The cofinality of an ultrafilter

The Tukey order, which finds its origins in the concept of Moore-Smith  convergence in topology, and is especially important when restricted to ultrafilters with reverse  inclusion. The Tukey order of ultrafilters over ω was studied intensively by Blass, Dobrinen, Isbell, Raghavan, Shelah, Todorcevic and many others, but still contains fundamental unresolved problems. In the first part of this talk, I will present a recent development in the theory of the Tukey order restricted to ultrafilters on measurable cardinals, and explain how different the situation is when compared to ultrafilters on ω. Moreover, we will see an important application to the Galvin property of ultrafilters. In the second

part of the talk we will demonstrate how ideas and intuition from ultrafilters over measurable cardinals lead to new results on the Tukey order restricted to ultrafilters over ω.