Descriptive Set Theory, Rigidity, and Classification

There are countless mathematical models constructed by mathematicians over the years. Understanding them and their chaotic behavior is an everlasting challenge for future generations. Generally, mathematical objects are better understood when organized according to some notion of equivalence. This common practice is called “classification”, and understanding the extent to which certain classifications are efficient is a common task for working mathematicians. Classification is closely related to rigidity phenomena. “Rigidity” is the study of distinguishing mathematical objects, even when they appear to be almost identical. The main goal of this project is to explore the interplay of classification and rigidity. As an essential part of this project, the PI will train and mentor graduate students at Rutgers University. More specifically, this project concentrates in classification and rigidity theorems in group theory and dynamics using techniques from descriptive set theory. The PI will study rigidity aspects of countable group actions with techniques from measured group theory and cocycle superrigidity. The PI will continue the investigation on the relationship between left-orderable groups, and the corresponding spaces of left-orders under the viewpoint of Borel classification theory. Implications in geometric topology, in particular in the theory of 3-manifolds, are further explored. Moreover, the PI will investigate the descriptive complexity of countable Borel equivalence relations that model classification problems in group theory. 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.