NSCPHD1220 Descriptive set theory (DesSet)
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.
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.
- Category
- Hours
- Exercises
- 14
- Lectures
- 28
- Preparation
- 164
- Total
- 206
Please register at: asgert@math.ku.dk
- Credit
- 7,5 ECTS
- Type of assessment
- Continuous assessmentContinuing evaluation based on three problem sets graded on the 7-point scale.
- Aid
- All aids allowed
- 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
- NSCPHD1220
- Credit
- 7,5 ECTS
- Level
- Ph.D.
- Duration
- 1 block
- Placement
- Block 4
- Schedule
- B (Mon 8-12 + Tues 13-17 + Fri 8-12)
- Course capacity
- No limit
- Study board
- Natural Sciences PhD Committee
Contracting department
- Department of Mathematical Sciences
Course responsibles
- Asger Dag Törnquist (6-637569677476426f63766a306d7730666d)
Lecturers
Postdoc David Schrittesser