Alexander Kechris


Greg Hjorth


Alain Louveau


Descriptive set theory, Spring 2012


Instructor: Christian Rosendal, room 416 SEO
Course number: Math 511.
Course hours: 1:00 PM - 1:50 PM, MWF.
Location: Room 301, 2TH.

Notes:
  • Baire category pdf
  • The Borel hierarchy pdf
  • A closure theorem for the Souslin operation pdf
  • The Kunen-Martin Theorem pdf
  • Determinacy and Lebesgue measurability pdf
  • The dichotomy Theorems pdf
  • Coanalytic ranks and reflection theorems pdf
  • Analytic determinacy and measurable cardinals pdf
  • Countable sections for locally compact group actions pdf
  • Applications to model theory pdf

    This is a course covering the core material of descriptive set theory with a focus on modern techniques such as infinite games, Ramsey theory and dichotomy theorems. Descriptive set theory concerns the fine structure and regularity properties of definable subsets of Polish spaces, e.g., definable subsets of the reals. Descriptive set theory thrives in its interactions with other branches of mathematics such as the study of models of set theory, the geometry of Banach spaces, ergodic theory, and harmonic analysis and has proved to be a useful tool in all of these domains. So the course will be of interest to the general analyst. Among the topics we will cover are:

  • Borel, analytic, and coanalytic sets.
  • Baire category.
  • Separation and uniformisation theorems.
  • Infinite games and determinacy.
  • Infinite-dimensional Ramsey theory.
  • Coanalytic ranks and reflection theorems.


    • No specific knowledge is required for the course, though much of the material will presuppose a certain maturity in analysis that can be gained from courses on real analysis, measure theory, functional analysis, or general topology. Knowledge of first order logic is also useful, but not required.

    Homework sets (unless otherwise stated, the exercises are from Kechris' book):
  • Due Friday February 3rd: 2.2, 2.3, 2.11, 3.12, 4.11, 4.12, 4.20, 4.21, 4.27, 4.29, 4.31.
  • Due Friday February 17th: Do exercises 3, 4, 10, 11, 13, 17, 18 in the set of notes on Baire category.
  • Due Monday March 12th: Do exercises 8.6, 9.6, 9.7, 14.14, 14.16, 28.4, 28.9.
  • Due Friday April 13th: Do exercises 7, 15 and 16 in the notes on Coanalytic Ranks and do (20.9) in Kechris' book.


    • Main reading:
  • Alexander S. Kechris: Classical descriptive set theory, Graduate texts in Mathematics 156, Springer Verlag 1995.

    • Other suggested reading:
  • J. Melleray: Theorie descriptive des groupes pdf.
  • D. Marker: Descriptive Set Theory ps.
  • S. M. Srivastava: A course on Borel sets, Springer 1998.
  • Y. Moschovakis: Descriptive set theory, American Mathematical Society 2009.
  • R. Mansfield & G. Weitkamp: Recursive aspects of descriptive set theory, Oxford 1985.
    • Back to Christian Rosendal's homepage.