True-Stage Systems

The main objective of this project is to prove results in descriptive set theory using tools that originated in computable structure theory. Descriptive set theory and computability theory are two areas of mathematical logic that study the complexity of mathematical objects. The former deals with objects of size continuum, while the latter deals with countably infinite objects. Despite this difference, results about the hyperarithmetic hierarchy in computability theory can often be generalized to results about the Borel hierarchy in descriptive set theory. The PI will contribute to the education of students and postdocs by advising them on their research, organizing seminars, and teaching classes. The tools in question are the iterated true-stage method and the topological game meta-theorem. Both allow one to build continuous functions using information that is far from continuous, lying high up in the Borel hierarchy. Given the recent success in applying these tools to descriptive set theory, the moment is ripe to develop them further and to pursue additional applications. 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.