Investigations into Set Theory and Descriptive Dynamics

该项目由两部分组成。第一个问题涉及对组合集合论的持续研究。这集中于使用强理想和泛型嵌入来证明反射性质和其他组合结果。它们的范围很广，从奇异基数的固定集反射到无限拉姆齐理论。这些工具包括强迫方法、PCF理论和大基数。二是利用描述集合论为遍历理论研究感兴趣的对象提供新的不变量。主要目的是能够对各种类型的转换的复杂性进行分类，并希望能够区分以前无法区分的类型(例如，一种可能是Borel，另一种可能是真正的分析)。一个可能导致这种技巧的重要问题的例子是证明了在紧致流形上存在一个遍历的有限熵保测度变换，它不同构于光滑保测度变换。这项工作是在数学的基础上进行的：数学是如何结合在一起的，以及它为什么会起作用。这些研究经常涉及到用通常的数学假设无法解决的问题的考虑：泽梅罗-弗兰克尔公理和选择公理。要使用的方法不包括一般的大基数，也就是数学宇宙的对称性，它揭示了经常解决棘手问题的强大规则。对数学基础的研究产生的考虑导致了根据问题的内在复杂性对问题进行分类。反过来，这些复杂性的测量可以应用于动力系统和遍历理论中的自然问题，正如本项目将追求的那样。

This project consists of two parts. The first involves continuing investigations into combinatorial set theory. This focuses on the use of strong ideals and generic embeddings to prove reflection properties and other combinatorial consequences. These are wide ranging, from stationary set reflection at singular cardinals to infinitary Ramsey theory. The tools include methods of forcing, the PCF theory and large cardinals. The second is the use of descriptive set theory to provide new invariants for studying objects of interest to ergodic theory. The main thrust is to be able to classify the complexity of various classes of transformations with the hope of being able to distinguish between previously indistinguishable classes (e.g one may be Borel and the other true analytic.) An example of an important problem that may yield to such a technique is to show that there is an ergodic, finite entropy, measure preserving transformation that is not isomorphic to a smooth measure preserving transformation on a compact manifold. This work is in the Foundations of Mathematics: how mathematics fits together and why it works. These studies often involve consideration of problems that are not solvable by the usual assumptions of mathematics: The Zermelo-Frankel Axioms with the Axiom of Choice. Methods to be used imclude generic large cardinals, or symmetries of the mathematical universe that reveal powerful regularities that often solve intractable problems. Considerations arising from studies of the foundations of mathematics have led to classifications of problems based on their inherent complexity. These measures of complexity, in turn, can be applied to natural problems in dynamical systems and ergodic theory, as will be pursued in this project.