February 23
Tom Benhamou,
Rutgers University
Commutativity of cofinal types of ultrafilters
The Tukey order 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 $\omega$ was studied intensively by Blass, Dobrinen, Isbell, Raghavan, Shelah, Todorcevic and many others, but still contains many fundamental unresolved problems. After reviewing the topological background for the Tukey order, 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 $\omega$. 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 $\omega$. This is joint with Natasha Dobrinen.