MATH 7870, Spring 2017

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

- 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.
- 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.
- 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.
- 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.
- Uniformization theorems of Luzin-Novikov and Jankov-von Neumann.
- 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.
- Infinite dimensional Ramsey theorems for regular partitions: the Galvin-Prikry theorem and Milliken's theorem.
- Determinacy of analytic games.

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

- Classical Descriptive Set Theory, A. Kechris.
- Lecture notes on descriptive set theory by S. Todorcevic.

The set of lecture notes up to the present.

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

**Week 15:**

- Monday (5/1): A theorem of Ghys and Carrier: a method for proving an equivalence relation is nonamenable. Lecture 33.
- Wednesday (5/3): A theorem of Ghys and Carrier, continued. Lecture 34.
- Friday (5/5): Gale-Stewart games and determinacy. Lecture 35.

**Week 14:**

- Monday (4/24): Countable Borel equivalence relations are generically hyperfinite; measure hyperfiniteness. Lecture 30.
- Wednesday (4/26): The Connes-Feldman-Wiess theorem. Lecture 31.
- Friday (4/28): The Connes-Feldman-Wiess theorem, continued. Lecture 32.

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

- Monday (4/17): A proof of Ellentuck's Theorem. Lecture 27.
- Wednesday (4/19): Smooth and hyperfinite equivalence relations; the Slaman-Steel Theorem. Lecture 28.
- Friday (4/21): the Slaman-Steel Theorem continued; hypersmooth equivalence relations and tail equivalence. Lecture 29.

**Week 12:**

- Monday (4/10): The \(G_0\)-dichotomy. Lecture 24.
- Wednesday (4/12): (two lectures) Borel equivalence relations, Borel reducibility, and dichotomy theorems for Borel equivalence relations. Lecture 25.
- Friday (4/14): Infinite dimensional Ramsey theory; the Galvin-Prikry Theorem. Lecture 26.

**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).

- Monday (3/27): Jankov-von Neumann Uniformization Theorem; set of unicity of a Borel set. Lecture 21.
- Wednesday (3/29): (first lecture) Luzin-Novikov Uniformization Theorem; Borel-on-Borel ideals; large sections uniformization. Lecture 22.
- Wednesday (3/29): (second lecture) countable group actions, countable Borel equivalence relations, and the Feldman-Moore theorem.
- Friday (3/31): Borel graphs and the Borel chromatic number. Lecture 23.

**Week 9:**

- Monday (3/20):
*class rescheduled* - Wednesday (3/22):
*class rescheduled* - Friday (3/24): finish analytic gaps theorem; uniformization. Lecture 20.

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

- Monday (3/13): Baire's characterization of Baire class 1 functions; analytic gaps. Lecture 18.
- Wednesday (3/15):
*Cornell closed due to snow.* - Friday (3/17): Todorcevic's analytic gaps theorems. Lecture 19.

**Week 7:**

- Monday (3/6): Hurewicz's theorem and the Kechris-Louveau-Woodin Theorem; Solecki's Closed Covering Property; characterizing being \(G_\delta\) within the coanalytic sets. Lecture 14.
- Wednesday (3/8): Solecki's Closed Covering Property and its consequences (double lecture). Lecture 15, Lecture 16.
- Friday (3/10): Baire Class 1 functions and Baire's characterization of them; separating disjoint sets by \(\Delta^0_2\)-sets. Lecture 17.

**Week 6:**

- Monday (2/27): measurability properties of analytic sets; Souslin's \(\mathcal{A}\)-operation. Lecture 11.
- Wednesday (3/1): Haar measure; 0-1 laws for measure and category. Lecture 12.
- Friday (3/3): Haar measure; Polish groups and Pettis's Lemma. Lecture 13.

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

- Monday (2/20): consequences of the separation theorem; a characterization of Borel functions; images of Borel sets under Borel injections are Borel. Lecture 8.
- Wednesday (2/22): trees and analytic sets; analytic and coanalytic sets as unions of \(\aleph_1\)-many Borel sets. Lecture 9.
- Friday (2/24): the Baire category theorem and the Kuratowski-Ulam theorem. Lecture 10.

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

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

- Monday (2/6): The Borel heirarchy; Borel functions; the separation theorem. Lecture 5.
- Wednesday (2/8): A universal analytic set; the boundedness principle for analytic sets. Lecture 6.
- Friday (2/10): Scattered and perfect sets; the Cantor-Bendixon derivative; the Open Coloring Dichotomy for analytic graphs; the perfect set property for analytic sets. Lecture 7.

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

- Monday (1/30): The injective universality of \(\mathbb{R}^\mathbb{N}\); compactifying Polish spaces. Lecture 2.
- Wednesday (2/1): Characterizing \(2^\mathbb{N}\) and \(\mathbb{N}^\mathbb{N}\) by their topological properties; surjective universality results. Lecture 3.
- Friday (2/3): Borel sets and change of topology. Lecture 4.

**Week 1: **

- Wednesday (1/25): Course overview (see above)
- Friday (1/27): Introduction to Polish spaces; Alexandroff's Theorem. Lecture 1.