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 I. Set theory and trigonometric series
After a brief introduction to the main concepts of descriptive set theory, I will discuss the classical problem of uniqueness for trigonometric series. This goes back to the work of Riemann and Heine in the mid-19th century and it was through his work on this problem that Cantor was led to the creation of set theory. After surveying some of the fascinating developments during the 20th century in harmonic analysis and number theory related to this subject, I will explain how methods of descriptive theory have been used to resolve some long standing problems in this area.