Classification and invariants for Borel equivalence relations

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A common thread through mathematics is the problem of classifying a collection of objects up to some notion of equivalence. A successful classification of these objects would be a simple list of properties, which are easy to observe, so that two objects which have the same properties will in fact be equivalent. A central aim of this project is to further develop the theory of "Borel equivalence relations". This is a field of study which provides a rigorous framework to analyze the complexity of various classification problems in mathematics and to determine when a successful classification is possible or not. This project will expand the theory, develop new methods, and apply these to study various classification problems in mathematics. The PI will work with undergraduate and graduate students, through teaching, directed reading, advising, and mentoring. The PI will also be involved in organizing conferences and seminars.This project will develop and expand general techniques to determine when certain classifying invariants are possible for a given classification problem. The analysis of classifying invariants will be facilitated by various techniques coming from axiomatic set theory, including symmetric models of set theory and cardinal characteristics of the continuum. Specifically, the PI will further develop the relationship between symmetric models of set theory, in which the axiom of choice fails, and Borel equivalence relations which are "classifiable by countable structures". This relationship will be applied to settle several problems about such equivalence relations and to further analyze their structure up to Borel reductions and Borel homomorphisms. Furthermore, generalized frameworks of classifying invariants, beyond "classification by countable structures", will be studied. These allow for a meaningful investigation of the possible classifying invariants for classification problems which were previously considered unclassifiable.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.