Model Theory Seminar-Nick Ramsey-Measures on PAC fields with free Galois group
Dec 2, 2022
2:30PM to 3:30PM
Date(s) - 02/12/2022
2:30 pm - 3:30 pm
Speaker: Nick Ramsey (Notre Dame University)
Title: Measures on PAC fields with free Galois group
Abstract: A beautiful result of Chatzidakis, van den Dries, and Macintyre describes how, in a pseudo-finite field, each definable set may be assigned, in a definable way, both dimension and measure. This measure is well-behaved under definable maps and gives a lot of meaningful information about the structure of definable sets in these fields. Pseudo-finite fields are the simplest examples of pseudo-algebraically closed (PAC) fields so it became natural to ask if similar measures could be constructed for PAC fields more generally. We show that, at least when the Galois group is free, the answer is yes. However, because these fields are no longer necessarily pseudo-finite, we have to use Galois theory instead of counting. As an application, we deduce that all groups definable in such fields are definably amenable. We will explain why both the relevant background on PAC fields and measures, and describe the construction in detail. This is joint with Zoé Chatzidakis.