v Advanced topics in mathematical logic, summer 2013
Bonn Mathematical Logic Group

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


Lecturers

Time and place

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

 

Last changed: 01 August 2013