Introduction to Descriptive Set Theory

Mathematicians in the early 20th century discovered that the Axiom of Choice implied the existence of pathological subsets of the real line lacking desirable regularity properties (for example nonmeasurable sets). This gave rise to descriptive set theory, a systematic study of classes of sets where these pathologies can be avoided, including, in particular, the definable sets. In the first half of the course, we will use techniques from analysis and set theory, as well as infinite games, to study definable sets of reals and their regularity properties, such as the perfect set property (a strong form of the continuum hypothesis), the Baire property, and measurability. Descriptive set theory has found applications in harmonic analysis, dynamical systems, functional analysis, and various other areas of mathematics. Many of the recent applications are via the theory of definable equivalence relations (viewed as sets of pairs), which provides a framework for studying very general types of classification problems in mathematics. The second half of this course will give an introduction to this theory, culminating in a famous dichotomy theorem, which exhibits a minimum element among all problems that do not admit concrete classification.