In this talk, we will study the following cardinals:

The ultrafilter number $\mathfrak{u}$, which is the smallest size of a base of an ultrafilter.

The almost disjointness number $\mathfrak{a}$, which is the smallest size of a MAD family.

Both are uncountable cardinals bounded by the cardinality of the real numbers. It is well known that it is consistent that $\mathfrak{a} < \mathfrak{u}$ (this inequality holds in the Cohen model). The consistency of $\mathfrak{u} < \mathfrak{a}$ is much harder and was proved by Shelah under the assumption that there is a measurable cardinal. In the model of Shelah, the inequalities $\omega_1 < \mathfrak{u}$ and $\mathfrak{u} < \mathfrak{a}$ hold. In spite of the beauty of this result, the following questions remain open:

(Shelah.) Does the consistency of ZFC imply the consistency of $\text{ZFC} + \mathfrak{u} < \mathfrak{a}$?

(Brendle.) Is it consistent that $\omega_1 = \mathfrak{u} < \mathfrak{a}$?

In this talk, we will answer both questions in an affirmative way. This is joint work with Damjan Kalajdzievski.