An ultrafilter on a cardinal $\kappa$ is uniform if it contains every cobounded subset of $\kappa$. The aim of this talk is to examine the Tukey order on the uniform ultrafitlers on $\omega_1$. We show that it is consistent that every uniform ultrafilter on $\omega_1$ is Tukey equivalent to the finite subsets of $2^{\omega_1}$, ordered by containment---the maximum possible Tukey type of a directed set of cardinality $2^{\omega_1}$. We will also show that PFA implies that Todorcevic's ultrafilter $\mathscr{U}_T$ associated to a coherent Aronszajn $T$ has maximum Tukey complexity.
This is joint work with Luke Serafin and Tom Behamou.