About Descriptive Set Theory
The origination of descriptive set theory can be traced back to the work of Borel, Baire and Lebesgue at the turn of the twentieth century, as they were beginning to understand the abstract notion of a function introduced by Dirichlet and Riemann. Descriptive set theory is the study of the sets of reals that can be explicitly defined or constructed, and so can be expected to have certain properties, such as Lebesgue measurability, not enjoyed by arbitrary sets. It is a central part of contemporary set theory and therefore dealing with major logical concepts, such as definability and undecidability. Moreover its results and methods are used in diverse fields of mathematics among which popology, real analysis, ergodic theory and functional analysis.
Abstracts and Slides
Alessandro Andretta
Descriptive Set Theory and Analysis
Descriptive set theory was created at the beginning of the twentieth century to secure on firm grounds the (then) novel theory of integration and real analysis, as developed by mathematicians like Baire, Borel, Lebesgue, Lusin, and Suslin. In the sixties the subject was revolutionized by methods of mathematical logic (most notably: recursion theory and infinite games) while in the nineties the focus shifted back on the interactions with mainstream mathematics. A key feature of descriptive set theory is that it provides the correct language to formulate (and often solve) classification questions in mathematics or theoretical computer science. Nowadays the subject has found important applications to other areas of mathematics, such as: real and harmonic analysis, ergodic theory, Banach spaces, operator algebras, to name a few. In my talk I will try to give an overview of the ideas and techniques (but not the technicalities) of the subject. If time permits, I will try to illustrate some recent applications to measure theory.
Jacques Duparc
Descriptive Set Theory and Games
Games are a very powerful tool in descriptive set theory. Starting from scratch, while playing easy games of finite length we will introduce the basic notions of strategy, winning strategy and determinacy. From there, we will move on to games of infinite length and show in particular that regularity properties such as the Lebesgue measurability, the Baire property, the perfect set property, all transcribe nicely in the game theoretical setting. The study of these infinite games leads to the formulation of an axiom that can be regarded as an alternative to the axiom of choice.