Investigations into Set Theory and Descriptive Dynamics

本项目由两部分组成。 第一个是继续研究组合集合论。这集中在使用强理想和一般嵌入证明反射性质和其他组合的后果。 这些都是广泛的，从固定集反射奇异基数无限拉姆齐理论。这些工具包括强迫方法，PCF理论和大型基数。 第二个是使用描述集理论提供新的不变量研究对象的兴趣遍历理论。 其主要目的是能够对各种变换类的复杂性进行分类，希望能够区分以前无法区分的类（例如，一个可能是Borel，另一个可能是真正的解析类）。一个例子的一个重要问题，可能会产生这样的技术是表明，有一个遍历，有限熵，措施保持变换，是不同构的一个光滑的措施保持变换在一个紧凑的流形。数学的基础：数学如何结合在一起，为什么它工作。 这些研究通常涉及考虑那些不能用通常的数学假设解决的问题：Zermelo-Frankel公理与选择公理。 使用的方法包括通用的大基数，或数学宇宙的对称性，揭示了强大的随机性，往往解决棘手的问题。 从数学基础的研究中产生的考虑导致了基于其内在复杂性的问题分类。 这些措施的复杂性，反过来，可以应用到自然问题的动力系统和遍历理论，将在这个项目中追求。

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.