Investigations into Set Theory and Descriptive Dynamics

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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.