Nikolai Lusin.

Descriptive set theory, Spring 2009

Instructor: Christian Rosendal, room 416 SEO
Course number: Math 512, Call number 28297 (notice that there are two 512 courses scheduled for spring).
Course hours: 12:00 - 12:50 PM, MWF.
Location: Room 636, SEO.

  • The dichotomy Theorems pdf

    This is a course covering the core material of descriptive set theory. Descriptive set theory concerns the structure and regularity properties of definable subsets of Polish spaces, e.g., definable subsets of the reals. It is well-known that using the axiom of choice bad sets of reals can be constructed, e.g., non-measurable sets, but if one only consider sets that are defined explicitly one can show that such sets do not occur. Descriptive set theory is the study of these explicitly defined sets. Descriptive set theory thrives in its interactions with other branches of mathematics such as the study of inner 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.
  • Baire class 1 functions.
  • Separation and uniformisation theorems.
  • Infinite games and determinacy.
  • Infinite-dimensional Ramsey theory.
  • Coanalytic ranks and scales.

  • 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.

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

  • Other suggested reading:
  • S. M. Srivastava: A course on Borel sets, Springer 1998.
  • Y. Moschovakis: Descriptive set theory, North Holland 1980.
  • Galen & Weitkamp: Recursive aspects of descriptive set theory, Oxford 1985.

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