩回 $M$，当超过临界值时，该空间在任何有限维CW复上都不会收缩（它在我们希望的高度上具有同调）。值得注意的是同伦的变化只发生在 $\pi_3$ 水平，所以这个临界值不可能被物理学家检测到。这就提出了三个问题：第一个问题是把这种现象从物理上理解为不确定性的表达，不确定性在我们的理论中是自然量子化的。我们正在寻找一个一般的框架来解释这一现象，严格地作为一个相变。第二个问题是试着用不同的技巧把这个推广到其他环面流形。第三个也是最有趣的问题是将这些表示不确定性的临界值与表示流形泊松反交换性水平的其他值联系起来。 按照Leonid Polterovich等人的说法，给出一个辛流形$(M, \omega)$，考虑一个由有限个数的开集覆盖的流形，以及一个单位的分割$f_1,..., f_k$，并取$\sum_i a_if_i$的泊松括号的范数在$a_1,...,a_k, b_1,...,b_k$上的sup with $\sum_i b_i f_i$，唯一的约束是$a_i, b_i \in [-1,1]$。然后对所有的分区取中值。这是一个号码，附在打开的盖子上。主要的结果是这个数是由一个常数限定的，这个常数只依赖于开集的数量。Polterovich推测应该存在一个正常数，它甚至不依赖于开放子集的数量。我们的目标是将这一理论推广到由开集连续体给出的覆盖，就像一个同伦类的代表在给定容量的球的辛嵌入空间$\pi_3$中给出的覆盖一样。对于具有相应函数的连续开子集族所构成的覆盖，我们确实发展了一个统一划分理论，其中和被积分所代替。如果Polterovich猜想成立，我们确实可以做到这一点。对于非交换性和不确定性的临界值还有待比较。该项目还包括与Atiyah-Floer猜想、Viterbo猜想中的簇复合体相关的其他三个实质性问题，以及确定辛拓扑基础的难度的问题。

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.