# Survey on the structure of the Tukey theory of ultrafilters

## Natasha Dobrinen

### University of Denver

The Tukey order on ultrafilters is a weakening of the well-studied Rudin-Keisler order, and the exact relationship between them is a question of interest. In second vein, Isbell showed that there is a maximum Tukey type among ultrafilters and asked whether there are others. These two questions are the main guiding forces of the current research. In this talk, we present highlights of recent work of Blass, Dobrinen, Mijares, Milovich, Raghavan, Todorcevic, and Trujillo (in various combinations for various papers). Further information about results mentioned in this talk can be found in a recent survey article by the speaker.

Professor Dobrinen earned her Ph.D. at the University of Minnesota under Karel Prikry in 1996, afterwards holding post-doctoral positions at Penn State and the University of Vienna before moving to the University of Denver. Her research interests mainly fall under the broad category of logic and foundations of Mathematics, and includes research in set theory, Ramsey theory, Boolean algebras, and measure theory. She has investigated relationships between random reals, eventually dominating functions, measure, generalized weak distributive laws, infinitary two-player games, and complete embeddings of the Cohen algebra into complete Boolean algebras. Currently, she is working on problems in Ramsey theory, problems regarding the structure of the Tukey types of ultrafilters, and problems involving both.