Math 223D, Topics in Descriptive Set Theory

Time and Place: MWF 1-2pm

The aim of this course is to develop the theory of Polish group actions, from the point of view of seeing it as a generalization of countable model theory. Initially we will be begin with introductory material on Polish spaces, Baire category methods, Borel sets, and topological groups. Afterwards we will discuss basic techniques in the study of Polish group actions, as found in Becker-Kechris, such as Birkhoff-Kakutani on the existence of left invariant metrics, Vaught transforms, the Becker-Kechris theorem on changes in topologies, the connections with infinitary logic, and universal spaces.

For students who want to get a letter grade, there will be three or four homework assignments; for people who don't wish to do the homeworks, I suggest taking the course on a Pass/No-Pass basis.

Here is a flyer containing some info. Further comments and additions will be posted on this web page.

 


Homework one is due on Monday, October 19, at the start of class.

Slightly at a tangent to the main direction of the course, here is a handout on the left invariant metrics and the completion of a Polish group with respect to its left uniformity. There are references to a number of papers and theorems which we will not have time to discuss in class.

Homework two is due on Wednesday, November 11, at the start of class.

Homework three is due on Monday, November 30, at the start of class. In all honesty, this homework is significantly more challenging than the last two; I did want to take the time to show people a rather non-trivial dichotomy theorem which can be proved for abelian Polish group actions.

A general handout, relating to turbulence: Examples of equivalence relations which are and are not classifiable by countable structures.