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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=656430
• 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，我和我的学生马丁·平索诺（Silvia Anjos后来加入了我们）发现了辛拓扑中可以被认为是第一个相变的东西：实际上，我们发现在某些规则的四维辛流形中，（M，\omega）$球容量的临界值$c_{crit}$使得低于该值，给定容量c < c_{crit}$的标准4-球的辛嵌入的无穷维空间（M，\omega）$在$M$上的辛框架的有限维流形上收缩，而超过该临界值，这个空间在任何有限维CW复形上都不收缩（它在我们所希望的那么高的维度上具有同调）。令人惊讶的是，同伦的变化只发生在$\pi_3$的水平上，所以这个临界值不可能被物理学家发现。这就提出了三个问题：第一，从物理上把这种现象理解为不确定性的表现，在我们的理论中，这种不确定性是自然量化的。我们正在寻找一个一般的框架来解释这种现象严格作为一个相变。第二个问题是试图推广到其他环面流形使用非常不同的技术。第三个也是最有趣的问题是将这些表示不确定性的临界值与表示流形的泊松反交换性水平的其他值联系起来。 根据Leonid Polterovich和al.，给定一个辛流形$（M，\omega）$，考虑一个有限开集的覆盖，和一个单位分划$f_1，...，f_k$，并将sup取过$a_1，...，a_k，b_1，.，B_k$的泊松括号的范数$\sum_i a_if_i$与$\sum_i B_i f_i$的唯一约束是$a_i，B_i \in [-1，1]$。然后在单位的所有分区上取inf。这给出了一个附在打开的盖子上的数字。主要的结果是，这个数字是由一个常数，只取决于开放集的数量下界。Polterovich解释说，应该有一个积极的常数，甚至不依赖于开放子集的数量。这里我们的目标是将这个理论推广到由连续开集给出的覆盖，就像由同伦类的代表在给定容量的球的辛嵌入空间中给出的覆盖一样。我们确实发展了一个理论分区的单位覆盖连续家庭的开放子集赋予相应的功能，其中总和取代积分。如果Polterovich猜想是正确的，我们确实可以做到这一点。在非对易性和不确定性中，仍然需要比较临界值。这个项目还包括其他三个实质性的问题有关的集群复杂的Atiyah-Floer猜想，在维泰博猜想和问题的确定有多难的基础辛拓扑。