Classification Using Sets of Sets of Reals as Invariants

Explore the complexity of classification problems in mathematics through a one-hour lecture on the theory of Borel equivalence relations. Delve into the Friedman-Stanley jumps and their role in capturing the intricacy of classification using invariants such as countable sets of reals and their iterations. Examine structural dichotomies for these jumps and their application as a tool for proving the difficulty of classification problems. Begin with an introduction to basic definitions and goals of Borel equivalence relations theory, progress through known structure and non-structure results, and culminate in the motivation behind new dichotomies. Gain insights into determining the feasibility of successful classification and identifying optimal classifying invariants in mathematical problems.