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

小行星9500494 福尔曼将在这个研究项目中研究两类问题。 第一种是遍历理论， 可逆保测度变换的共轭性。 特别是， 福尔曼将使用描述性集合论的工具，试图 确定是否每个可逆遍历变换有限 熵与紧光滑上的光滑变换共轭 歧管 第二类问题涉及到力量和后果 第二不可数基数上的非常大的理想。 福尔曼打算用这些理想来研究划分关系。 在世纪早期，遍历理论作为一种研究物理现象的复杂模型的技术出现， 相关流的统计行为。 动力系统 在物理学中产生的，否则难以处理的行为， 这种分析。 遍历理论中的一个突出问题是 在某种程度上，测量的一般统计行为 保存系统模拟物理现实情况的行为。 福尔曼将继续调查这些问题， 描述集合论的工具，一种非传统的方法， 这个问题的逻辑性。 ***

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. ***