Dynamics Beyond Turbulence and Obstructions to Classification

超越湍流和分类障碍的动力学

• 批准号: 2154258
• 负责人: Clinton Conley
• 金额: $ 18万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154258&HistoricalAwards=false
• 关键词: Dynamics; Beyond; Turbulence; Obstructions; Classification

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.

这是我们数学实践的经验现实——也许是任何适当的数学形式主义的不可避免的产物——复杂的数学对象有许多不同的，但等效的，具体的描述。随着相关研究对象的复杂化，这种等价描述的多元宇宙也会变得越来越复杂，而解决相关的“分类问题”，即设计一种有效的方法来判断两个具体描述是对应于同一个对象还是对应于两个不同的对象，往往变得非常困难。这是一个众所周知的难题，例如，决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，以及找到在任意坐标变化下不变的可测量量是一个开放的问题。不变量描述性集合论是数学逻辑的一个领域，它提供了一个形式化的框架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，最好的例子是Hjorth的湍流理论。不幸的是，除了湍流之外，几乎没有什么是已知的。事实上，超过湍流点的动力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，这使得经典的调和分析方法和离散模型理论不足。然而，PI和其他人最近取得的一系列突破为处理这种动态提供了一些新的策略。通过这个项目，我们开始系统地研究这种比湍流更“疯狂”的动力现象，它可以作为更一般分类形式的障碍。拟议的研究计划包括四个独立但相互影响的项目。第一个项目解决了无限维可分希尔伯特空间的酉群的作用是否存在分类障碍的问题。在此过程中，它将研究Borel约简层次与度量模型理论中关于稳定性/NIP与自反性/罗森塔尔可表征性之间的对应关系的一些最新发展之间的联系。第二个项目侧重于同胚群的动力学。其中，它解决了区间的同胚群是否允许湍流作用这一长期悬而未决的问题，并提出了一些“高维”的湍流变体，这些变体可以作为阻碍n维紧构的同胚群通过作用进行分类的障碍。第三个项目提出了一个统一的框架，用于提取与巴拿赫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群体动态的问题，这些群体不承认来自PI和其他人最近工作的双面不变度量。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。