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 p