# Math 512: Descriptive Set Theory

### Fall 2002

MWF 11:00 216 Taft Hall

### Instructor David Marker

- Office: 411 SEO
- Office Phone: (312) 996-3069
- Office Hours: M 9-11, W 12-1 and by appointment
- Fax: (312) 996-1491
- e-mail: marker@math.uic.edu
- web page: http://www.math.uic.edu/~marker
- course web page: http://www.math.uic.edu/~marker/math512

## Description

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
we try 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. Lately there have been
many intersting connections with dynamical systems, through the study
of orbit equivalence relations.

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:
- Infinite games and determinacy
- Borel equivalence relations
- Polish group actions and connections to model theory

### 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.
### Homework

I will be distributing lecture notes containing a number of exercises.
Over the course of the semester you should complete and turn in 15
exercises.
### Lecture Notes

### References

- 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Relations*, unpublished notes.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.
- Y. Moschovakis,
* Descriptive Set Theory*, North-Holland, 1980.

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