## Advanced Topics in Mathematical Logic (V5A7) - Descriptive set theory

### Lecturers

- Prof. Dr. Stefan Geschke
- Dr. Philipp Schlicht

### Time and place

- Lecture: Tuesday 14.15-16.00 and Thursday 14.15-16.00, both in seminar room 1.007.

### Contents

Most sets and structures appearing in mathematics are definable in some sense (for example Borel in an appropriate space). For definable sets, many well-known independence results can be avoided, for example the continuum hypothesis holds for Borel sets and for quotients of the reals (and other Polish spaces) modulo Borel equivalence relations. Moreover, there are deep theorems (so-called dichotomy theorems) about the structure of Borel equivalence relations. We cover basic properties of Polish spaces, the Urysohn space, Borel and analytic sets, measurability, Borel equivalence relations, dichotomy theorems, actions of countable groups and the symmetric group, projective sets and Shoenfield absoluteness.

### References

Most of the introductory material can be found in *Invariant descriptive set theory* by Gao and *Classical descriptive set theory* by Kechris. For the dichotomy theorems, we follow Ben Millers Paris lecture notes which can be found online.

Lecture notes 09-16 April, 02-07 May, 06 June

### Schedule

**09 April**Schlicht*Polish spaces (Lecture notes)*

**11 April**Schlicht*Hyperspaces, Urysohn space (Lecture notes)*

**16 April**Schlicht*Urysohn space, Ultrametric spaces (Lecture notes)*

**18 April**Geschke*Borel and analytic sets (Kechris sections 10-14)*

**23 April**Geschke*Borel and analytic sets (Kechris sections 10-14)*

**25 April**Geschke*Borel and analytic sets (Kechris sections 10-14)*

**30 April**Geschke*Separation of analytic sets (Kechris section 14)*

**02 May**Schlicht*Perfect set property, Borel ismorphisms (Lecture notes, Kechris sections 14-15)*

**07 May**Schlicht*Injective images of Borel sets (Lecture notes, Kechris sections 15, 29)*

09 May holiday

**14 May**Geschke*The G_0 dichotomy (Ben Miller's lecture notes)*

**16 May**Geschke*The G_0 dichotomy (Ben Miller's lecture notes)*

21 May to 25 May holidays

**28 May**Geschke*The G_0 dichotomy (Ben Miller's lecture notes)*

30 May holiday

**04 June**Geschke*The G_0 dichotomy (Ben Miller's lecture notes)*

**06 June**Schlicht*Silver's theorem (lecture notes)*

**11 June**Schlicht*Lusin-Novikov, Feldman-Moore (lecture notes)*

**13 June**Schlicht*The E_0 dichotomy (Ben Miller's lecture notes)*

**18 June**Schlicht*The E_0 dichotomy (Ben Miller's lecture notes)*

**20 June**Schlicht*Countable group actions (Gao)*

**25 June**Geschke*Countable group actions (Gao)*

**27 June**Geschke*Countable group actions (Gao)*

**02 July**Geschke*Universal group actions (Gao)*

**04 July**Geschke*Universal group actions (Gao)*

**09 July**Schlicht*Subgroups of the symmetric group (Gao)*

**11 July**Schlicht*Classification problems (Gao)*

**16 July**Geschke*Shoenfield absoluteness*

**18 July**Geschke*Shoenfield absoluteness*