Fundamental and Statistical Symplectic Topology

基本和统计辛拓扑

• 批准号: RGPIN-2017-06566
• 负责人: Lalonde, François
• 金额: $ 1.75万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759325
• 关键词: Fundamental; Statistical; Symplectic; Topology

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.

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。根据Leonid Polterovich等人的说法，给定一个辛流形$(M，omega)$，考虑由有限个开集组成的覆盖，以及一个单位划分$f_1，…，f_k$，并且取$a_1，…，a_k，b_1，...，b_k$的泊松括号范数的超，其中$\sum_i a_if_i$，$\sum_i b_i_f_i$，唯一的约束是$a_i，[-1，1]$中的B_i\。然后，将信息覆盖所有统一的分区。打开的盖子上会有一个数字。主要结果是，这个数字由一个仅取决于开集数目的常量来限定。Polterovich猜想，应该有一个甚至不依赖于开子集的数量的正常数。我们的目标是将这一理论推广到由开集的连续体给出的覆盖，就像给定容量的球的辛嵌入空间的$\pi_3$中的同伦类的代表所给出的覆盖一样。我们确实发展了一种由赋予相应函数的开子集族构成的覆盖的单位分解理论，其中和被积分代替。如果Polterovich猜想是真的，我们确实可以证明这一点。在非对易性和不确定性方面的临界值仍有待比较。这个项目还包括另外三个与Atiyah-Floer猜想中的团簇复合体有关的实质性问题，Viterbo猜想和确定辛拓扑的基础有多难的问题。