**Instructor:** Itay Neeman.

Office: MS 6334.

Email:

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.