# NMAK17004U  CANCELLED: Introduction to Descriptive Set Theory (DesSet)

Volume 2017/2018
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. 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.

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.

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.

General topology and measure theory.
4 hours of lectures/week + 2 hours of exercises per week for 8 weeks.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Continuing evaluation based on three problem sets graded on the 7-point scale. Each problem set caries equal weight towards the final grade.
Aid
All aids allowed
Marking 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.

• Category
• Hours
• Lectures
• 28
• Exercises
• 14
• Preparation
• 164
• Total
• 206