Math 223S, Topics in Set Theory, on Determinacy.

Instructor: Itay Neeman.
Office: MS 6334.
Phone: 794-5317.
Office hours: Mondays and Wednesdays, 3pm to 3:50pm.

Time and Place: Mondays and Wednesdays, 4-5:15pm, in MS 5128.

Material: For a set A contained in the closed interval [0,1], the game G(A) is played as follows: players I and II take turns writing digits, to produce together (after infinitely many rounds) a real number x in [0,1]. Player I wins if x belongs to A, and otherwise player II wins. The game G(A) is determined if one of the players has a winning strategy. Determinacy for a class of sets asserts the existence of such strategies for all A in the class. Determinacy was initially considered in the '30s and '50s, and shown for example to imply the Baire property and Lebesgue measurability for sets within its class. Today it is known to hold for definable sets of reals, and indeed it is central to their study. The course will cover applications of determinacy to the study of definable sets of reals, including regularity properties, the Wadge hierarchy, propagation of scales, uniformization, definable prewellorders, and large cardinal properties under the axiom of full determinacy.

Prerequisites: Familiarity with basic analysis and topology, and transfinite induction. We will start from the basics, and progress quickly.

Text: We will refer to the books Descriptive set theory by Yiannis Moschovakis and Classical descriptive set theory by Alexander Kechris, and to several papers from the Cabal seminar volumes.

Grading and assignments: Students will be asked to present some results, either from papers or from exercises, in class. Grading will be based on the presentations.