lecture: Monday 10:00 - 12:00, ESI lecture hall
seminar: to be announced
The course starts on October 5, 2004

 
It is well know that when one studies arbitrary subsets of the real numbers one runs into many pathologies (non-measurable sets) and independent problems (the Continuum Hypothesis). In Descriptive Set Theory one tries to avoid these pathologies by concentrating on natural classes of well-behaved sets of reals, like Borel sets or projective sets (the smallest class of sets containing Borel sets and closed under projections from higher dimenional spaces). While this is a restricted class of sets it includes most of the sets that arise naturally in mathematical practice. Classical descriptive set theory has developed a detailed theory of these sets and most natural questions about them have been settled.

Lately there have been many new developments and intersting connections with dynamical systems, ergodic theory, etc through the study of orbit equivalence relations and induced quotient spaces. These spaces are typically nonstandard, i.e. the induced Borel structure is degenerate, nevertheless one can study their properties by lifting them to the original space and using advanced methods of effective decriptive set theory and other techniques.. While the question of definable cardinality is completely settled and easy for classical spaces for quotient spaces it leads to some very deep and important considerations about classification problems in various areas of mathematics. Descriptive set theory provides a general framework and tools for studying these problems and it allows us to properly formulate and give precise answers to some previously vague classification problems in various areas of mathematics.

The first half of the course will be devoted to developing the fundamental results and techniques of descriptive set theory. In the second half we will look at more recent developments. The exact topics covered will depend on the background and interest of the class.

The topics covered may include:

• Classical descriptive set theory: Borel and projective sets
• Infinite games and determinacy
• Effective descriptive set theory
• Borel equivalence relations
• Polish group actions and connections to ergodic theory and dynamical systems

Prerequisities
Basics of topology of metric spaces, ideally students should be familiar with very basic set theory (cardinals and ordinals) and elementary measure theory, but these topics can be picked up as we go along The course should be accessible to advanced undergraduate and graduate students in logic, ergodic theory and dynamical systems.

Homework
I will be distributing lecture notes containing a number of exercises. Over the course of the semester students should complete and turn in 15 exercises.

Literature:
A. Kechris: "Classical Descriptive Set Theory", Graduate Texts in Mathematics 156, Springer-Verlag, 1995.

H. Becker and A. Kechris: "The Descriptive Set Theory of Polish Group Actions", Cambridge U. Press, 1996.

A. Kechris: "Lectures on Definable Group Actions and Equivalence Relations", unpublished notes.

R. Mansfield and G. Weitkamp: "Recursive Aspects of Descriptive Set Theory", Oxford, 1985.