Adrian Mathias, University of Freiburg
Linking descriptive set theory to symbolic dynamics
1. I'll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.
2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.
My Réunion colleague Chrstian Delhommé simplified and extended my ideas.
I'll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.
3. I will outline more recent work using ideas of Blass and Fremlin to to study 'uniform' versions of the results of 1993-96.
I'll end with listing some open problems which I hope will be found interesting.