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。

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.