The course. The course "Topics in descriptive set theory" has code MAT6932, section number 5731.This is the first part of a two semester sequence in descriptive set theory. The sequence will familiarize the students with the basic notions of descriptive set theory and the more recent theory of Borel equivalence relations. The material finds many applications throughout mathematics and mainly in abstract analysis, where it is used to rate many problems in terms of their intuitive difficulty. The course meets MWF 8th period, LIT 0219 The instructor is Jindrich Zapletal , 468 Little Hall, phone 352-392-0281x277. The office hours are MWF 9th period. Do not hesitate to contact me by e-mail to set up an appointment.
Textbook and syllabus. The recommended textbook is Alexander S. Kechris: Classical Descriptive Set Theory, Springer-Verlag 1991. ISBN 0-387-94374-9. The instructor will also produce notes. There will be five main topics, I will devote about three weeks to each. Polish spaces. Basic topological definitions, such as compactness, continuity etc. Definition of Polish spaces with basic examples: Cantor space, Baire space, Hilbert cube, Urysohn space. Universal spaces. More involved constructions of Polish spaces, such as K(X), C(X), Hom(X), P(X). The meager ideal. The Borel hierarchy. Definition of Borel and analytic sets. Suslin’s theorem: every analytic coanalytic set is Borel. Computation of Borel complexity for basic examples of sets from mathematical analysis. Hurewitz separation theorem. Uniformization theorems. One-to-one Borel images of Borel sets are Borel. All uncountable Polish spaces are Borel isomorphic. Better Polish topologies. Analytic and coanalytic sets. Examples of complete and universal analytic sets from mathematical analysis. Coanalytic ranks with examples. Reflection theorems. Projective hierarchy. Polish groups and their actions. Basic examples. Metrizability. Amenability and Banach-Tarski paradox. Fixed point theorems and extreme amenability. Infinite games. The concept. Borel determinacy—the statement. Examples: perfect set game, measurability game, Wadge game, Wadge hierarchy. Borel determinacy—the proof. Axiom of Determinacy.
Grading. There will be five takehome exams including the final corresponding to the five main topics outlined above.