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 III. Dynamics of non-archimedean groups, logic, and Ramsey theory.
A topological group is non-archimedean if it admits a basis at the identity consisting of open subgroups. Examples of such groups are the profinite groups, the additive group of the p-adic numbers, the infinite symmetric group and more generally the automorphism groups of countable structures. Recently there has been considerable activity in the study of the dynamics of these groups and in particular in the problem of classifying their minimal actions and this work has led to interesting connections between logic and set theory, finite combinatorics, topological dynamics and ergodic theory. In this lecture I will give an overview of some of the main directions in this area of research.