Descriptive Set Theory

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DMS-998-7437ABSTRACTThis general aim of this project is the development of the theory ofdefinable equivalence relations, and the closely related study of definableactions of Polish groups, and the structure and classification of theirorbit spaces. This theory takes place within the context of descriptive settheory, which provides the basic underlying concepts and methods. On theother hand, in view of the broad scope of this theory, there are naturalinteractions with other areas of logic and mathematics, such as modeltheory, recursion theory, the theory of topological and transformationgroups, topological dynamics, ergodic theory, and operator algebras.One of the fundamental questions that arises in many areas of mathematics isthat of classifying a given collection of objects that is being studied insome particular field. This amounts to providing a "catalog" or "listing" ofthese objects, in principle not unlike that of cataloging species in biologyor 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 veryimportant to understand under what circumstances a classification ispossible. This difficult foundational question is further complicated by thefact that what constitutes an acceptable classification is very muchdependent on the particular field of mathematics studied, so the criteriafor a "good" classification in one area might not be appropriate in another.At its basic level, this project aims to develop a general quantitativetheory, which in many situations can precisely measure the complexity of aclassification problem and thus provide objective means by which one candecide, in any given field, whether a satisfactory classification of theobjects in question is possible. This is achieved by associating with eachcollection of objects to be classified an appropriate concept of "magnitude"or "size", which in a precise sense measures the difficulty of itsclassification problem. This new theory of "magnitude" is investigated inthis project.