Rigidity of Quotient Structures

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The project to be undertaken is a research of quotients of structures based on the set of real numbers. Certain categories of such quotients exhibit a strong permanence property: a structure reducible to another one is either very simple, or it inherits characteristic properties of the more complex structure. Via a theory of liftings previously developed by the investigator, this phenomenon is now well understood for some classes of groups and Boolean algebras, in the contexts of both decriptive and combinatorial set theory. (The former corresponding to the situation in which we study only the simply definable sets of real numbers and the latter corresponding to the situation in which we accept the Axiom of Choice and study arbitrary sets of reals in an appropriate ambient theory.) The investigator plans to further this research into the realm of Borel equivalence relations induced by Polish group actions. An example of an important problem that may yield to such a technique is an isolation of a simple basis for turbulent orbit equivalence relations. Quotient of a mathematical structure is formed by identifying its elements that differ "in an inessential way." Different applications require different understanding of what is "inessential," and therefore single structure gives rise to a multitude of quotients with rather diverse properties. Some of them are reducible to the original structure, but the resulting quotient is frequently, yet somewhat paradoxically, much more complex than the underlying space. One of the goals of this project is in finding a simple basis for a class of quotients corresponding to the new notion of "magnitude" introduced in order to study difficulties of classification problems in mathematics and extensively studied during the last decade. Basis is a set, typically small, of objects from a given class that are "critical" in the sense that every object reduces one from the basis. Canonical objects consisting the basis are easier to understand than the arbitrary "generic" objects, yet the intrinsic properties of the latter are reflected in the former. Therefore the basic objects provide a paradigm for studying the whole class.