In joint work with Saharon Shelah, we develop a new method for proving consistency results on cardinal invariants, particularly results involving the invariant 𝔞. This method can be used with a wide range of forcing notions, including arbitrary ccc posets. However the method always requires a supercompact cardinal κ in the ground model and produces forcing extensions in which the desired invariants sit above κ. Another feature of our method is that it generalizes to cardinal invariants above ω, and can be used to give uniform consistency proofs that work at any regular cardinal. It can also be used to treat situations where three cardinal invariants must be separated. In particular, our technique solves various long standing open problems about cardinal invariants at uncountable regular cardinals. All the results use Boolean ultrapowers, studied by Keisler and other model theorists in the 1960s. I will aim to give a fairly self contained introduction to this method and to some to its applications to the theory of cardinal invariants.
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