Dynamics Beyond Turbulence and Obstructions to Classification

复杂的数学对象有许多不同但相同的具体描述，这是我们数学实践的经验现实--也许也是任何适当的数学形式主义的必然产物。随着相关的研究对象变得更加复杂，这种等价描述的多重宇宙往往会变得更加复杂，而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对应于同一对象还是两个不同的对象。例如，确定爱因斯坦方程的两个不同的具体解是否描述了完全相同的物理现实，以及寻找在任意坐标变化下不变的可测量是一个悬而未决的问题，这是一个出了名的困难问题。不变描述集合论是一个数理逻辑领域，它提供了一个形式化的框架，用来衡量这类分类问题的内在复杂性，并在每种情况下决定哪些类型的不变量“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元数学之间的重要联系，Hjorth的湍流理论就是最好的例证。不幸的是，除了湍流点之外，几乎没有什么事情是已知的。事实上，超过湍流点的动力学现象存在于既不是局部紧致的也不是非阿基米德的对称组中，这使得经典的调和分析方法和离散模型理论不够充分。然而，最近从国际和平研究所和其他机构的工作中取得的一系列突破表明，应对这种动态的一些新战略。通过这个项目，我们开始了对这些比湍流更“狂野”的动力现象的系统研究，这些现象可以作为更一般形式的分类的障碍。第一个方案讨论了无限维可分Hilbert空间的酉群的作用分类是否存在障碍的问题。在这样做的过程中，它将考察Borel约化等级与度量模型理论中关于稳定性/NIP和自反/Rosenthal表征之间的对应关系的一些最新发展之间的联系。第二个项目主要研究同胚群的动力学。其中，它解决了一个长期未解决的问题，即区间的同胚群是否允许湍流作用，并提出了一些湍流的“高维”变体，它们可以用作n维紧空间的同胚群作用分类的障碍。第三个项目提出了一个统一的框架，用于提取与Banach空间的动力学有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的问题，这些团体不承认源于PI和其他人最近的工作的双边不变指标。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.