For more information about this meeting, contact Stephen Simpson.

Title: | Proof mining in topological dynamics |

Seminar: | Logic Seminar |

Speaker: | Philipp Gerhardy, University of Oslo |

Abstract: |

The Multiple Birkhoff Recurrence theorem by Furstenberg and Weiss
in 1978 is a seminal result for the interaction
between topological dynamics and combinatorics, establishing Ramsey-type
theorems through corresponding recurrence
results. However, while combinatorial proofs often contain explicit
quantitative information, topological proofs usually
do not contain realizers, bounds or similar data. E.g. for van der Waerden's
theorem -- for every finite colouring of
the integers one colour contains arbitrarily long arithmetic progressions --
one may ask for a number N = N(q; k) such
that for every q-colouring of [0;N] one colour contains a progression of
length k. The combinatorial proof contains an
explicit upper bound on N(q; k), while Furstenberg and Weiss' topological
proof does not. Thus one may ask: what is
the algorithmic content of the topological proofs of Ramsey-type theorems.
We will present an analysis of Furstenberg
and Weiss' Multiple Birkhoff Recurrence theorem which generalizes a previous
analysis by Girard. We will also discuss
the use of compactness in proofs of the Multiple Birkhoff Recurrence
theorem, i.e. the concept of minimality in topological
dynamics, sketch the treatment of generalizations of the Multiple Birkhoff
Recurrence theorem, and, if time permits,
survey some related proof mining results in ergodic theory. |

### Room Reservation Information

Room Number: | MB315 |

Date: | 01 / 13 / 2009 |

Time: | 02:30pm - 03:45pm |