Mathematical Sciences: Problems in Descriptive Set Theory, Ergodic Theory and Set Theory

在这个研究项目中，工头将研究两类问题。第一类属于遍历理论，与可逆保测度变换的共轭性有关。特别是，Foreman将使用描述集合论的工具来尝试确定是否每个具有有限熵的可逆遍历变换都共轭于紧致光滑流形上的光滑变换。第二类问题涉及非常大的理想对基数的强度和后果，例如第二个不可数基数。Foreman打算用这些理想来研究划分关系。在二十世纪早期，遍历理论作为一种通过观察相关流的统计行为来研究物理现象的复杂模型的技术而兴起。物理学中出现的动力系统，在其他方面具有难以解决的行为，可适用于这种分析。遍历理论中的一个突出问题是，测度保持系统的一般统计行为在多大程度上模拟了物理现实情况的行为。福尔曼将继续使用描述性集合理论的工具来研究这些问题，这是一种非常规的方法，它将逻辑带到问题上。＊＊＊

9500494 Foreman Foreman will work on two types of problems in this research project. The first kind lies within ergodic theory and has to do with conjugacy of invertible measure-preserving transformations. In particular, Foreman will use the tools of descriptive set theory to attempt to determine whether every invertible ergodic transformation with finite entropy is conjugate to a smooth transformation on a compact smooth manifold. The second kind of problem concerns the strength and consequences of very large ideals on cardinals such as the second uncountable cardinal. Foreman intends to use these ideals to investigate partition relations. In the early twentieth century, ergodic theory arose as a technique for studying complicated models of physical phenomenon by looking at the statistical behavior of the associated flows. Dynamical systems arising in physics with otherwise intractible behavior were amenable to this kind of analysis. A prominent problem in ergodic theory deals with the extent to which the general statistical behavior of measure preserving systems models the behavior of physically realistic situations. Foreman will continue to investigate these problems using the tools of descriptive set theory, an unconventional approach which brings logic to bear on the problem. ***