This general aim of this project is the development of the theory of definable equivalence relations, and the closely related study of definable actions of Polish groups, and the structure and classification of their orbit spaces. This theory takes place within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions with other areas of logic and mathematics, such as model theory, recursion theory, the theory of topological and transformation groups, topological dynamics, ergodic theory, and operator algebras.
One of the fundamental questions that arises in many areas of mathematics is that of classifying a given collection of objects that is being studied in some particular field. 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" is investigated in this project.