Some Problems on the Edge of Descriptive Set Theory

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The specifics of this project concern three directions which arose out of a general interest in definable equivalence relations. The first of these directions relates to the treeable equivalence relations. Following work of Adams and Kechris, we know that there is a mass of countable Borel equivalence relations which are mutually incomparable. No such result is known for the treeable Borel equivalence relations. We do not know whether there are infinitely many distinct examples, and we basically have only one established example which is not hyperfinite. More generally we do not know whether the implicit involvement of measure theoretic examples involving free actions of the free group is the sole obstruction to hyperfiniteness. The second direction of Hjorth's project concerns issues in the fine study of Borel complexities of countable isomorphism types in the topology of quantifier free logic, and may be connected with a translation of some basic concepts from first order logic into a quantifier free context. The third direction of the proposal is to investigate some combinatorial questions, such as having a model with a certain partition property for definable partitions, for infinitary sentences, especially those arising as the Scott sentence of some countable structure; this may be related to a still open problem posed by Shelah in the 1970's on the Hanf number up to the continuum for countably infinitary logic. In very general terms, this project can be located inside the branch of mathematics known as "descriptive set theory". This area arose around the end of 19th century as part of an effort to better understand the basic objects -- such as the real number line, real valued functions, subsets of the reals, subsets or regions of two dimensional and three dimensional space, the area or volume of such subsets -- which appear in calculus, and which are needed for applications in engineering, physics, and differential equations. Descriptive set theory does not itself actually address any of these eventual applications, but is rather preoccupied with purely foundational issues. Following Silver's theorem in the 1970's, many descriptive set theorists have become interested in equivalence relations on spaces such as the real number line, or two dimensional space, or similar classes of "topological spaces". The study of such equivalence relations leads to quotient objects which arise by considering the collection of all equivalence classes. For instance if we set two real numbers to be equivalent when the result of subtracting one from the other is an integer (i.e. a "whole number"), then the collection of equivalence classes may be naturally identified with the result of basically wrapping the real number line around itself, to obtain circle of circumference one. In this simple example the quotient object is easily understood, and has a geometrical representation. Most of the work in Hjorth's area deals with the so called "non-smooth" equivalence relations whose quotient objects do not admit such a representation, and the study of these quotient spaces is known to have connections with a variety of mathematical disciplines, such as "dynamics", and "ergodic theory", and some of the more abstract branches of "analysis", such as "infinite dimensional group representations".