Applications of descriptive set theory in Ergodic theory and investigations into singular cardinals combinatorics

描述性集合论在遍历理论中的应用及奇异基数组合学的研究

• 批准号: 0701030
• 负责人: Matthew Foreman
• 金额: $ 37.5万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701030&HistoricalAwards=false
• 关键词: Applications; descriptive; set; theory; Ergodic

Foreman's proposal involves applying tools from mathematical logic to questions in dynamical systems. Many dynamical systems, while completely determinate, appear to have elements of random behavior. This phenomenon can be described explicitly when there is a time-invariant probability measure on the system. Such a description could state that the system is measure theoretically isomorphic to a particular known process, such as a Bernoulli process. From this point of view it is natural to try to attempt to classify dynamical systems measure-theoretically. The hope would be to have a "library" of possible measure preserving systems and be able to describe an arbitrary system measure theoretically as one in the library.This project, while very successful in its early stages, runs into insuperable obstacles for deep logical reasons. Foreman's previous work with his co-authors showed that the isomorphism problem for ergodic measure preserving systems is inherently too complex from a logical point of view to admit a classification. Foreman's proposed work involves extending these anti-classification results to differentiable systems on compact manifolds and to identify those systems for which the isomorphism problem is tractable.Many natural systems evolve over time according to definite rules. Much of science involves discovering these rules and describing them, perhaps by a system of equations. These rules discuss how individual points in a system behave, and are often completely deterministic. However, what can be actually observed (for example due to round-off error) in these systems are sets of points. At this level the qualitative behavior of a dynamical system can be apparently random. This led to the project of classifyingthe possible behavior statistically so that the qualitative behavior ofnatural systems could be catalogued. There were many successes in the program in its early stages. Recently however, it turns out that there are reasons related to mathematical logic that the program cannot, in principle, work. Foreman's proposed research explores extending the impossibility results to concrete settings and finding large collections of systems for which there are good classifications of their statistical behavior.

工头的建议涉及将数学逻辑中的工具应用于动态系统中的问题。许多动态系统虽然完全确定，但似乎具有随机行为的元素。当系统上存在时间不变的概率度量时，可以明确描述此现象。 这样的描述可以指出，该系统在理论上是针对特定已知过程（例如伯努利过程）的同构。从这个角度来看，自然要尝试以理论方式对动态系统进行分类。希望拥有一个可能的“库”来保存系统的“库”，并能够将任意系统从理论上描述为图书馆中的系统。该项目虽然在早期阶段非常成功，但出于深刻的逻辑原因，该项目却遇到了不可避免的障碍。工头以前与他的合着者的工作表明，从逻辑的角度来看，维护奇异措施的同构问题本质上太复杂了，无法接受分类。工头的提议工作涉及将这些反分类结果扩展到紧凑型歧管上的可区分系统，并确定同构问题是可以处理的那些系统。许多自然系统根据确定的规则随时间而变化。许多科学都涉及发现这些规则并描述它们，也许是通过方程式系统来描述这些规则。这些规则讨论了系统中各个要点的行为方式，并且通常是完全决定性的。但是，这些系统中实际上可以观察到的（例如由于圆形误差）是一组点。在这个层面上，动态系统的定性行为显然可以是随机的。这导致了从统计学上对可能行为进行分类的项目，以便可以对自然系统的定性行为进行分类。该计划的早期阶段取得了许多成功。但是，最近，事实证明，与数学逻辑有关的原因是该程序原则上不能起作用。工头提议的研究探讨了将不可能结果扩展到具体设置并找到大量系统的大量系统，这些系统对其统计行为进行了良好的分类。