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

福尔曼的建议包括应用数学逻辑的工具来解决动力系统中的问题。许多动力系统，虽然完全确定，似乎有随机行为的元素。当系统上存在定常概率测度时，可以明确地描述这种现象。这样的描述可以说明系统在理论上是测量同构于一个特定的已知过程，如伯努利过程。从这个观点出发，很自然地试图从理论上对动力系统进行测量分类。我们的希望是拥有一个可能的测度保存系统的“库”，并且能够从理论上将任意的系统测度描述为库中的一个测度。这个项目虽然在早期阶段非常成功，但由于深层次的逻辑原因，它遇到了不可逾越的障碍。Foreman和他的合作者先前的工作表明，遍历测度保持系统的同构问题从逻辑的角度来看本质上太复杂，无法进行分类。Foreman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系统，并识别那些同构问题可处理的系统。许多自然系统根据确定的规则随着时间的推移而进化。许多科学研究都涉及发现这些规律并描述它们，也许是通过一套方程系统。这些规则讨论了系统中各个点的行为，并且通常是完全确定的。然而，在这些系统中可以实际观察到的（例如由于舍入误差）是点的集合。在这个层次上，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，以便可以对自然系统的定性行为进行编目。该计划在早期阶段取得了许多成功。然而最近，事实证明，有一些与数学逻辑有关的原因，程序在原则上不能工作。福尔曼提出的研究将不可能的结果扩展到具体的环境中，并找到大量的系统集合，这些系统的统计行为有很好的分类。

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.