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)发现了可以被认为是辛拓扑中的第一个相变：事实上，我们在一些规则的4维辛流形$(M，\omega)$中发现了球容量的临界值$c_{crit}$，在该值下，给定容量的标准4-球的辛嵌入的无限维空间$c&lt；C_(Crit)$in$(M，\omega)$收缩在$M$上的辛框架的有限维流形上，而超过该临界值，该空间不会收缩到任何有限维CW复形上(它在维度上的同调就像我们希望的那样高)。引人注目的是，同伦的变化只发生在$\pi_3$水平上，所以这个临界值不可能被物理学家检测到。这引发了三个问题：第一个问题是将这种现象理解为不确定性的物理表达，这在我们的理论中自然是量化的。我们正在寻找一个总体框架，将这一现象严格地解释为一种相变。第二个问题是试图通过使用非常不同的技术将其推广到其他环形流形。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的泊松反对易程度的其他值联系起来。

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.