Topological Dynamics and Countable Combinatorics

动力学是对运动物体的研究。在这里，一个“物体”被表示为某个数学结构中的一个点，而“运动”被理解为这个结构的一组对称性。通过考虑不同的数学结构，来自动力学的思想与数学的其他几个领域相互作用，特别是逻辑学、群论和组合学。反过来，这些领域的思想产生了动力学中的新现象。这个项目将集中于来自“可数组合学”的想法。组合学研究的是离散结构，而不是连续结构，而对可数组合学的强调意味着这些离散结构将是无限的，但仍然足够小，我们可以写下它们的元素列表或枚举。过去二十年的大量研究已经阐明了可数结构的鸽子洞原理如何产生具有简单动力学的群的例子；在这里，对称组将是很大的，即不可数的。另一方面，最近注意力已经转移到考虑可数群本身的组合论证；这些可以用来证明可数群总是具有野动态。对于这个项目，PI有两个主要目标。第一个是继续在可数群和更一般的波兰群的组合原理和动态特性之间来回发展字典。在可数群的情况下，群上某些模式的存在可以导致具有有趣行为的子位移的构造，例如双最小位移或与另一个给定动力系统不相交的位移。对于更一般的波兰群，类似拉姆齐的原理可以用来产生有趣的超滤物体空间，群可以在超滤物体上行动。然后，这些动力系统将具有重要的通用特性，例如对于最小流是通用的，或者是一个通用的“完成流”，一个由PI先前定义和研究的动态对象。第二是发展证明可数组合学新定理的技术。特别有趣的是断言各种可数一阶结构具有有限大拉姆齐度的定理。虽然感兴趣的属性是关于可数对象的，但所使用的技术可以处理非常大的对象，例如超过滤器的空间或强制偏置集。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。

Dynamics is the study of objects in motion. Here, an "object" is represented by a point in some mathematical structure, and "motion" is understood to mean a group of symmetries of this structure. By considering different mathematical structures, ideas from dynamics interact with several other areas of mathematics, in particular logic, group theory, and combinatorics. In turn, ideas from these fields give rise to new phenomena in dynamics. This project will focus on ideas coming from "countable combinatorics," broadly construed. Combinatorics is the study of discrete structures, as opposed to continuous ones, while the emphasis on "countable" combinatorics means that these discrete structures will be infinite, but still small enough that we can write down a list, or enumeration, of their elements. Much research in the past two decades has elucidated how pigeon-hole principles on countable structures can yield examples of groups with simple dynamics; here, the group of symmetries will be large, i.e. uncountable. On the other hand, attention has shifted recently to considering combinatorial arguments on countable groups themselves; these can be used to show that countable groups always have wild dynamics.The PI has two primary goals for this project. The first is to continue to develop the dictionary back and forth between combinatorial principles and dynamical properties of countable groups and more general Polish groups. In the case of countable groups, the existence of certain patterns on the group can lead to the construction of subshifts with interesting behavior, such as doubly minimal shifts or shifts disjoint from another given dynamical system. For more general Polish groups, Ramsey-like principles can be used to produce interesting spaces of ultrafilter-like objects on which the group can act. These dynamical systems will then have important universal properties, such as being universal for minimal flows, or being a universal "completion flow," a dynamical object previously defined and investigated by the PI. The second is to develop techniques to prove new theorems in countable combinatorics. Of particular interest are theorems asserting that various countable first-order structures have finite big Ramsey degrees. While the properties of interest are about countable objects, the techniques used can deal with extremely large objects, such as spaces of ultrafilters or forcing posets.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.