NMAK13032U Descriptive Set Theory, Group Actions and Equivalence Relations (DisMæL)
Volume 2013/2014
Education
MSc Programme in
Mathematics
Content
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. Borel reducibility, and the dichotomy theorems of Silver and Harrington-Kechris-Louveau, and possibly other topics.
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. Borel reducibility, and the dichotomy theorems of Silver and Harrington-Kechris-Louveau, and possibly other topics.
Learning Outcome
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
Harrington-Kechris-Louveau.
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.
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.
Academic qualifications
General topology and
measure theory.
Teaching and learning methods
4 hours of lectures/week + 2
hours of exercises.
Workload
- Category
- Hours
- Lectures
- 28
- Preparation
- 164
- Theory exercises
- 14
- Total
- 206
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Exam
- Credit
- 7,5 ECTS
- Type of assessment
- Continuous assessmentContinuing evaluation based on three problem sets graded on the 7-point scale.
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
one internal examiner
- Re-exam
- 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.
Course information
- Language
- English
- Course code
- NMAK13032U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 2
- Schedule
- C
- Course capacity
- No limit
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Course responsibles
- Konstantin Slutsky (7-766f7877766e7c437064776b316e7831676e)
Saved on the
30-04-2013