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.

福尔曼的提议涉及将数理逻辑工具应用于动力系统中的问题。许多动力系统虽然是完全确定的，但似乎具有随机行为的元素。当系统上存在时不变概率测度时，可以明确地描述这种现象。 这样的描述可以表明该系统在理论上与特定的已知过程（例如伯努利过程）是同构的。从这个角度来看，尝试从测度理论上对动力系统进行分类是很自然的。我们希望拥有一个可能的测度保存系统的“库”，并且能够像库中的系统测度一样从理论上描述任意系统测度。这个项目虽然在早期阶段非常成功，但由于深刻的逻辑原因遇到了不可克服的障碍。福尔曼与他的合著者之前的工作表明，从逻辑角度来看，遍历测度保持系统的同构问题本质上过于复杂，无法进行分类。福尔曼提出的工作包括将这些反分类结果扩展到紧流形上的可微分系统，并识别那些同构问题易于处理的系统。许多自然系统根据确定的规则随着时间的推移而演化。许多科学涉及发现这些规则并描述它们，也许通过方程组。这些规则讨论系统中各个点的行为方式，并且通常是完全确定性的。然而，在这些系统中实际可以观察到的（例如由于舍入误差）是点集。在这个层面上，动力系统的定性行为显然是随机的。这导致了对可能的行为进行统计分类的项目，以便对自然系统的定性行为进行分类。该计划在早期阶段取得了许多成功。然而最近发现，由于数理逻辑方面的原因，该程序原则上无法运行。福尔曼提出的研究探索了将不可能结果扩展到具体环境，并找到对其统计行为有良好分类的大量系统。