Descriptive Set Theory

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The general aim of this project is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, it has natural interactions with many other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, operator algebras, and combinatorics. Within this general program it is proposed to study: (i) problems arising in the theory of countable Borel equivalence relations, particularly concerning hyperfiniteness and treeability, including descriptive aspects of free actions of free groups; (ii) newly developed connections between the topological dynamics of automorphism groups of countable structures and finite Ramsey theory as well as a related semigroup framework for such connections with infinite Ramsey theory; (iii) the concepts of genericity and ample genericity in Polish groups and their relation to other structural properties of groups such as the small index property, uncountable cofinality, the Bergman finite generation property, fixed point properties for actions on trees and automatic continuity; (iv) complexity of classification problems concerning the isometric or topological classification of various kinds of metric, topological or Banach spaces. A fundamental question that arises in many fields of mathematics is that of classifying a given collection of objects under study. This amounts to providing a "catalog" or "listing" of these objects, in principle not unlike that of cataloging species in biology or stars and galaxies in astronomy. If such a classification is possible, one has a "complete" understanding of the mathematical structures involved. Otherwise a more or less "chaotic" behavior is expected. It is thus very important to understand under what circumstances a classification is possible. This difficult foundational question is further complicated by the fact that what constitutes an acceptable classification is very much dependent on the particular field of mathematics studied, so the criteria for a "good" classification in one area might not be appropriate in another. At its basic level, this project aims to develop a general quantitative theory, which in many situations can precisely measure the complexity of a classification problem and thus provide objective means by which one can decide, in any given field, whether a satisfactory classification of the objects in question is possible. This is achieved by associating with each collection of objects to be classified an appropriate concept of "magnitude" or "size", which in a precise sense measures the difficulty of its classification problem. This new theory of "magnitude" as well as problems in different directions that arise in the course of the development of this theory are investigated in this project.