NMAK17004U CANCELLED: Introduction to Descriptive Set Theory (DesSet)
MSc Programme in Mathematics
Polish spaces and standard Borel spaces. The Borel hierarchy. Analytic sets and tree representations. Lusin's separation theorem. Baire and Lebesgue measurability, Kuratowki-Ulam theorem and other regularity properties. Selection theorems for Borel relations. Borel and analytic equivalence relations. Polish groups and their actions; orbit equivalence relations. As time allows, topics such as infinitary Ramsey theory, the dichotomy theorems of Silver and Harrington-Kechris-Louveau, and connections to ergodic theory will be discussed at the end of the course.
Knowledge: The student should know the definitions of
Polish spaces, standard Borel spaces, and examples of these, as
well as the definition of the Borel hierarchy, of analytic sets,
and for their tree analysis; Lusin's separation theorem and its
consequences, and the regularity properties of analytic sets; the
selection problem for Borel relations, as well as the Jankov-von
Neumann selection theorem, and the selection principle for Borel
relations with countable sections; the concenpt of genericity
together with the Kuratowski-Ulam theorem; the concept of Borel
reducibility, and the basic dichotomies of Silver and
Skills: The student should be able to apply descriptive set theoretic concepts and result mentioned in the previous paragraph to prove borelness/analyticity of a relation/function, check whether a given set is generic/meager, apply basic dichotomies to equivalence relations and solve other problems related to the material of the course.
Competences: The student should be able to formulate the main results of the course, check whether they are applicable in a concrete problem and use them to solve it.
Example of course litterature:
A. Kechris: Classical Descriptive Set Theory.
Note that this book is available as a pdf for free from the Springer website.
- 7,5 ECTS
- Type of assessment
- Continuous assessmentContinuing evaluation based on three problem sets graded on the 7-point scale. Each problem set caries equal weight towards the final grade.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
One internal examiner
30 min oral examination, no preparation time.
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.