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

福尔曼的建议涉及应用工具，从数学逻辑的问题，在动力系统。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的概率测度时，这种现象可以被明确地描述。 这样的描述可以说明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。从这一点来看，试图从理论上对动力系统进行分类是很自然的。希望将有一个“图书馆”的可能措施保持系统，并能够描述一个任意的系统措施理论上作为一个在library.This项目，而非常成功的早期阶段，运行到不可逾越的障碍深层次的逻辑原因。福尔曼以前的工作与他的合著者表明，同构问题的遍历措施保持系统本质上是太复杂，从逻辑的角度来看，承认一个分类。福尔曼提出的工作涉及到扩展这些反分类结果可微系统的紧凑型流形，并确定这些系统的同构问题是tractable.Many自然系统随着时间的推移演变根据明确的规则。许多科学都涉及到发现这些规则并描述它们，也许是通过一个方程组。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，在这些系统中实际上可以观察到的（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可以是明显随机的。这导致了对可能的行为进行统计分类的项目， 自然系统的定性行为可以被分类。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，该程序原则上无法工作。福尔曼的研究计划探索将不可能性结果扩展到具体的环境，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.