Oregon mathematics department colloquium, Eugene, October 2017. (Eugene slides)
Abstract: Borel complexity theory is the study of the relative complexity of classification problems in mathematics. At the heart of this subject is invariant descriptive set theory, which is the study of equivalence relations on standard Borel spaces and their invariant mappings. The key notion is that of Borel reducibility, which identifies when one classification is just as hard as another. Though the Borel reducibility ordering is wild, there are a number of well-studied benchmarks against which to compare a given classification problem. In this talk we will introduce Borel complexity theory, present several concrete examples, and explore techniques and recent developments surrounding each.