Investigations into Set Theory and Ergodic Theory

Foreman将研究遍历测度保存系统分类问题的描述集理论方面。他的部分结果表明，例如，它们不能通过可数结构进行分类。他推测同构问题是可证明的复杂问题，例如解析问题而不是Borel问题。福尔曼建议的第二个方面是研究集合论中的“规范结构”。这是一种结构，它的存在需要选择公理，但独立于其结构中的选择。特别感兴趣的是PCF对象及其与正方形和菱形的关系。福尔曼将研究数学的两个领域。第一个与复杂系统随时间的演化有关。这些系统似乎表现出随机行为，但性质不同。福尔曼的项目是要证明各种行为类型是无法通过具体的观察来区分的。第二个项目与数学研究所基于的假设有关。通常的假设还不足以找到所有数学问题的答案。出于这个原因，有必要研究一下普通假设的强化情况。福尔曼建议通过关注集合论中的“规范结构”来做到这一点。

Foreman will investigate the descriptive set theoreticaspects of the classification problem for ergodic measure preservingsystems. He has partial results showing, for example, that they cannot beclassified by countable structures. He conjectures that the isomorphismproblem is provably complex, e.g. analytic but not Borel. The secondaspect of Foreman's proposal is to study ``canonical structure" in settheory. This is structure that requires the Axiom of Choice for itsexistence, but is independent of the choices made in it construction. Ofparticular interest are PCF objects and relations with squares anddiamonds.Foreman will investigate two areas of mathematics. The first has to dowith the evolution of complex systems in time. These systems appear toshow random behaviour, but of qualitatively different sorts. Foreman'sproject is to show that various behavior types are not distinguishable byconcrete observations. The second project has to do with the assumptionsthat the study of mathematics is based on. The usual assumptions are notstrong enough to be able to find the answers to all mathematicalquestions. For this reason it is necessary to investigate strengtheningsof the ordinary assumptions. Foreman proposes to do this by focussing on``canonical structure" in set theory.