Determinacy and Definable Equivalence Relations

About This Grant

The project involvew research in two main directions in descriptive set theory, which is a branch of mathematics in which modern set-theoretic methods are used to develop the theory of subsets of the real line and related structures which are in turn the fundamental objects of mathematical analysis used throughout mathematics and its applications. One of the directions concerns developing the structural theory of sets in models of the axiom of determinacy. This is important as this axiom holds in various mathematical universes which interact with the ``true'' universe and also because it is giving us the theory of definable objects which is a central goal of descriptive set theory. A second direction concerns the theory of definable equivalence relations. This is a relatively recent area of descriptive set theory which has been investigated extensively the past several decades. This area interacts heavily with a number of different areas of mathematics and provides a unifying framework for them. There are a number of fundamental problems that remain open in this area concerning the structure of countable Borel equivalence relations in particular, but significant progress has been made in the last few years. The project will further develop these methods with a goal of attacking some of these problems. This project will involve graduate students. Specifically, the combinatorics of non-wellordered sets in models of determinacy is a main line of the first direction of the proposal. An example is Chan's recent proof of the ``ABCD'' conjecture describing the relation between cardinalities of the form A^B for ordinals A and B (the conjecture roughly speaking asserts that the only relations between these cardinalities are the obvious ones). The principal investigator along with Chan and Trang have isolated a general principle about infinity Borel sets which also proves the ABCD conjecture and seems likely to have other applications. There are, however, many fundamental questions concerning these non-wellordered cardinalities that remain open which the proposal plans to investigate. The theory of countable Borel equivalence relations has shown much progress in recent years. For example, one of the central questions is the hyperfiniteness question which asks which groups have the property that all of their Borel actions generate only hyperfinite (an increasing union of equivalence relations with finite classes) equivalence relations. Recent work has shown that this class extends to include the polycyclic groups, so, in particular contains finitely generated groups of exponential growth. How much further this extends is an open problem. Some new techniques have been introduced recently by the principal investigator and co-authors which will help answer several structuring questions for actions of Z^n and other fairly simple groups. This hopefully leads to marking the boundary of what types of Borel or continuous structurings can be done in an invariant manner. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.