Investigations into Set Theory and Ergodic Theory

Foreman将研究遍历度量保持系统分类问题的描述集论方面。例如，他的部分结果表明，它们不能用可数结构来分类。他推测同构问题可以证明是复杂的，例如，解析而不是Borel。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，但它独立于在它构建中所做的选择。特别感兴趣的是PCF对象以及与正方形和钻石的关系。Foreman将研究两个数学领域。第一个问题与复杂系统的时间演化有关。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，不同的行为类型不能通过具体的观察来区分。第二个项目与数学学习所依据的假设有关。通常的假设不足以找到所有数学问题的答案。出于这个原因，有必要研究普通假设的强化。福尔曼建议通过关注集合论中的“规范结构”来实现这一点。

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.