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理论。重点是使用强理想和泛型嵌入来证明反射特性和其他组合结果。这些是广泛的，从固定集反射在奇异基数到无限拉姆齐理论。这些工具包括强迫方法、PCF理论和大基数。二是利用描述集理论为遍历理论研究感兴趣的对象提供新的不变量。其主要目的是能够对各种类型的变换的复杂性进行分类，并希望能够区分以前无法区分的类别（例如，一个可能是Borel，另一个可能是真正的分析）。一个可能产生这种技术的重要问题的例子是证明在紧流形上存在一个遍历的、有限熵的、保持测度的变换，它与光滑保持测度的变换不同构。这项工作是在数学基础：数学是如何结合在一起的，为什么它工作。这些研究通常涉及到用通常的数学假设无法解决的问题：Zermelo-Frankel公理和选择公理。要使用的方法包括通用的大基数，或者数学宇宙的对称性，它们揭示了强大的规律，通常可以解决棘手的问题。从数学基础的研究中产生的考虑导致了基于其固有复杂性的问题分类。反过来，这些复杂性的度量可以应用于动力系统和遍历理论中的自然问题，正如本项目所追求的那样。

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.