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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711999
• 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 后来加入我们）发现了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 $(M, \omega )$ 球容量的临界值 $c_{crit}$ ，低于该值，标准辛嵌入的无限维空间$(M, omega)$ 内给定容量 $c < c_{crit}$ 的 4 球在 $M$ 上辛框架的有限维流形上收缩，而超过该临界值时，该空间不会在任何有限维 CW 复合体（它在维度上具有我们希望的高同源性）。令人惊奇的是，同伦的变化只发生在$\pi_3$级别，因此物理学家无法检测到这个临界值。这就提出了三个问题：第一个是将这种现象在物理上理解为不确定性的表达，这在我们的理论中自然地被量化。我们正在寻找一个通用框架来严格地将这种现象解释为相变。第二个问题是尝试通过使用非常不同的技术将其推广到其他复曲面流形。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。 根据 Leonid Polterovich 等人的说法，这按照以下方式进行，给定辛流形 $(M, omega)$，考虑有限数量的开集覆盖，以及统一的划分 $f_1,... , f_k$，并对 $\sum_i a_if_i$ 的泊松括号范数的 $a_1,...,a_k, b_1,...,b_k$ 取 SU $\sum_i b_i f_i$ 唯一的约束是 $a_i, b_i \in [-1,1]$。然后对 Unity 的所有分区进行 inf 处理。这给出了附在打开的盖子上的编号。主要结果是这个数字的边界是一个仅取决于开集数量的常数。波尔特罗维奇推测应该存在一个正常数，该常数甚至不依赖于开子集的数量。我们的目标是将这一理论扩展到由开集连续统给出的覆盖，就像给定容量的球的辛嵌入空间 $\pi_3$ 中同伦类的代表给出的覆盖。我们确实开发了一种统一划分理论，用于由连续的开子集族组成的覆盖，赋予相应的函数，其中和被积分代替。如果波尔特罗维奇猜想成立，我们确实可以实现这一点。仍然需要比较非交换性和不确定性的临界值。该项目还包括与 Atiyah-Floer 猜想、Viterbo 猜想中的簇复形相关的其他三个实质性问题以及确定辛拓扑基础有多难的问题。