We discuss how BMM affects the large cardinal
structure of V as well as the size of \theta^{L(R)}. BMM proves
that V is closed under sharps (and more), and BMM plus the
existence of a precipitous ideal on \omega_1 proves that
\delta^1_2 = \aleph_2. Part of this is joint work with my
student Ben Claverie.

We present some exact equiconsistency results on the preservation
of the property of L(R) being a Solovay model under various classes of
projective forcing extensions. As an application we build models in which MA
holds for $\Sigma^1_n$ partial orderings, but it fails for the
$\Sigma^1_{n+1}$.