          
          
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 KunenMartin 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.
Infinitedimensional 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.
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