Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric spaces), like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of topology, measure theory and functional analysis.

Lecture II. The complexity of classification problems in ergodic theory.

The last few decades have seen the emergence of a theory of set theoretic complexity of classification problems in mathematics. In this lecture I will survey recent developments concerning the application of this theory to classification problems in ergodic theory.