CAREER: Uncountable topological dynamics

职业：不可数的拓扑动力学

• 批准号: 2144118
• 负责人: Dana Bartosova
• 金额: $ 40万
• 依托单位: University of Florida
• 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: 2022-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2144118&HistoricalAwards=false
• 关键词: CAREER; Uncountable; topological; dynamics

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).Dynamics has its origins in physics: it studies motion in space, typically in discrete or continuous time. Its abstractions find numerous applications throughout mathematics. For example, objects can be understood or classified by their groups of symmetries and their actions (such as crystals or systems of differential equations), or they can be enclosed into a space and studied via its motions in it (such as flow of water on Earth and in its atmosphere). This project is in the realm of abstract topological dynamics, which extends the notion of time to any topological group and restricts spaces to compact Hausdorff. Its applications lie mostly in combinatorics, number theory, and measure theory. A priori, there is no bound on the size of the groups or spaces, however the majority of results have countability restrictions on both. The purpose of this project is to relax the countability conditions and to lay the foundations of uncountable topological dynamics. There are two major motives to do so: first, generalizations often bring clarification and simplification of a theory by revealing its true core and requires novel ideas that lead to new discoveries. Second, the uncountable is unavoidable as soon as we accept existence of countable infinity and even very natural dynamical systems of the discrete group of integers have complexity beyond any notion of countability. The core objects of abstract topological dynamics are minimal dynamical systems that can be thought of as building blocks of dynamical systems. In particular, the universal minimal flow captures the complexity of minimal dynamics of a given group since it quotients onto every minimal dynamical system. It is known that for infinite locally compact, non-compact groups, the universal minimal flow is non-metrizable. Yet, even the universal minimal flow of the discrete group of integers is poorly understood, in contrast to an extensively developed theory of its metrizable minimal flows. The aim of this project is to fill this gap and to provide explicit descriptions of universal minimal flows of foundational groups such as the integers, the integer lattice, or the general linear group. More generally, the PI will continue recently initiated investigation of interactions between universal minimal flows and operations on groups (such as products, short exact sequences, or inverse limits). On the other side of the spectrum, remarkable discoveries from the verge of the century of connections between finite combinatorics (Ramsey theory) and topological dynamics showed that universal minimal flows of non-locally compact (called infinite dimensional) groups that are essentially countable can be metrizable or even trivial. The PI proved analogous results for infinite dimensional groups with the countability assumption removed and will build further upon these results. The ultimate goal is to find a unifying theory for these two seemingly orthogonal dynamical behaviors of locally compact and infinite dimensional groups in the joint framework of ultrafilter dynamics established in discrete dynamics and extended to topological dynamics by the PI. The project lies on the boundary of set theory, topology, model theory, combinatorics, and algebra and offers ample training opportunities for graduate, undergraduate, and even talented high school students.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.

该奖项全部或部分由2021年美国救援计划法案（公法117-2）资助。动力学起源于物理学：它研究空间中的运动，通常是离散或连续的时间。它的抽象在整个数学中找到了许多应用。例如，物体可以通过它们的对称性和它们的作用来理解或分类（如晶体或微分方程组），或者它们可以被封闭在一个空间中，并通过它在其中的运动来研究（如地球上和大气中的水流）。这个项目属于抽象拓扑动力学领域，它将时间的概念扩展到任何拓扑群，并将空间限制为紧致豪斯多夫。它的应用主要在于组合学，数论和测度论。先验地，群或空间的大小没有限制，然而大多数结果都对两者有可数性限制。本项目的目的是放宽可数性条件，为不可数拓扑动力学奠定基础。这样做有两个主要动机：第一，概括经常通过揭示理论的真正核心来澄清和简化理论，并需要导致新发现的新想法。第二，一旦我们接受可数无穷大的存在性，不可数就不可避免了，即使是非常自然的整数离散群的动力系统，其复杂性也超出了任何可数性的概念。抽象拓扑动力学的核心对象是最小动力系统，可以被认为是动力系统的构建块。特别地，泛最小流捕捉到了给定群的最小动力系统的复杂性，因为它服从于每个最小动力系统。已知对于无限局部紧的非紧群，泛极小流是不可度量化的。然而，即使是离散整数群的泛最小流也知之甚少，与其可度量化的最小流的广泛发展的理论相反。这个项目的目的是填补这一空白，并提供明确的描述通用的最小流的基础组，如整数，整数格，或一般的线性组。更一般地说，PI将继续最近发起的泛最小流和群运算（如乘积、短精确序列或逆极限）之间相互作用的研究。在光谱的另一边，从世纪的有限组合学（拉姆齐理论）和拓扑动力学之间的联系的边缘显着的发现表明，非局部紧（称为无限维）基本上可数的群的泛极小流可以是可度量化的，甚至是平凡的。PI证明了无穷维群的类似结果，去掉了可数性假设，并将在这些结果的基础上进一步建立。最终目标是在建立于离散动力学并由PI推广到拓扑动力学的超滤动力学联合框架下，为局部紧和无限维群这两种看似正交的动力学行为找到一个统一的理论。该项目涵盖了集合论、拓扑学、模型论、组合学和代数等学科，为研究生、本科生甚至有才华的高中生提供了充足的培训机会。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。