MATH40350 Introduction to Descriptive Set Theory

Academic Year 2021/2022

Descriptive set theory is the study of certain classes of mathematical objects that can be defined or `described' explicitly. Lebesgue non-measurable subsets of real numbers and unbounded operators defined everywhere on Banach spaces are examples of objects that exist by virtue of the Axiom of Choice, but have no such explicit description.

Some key topics of classical descriptive set theory are covered, including

(1) metric spaces - review of fundamental concepts, Polish spaces and ways to construct them;
(2) trees - the tree A^{'less then N'} and the metric space A^N, the Baire and Cantor spaces, Lusin schemes, subtrees of A^{'less then N'} and retractions;
(3) Borel sets - sigma-algebras and measurable functions, ordinal numbers and the Borel hierarchy, universal sets;
(4) analytic and coanalytic sets - the separation theorem, complete analytic and coanalytic sets, e.g. IF, WF, DIFF, coanalytic ranks; (5) applications (time allowing).

The theory can be applied to many fields such as probability, functional analysis, potential theory, group representation theory, harmonic analysis and ergodic theory. It is seen, in an expanding area of research, as a natural framework for comparing the `complexity' or `difficulty' of various classification problems, in apparently very disparate fields of mathematics.