Dynamics Beyond Turbulence and Obstructions to Classification

这是我们数学实践的一个经验现实，也可能是任何适当的数学形式主义的一个无情的产物，复杂的数学对象有许多不同的，但等价的，具体的描述。随着相关的研究对象变得更加复杂，这种多元的等效描述往往变得更加复杂，并且通常很难解决相关的“分类问题”，即，设计一种有效的方法来区分两个具体描述是对应于同一对象还是对应于两个不同的对象。这是一个出了名的困难的问题，例如，决定是否两个不同的具体解决方案爱因斯坦的方程是描述完全相同的物理现实和一个开放的问题，找到可测量的数量是不变的任意变化的坐标。不变量描述集合论是数理逻辑的一个领域，它提供了一个正式的框架来衡量这种分类问题的内在复杂性，并决定在每种情况下，哪些类型的不变量“太简单”而不能用于完整的分类。它还提供了拓扑动力学和分类的元数学之间的重要联系，最好的例子是Hjorth的湍流理论。不幸的是，除了湍流之外，几乎没有什么东西是已知的。事实上，超越湍流点的动力学现象存在于既不是局部紧致也不是非阿基米德的对称群中，使得调和分析和离散模型理论的经典方法不足。然而，PI和其他人的工作最近取得了一系列突破，提出了一些处理这种动态的新策略。通过这一计划，我们开始系统地研究这种动力学现象，这是“怀尔德”比湍流，它可以作为更一般形式的分类障碍。拟议的研究计划的特点是四个独立的，但相互作用的项目。第一个项目解决了无限维可分Hilbert空间的酉群的作用是否存在分类障碍的问题。在这样做的时候，它将检查之间的联系，博雷尔减少层次结构和一些最近的发展度量模型理论的稳定性/NIP和自反/罗森塔尔表示之间的对应关系。第二个项目的重点是同胚群的动力学。除其他外，它解决了长期存在的开放问题，是否同胚群的间隔承认一个动荡的行动，它提出了一些“高维”的变种动荡，可用于作为障碍的分类行动同胚群的n-维arcta。第三个项目提出了一个统一的框架，用于提取与Banach空间动力学相关的各种湍流现象的几何内容。第四个项目解决了几个关于波兰团体的动态问题，这些团体不承认PI和其他人最近的工作产生的双边不变度量。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

It is an empirical reality of our mathematical practice—and perhaps an inexorable artifact of any adequate mathematical formalism—that complex mathematical objects come with many different, yet equivalent, concrete descriptions. This multiverse of equivalent descriptions tends to grow in complexity as the associated objects of study become more complex, and it often becomes very difficult to solve the associated “classification problem”, i.e., to design an efficient method for telling whether two concrete descriptions correspond to the same object or to two different objects. It is a notoriously hard problem for example to decide whether two different concrete solutions of Einstein’s equations are descriptions of the exact same physical reality and an open problem to find measurable quantities which are invariant under the arbitrary change of coordinates. Invariant descriptive set-theory is an area of mathematical logic which provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides an important link between topological dynamics and the meta-mathematics of classification as best exemplified by Hjorth’s theory of turbulence. Unfortunately, few things are known beyond the point of turbulence. Indeed, dynamical phenomena beyond the point of turbulence reside in groups of symmetries which are neither locally-compact nor non-archimedean, rendering classical methods of harmonic analysis and discrete model theory insufficient. However, a series of recent breakthroughs coming from the work of the PI and others suggest some new strategies for dealing with such dynamics. Through this program we initiate the systematic study of such dynamical phenomena which are “wilder” than turbulence and which can serve as obstructions to more general forms of classification.The proposed research program features four independent—yet mutually interacting—projects. The first project addresses the question of whether there exist obstructions to classification by actions of the unitary group of the infinite dimensional separable Hilbert space. In doing so it will examine the connections between the Borel reduction hierarchy and some recent developments in metric model theory regarding the correspondence between stability/NIP and reflexive/Rosenthal representability. The second project focuses on the dynamics of homeomorphism groups. Among others, it addresses the long-standing open problem of whether the homeomorphism group of the interval admits a turbulent action and it proposes some “higher dimensional” variants of turbulence which could be used as obstructions to classification by actions of homeomorphism groups of n-dimensional compacta. The third project proposes a unified framework for extracting the geometric content of various turbulent phenomena which are associated with dynamics of Banach spaces. The fourth project addresses several questions regarding the dynamics of Polish groups that do not admit two-sided invariant metrics which stem from the recent work of the PI and others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.