Descriptive set theory and computability

A common problem in mathematics is to completely classify some type of mathematical object by invariants. The field of descriptive set theory provides a general framework for studying these types of classification problems, comparing their relative difficulties, and determining when a complete classification is provably impossible. The proposed research uses these tools from descriptive set theory to understand classification problems in computability, operator algebras, topological dynamics, and ergodic theory. A key part of the project will consist of educating graduate students and other young researchers. The project has several parts. First, studying Weiss's question on amenability and hyperfiniteness using tools from Gromov's theory of asymptotic dimension. This investigation has applications to topological dynamics and operator algebras. Second, research on classical geometrical paradoxes such as the Banach-Tarski paradox and Tarski's circle squaring problem using recent advances in measurable combinatorics. Finally, the project will use Montalban's framework of true stages from computability theory to prove dichotomy theorems for definable sigma-ideals. 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.