Grant: Discovery Launch Supplement

A common thread through mathematics is the problem of classifying a collection of objects up to some notion of equivalence. That is, assigning to each object an invariant, so that two objects are equivalent if and only if they are assigned the same invariant. These invariants need to be meaningful and easy to compute. One famous example is the classification of compact orientable surfaces, up to homeomorphism, by their genus. The theory of Borel Equivalence Relations 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 field has become a central topic of research in Descriptive Set Theory in recent decades and has deep connections with many other fields of mathematics. At the heart of the field is the notion of Borel reducibility, with which we compare the complexity of two equivalence relations. The field advances simultaneously on two fronts: on one side, the abstract study of Borel equivalence relations, up to Borel reducibility, and the identification of benchmark equivalence relations; on the other side, studying particular classification problems, arising from various fields of mathematics, and comparing them to the benchmark equivalence relations to measure their complexity. A central aspect of this program is to develop new methods to construct Borel reductions and to prove Borel irreducibility results. These will be used to advance the abstract theory of Borel equivalence relations, and applied to study concrete classification problems, as well as orbit equivalence relations. Furthermore, the proposal suggests a new generalized framework of classifying invariants, beyond the standard "classification by countable structures". This will open the door for a meaningful investigation of the possible classifying invariants for classification problems which were previously considered unclassifiable. A long-term goal for developing and expanding these new techniques is to tackle several big open problems, dating to the early days of the field, which resisted resolution via existing methods for many years. A common theme of the methods proposed in this program is the use of intuitions and machinery from Set Theory, such as forcing, absoluteness, symmetric models, and cardinal characteristics of the continuum. At the heart of this program is a recently discovered connection between Borel equivalence relations and choiceless models of set theory. The study of Borel equivalence relations is quite a new and extremely influential topic, while choiceless models have been studied for over a 100 years, starting from Fraenkel's seminal work on permutation models in 1922. The program will further investigate and expand this relationship, deepening our understanding of different mathematical ideas and their connections. Furthermore, these connections will bring closer together two parts of the set theoretic community.