On the Cofinality of Ultrafilters

In the first part of the talk we will present background and motivation from topology for the study of the cofinality of ultrafilters. Then we will present Kunen's problem asked in the 70's regarding the existence of ultrafilters with a small character on uncountable sets, followed by our recent contribution to the study of this question (Joint with Moore and Serafin), and an answer to a related question of Raghavan and Shelah (Joint with Eskew). We will present some ideas from the proof and related results. Only undergraduate background in set theory and topology is assumed for the first part. In the second part of the talk, motivated by an ultrapower characterization for the character of an ultrafilter, we will present a related results about the Tukey-type of a general poset, involving tools from PCF theory.