Descriptive Set Theory
MATH 7870, Spring 2017
10:10-11:00 MWF in 206 Malott

This course will provide an introduction to classical descriptive set theory. The following is an overview of the material:

  1. Introduction to Polish spaces and Borel sets: a characterization of which subspaces of a Polish space are Polish; examples of Polish spaces including those which are surjectively and injectively universal; change of topology results.
  2. Baire category and measure: The Baire Category theorem; Fubini's theorem and the Kuratowski-Ulam characterization of 1st category subsets of products of Polish spaces; Pettis's theorem.
  3. Analytic sets: Souslin's \(\mathcal{A}\)-operation and the measurability of analytic sets; the perfect set property of analytic sets and the open coloring dichotomy; tree representations of analytic sets and their complements; the boundedness theorem for analytic sets; the separation theorem for analytic sets.
  4. Hurewicz phenomena: characterizing when an analytic set \(A\) can be separating from a set \(B\) by an \(F_\sigma\)-set; Todorcevic's analytic gaps theorem; the \(G_0\)-dichotomy.
  5. Uniformization theorems of Luzin-Novikov and Jankov-von Neumann.
  6. Borel and analytic equivalence relations: countable Borel equivalence relations and the Feldman-Moore theorem; Borel reducability and the dichotomy theorems of Silver and Glimm-Effros; obstructions an equivalence relation being induced by a Polish group action.
  7. Infinite dimensional Ramsey theorems for regular partitions: the Galvin-Prikry theorem and Milliken's theorem.
  8. Determinacy of analytic games.
Some of the later topics may only be covered peripherally.

While there is no official text for the course, the lectures will draw on material from the following sources:

This course should be enrolled in with the S/U option. To receive an S, students are expected to complete the regular homework assignments.

The set of lecture notes up to the present.

Final lecture: 10am-noon on May 19, 2017 (206 Malott).

Week 15:

Week 14:

Week 13: Homework 5 is due on 4/28.

Week 12:

Week 11: spring break.

Week 10: note that the typed notes for lectures 21-23 coincides with the lectures for week 10 and amount to a reorganization of the week's lectures (there is not a day-by-day correspodence between the actual lectures and the notes this week).

Week 9:

Week 8: Homework 4 is due on 3/27.

Week 7:

Week 6:

Week 5: Homework 3 is due on 3/15.

Week 4: No lectures this week (2/13, 2/15, 2/17); classes rescheduled.

Week 3: Homework 2 is due on 3/3.

Week 2: Homework 1 is due on 2/8.

Week 1: