Strongly unfoldable cardinals are relatively low in the hierarchy of large
cardinals, they lie well below measurable cardinals and are consistent with
V=L. In this talk I will discuss recent results that have shown how strong
unfoldability can serve as a highly efficient substitute for supercompactness
in several large cardinal phenomena.
In particular, I will discuss a Laver-like indestructibility theorem for strong
unfoldability and a Baumgartner-like relative consistency proof of a fragment
PFA: If kappa is a strongly unfoldable cardinal, then there is a model in which
kappa is indestructible by all <kappa-closed, kappa^+ preserving forcing
notions; and there is a model in which PFA holds for forcing notions that
preserve either aleph_2 or aleph_3.
This is joint work with Joel David Hamkins.