Given a supercompact cardinal $\kappa$ and a regular cardinal
$\lambda

The geometric theory of Banach spaces underwent a tremendous
development in the decade 1990-2000 with the solution of several
outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the
class of separable Banach spaces and also laid the beginnings of a
classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was
made on the classification program. We shall discuss some recent
advances due to V. Ferenczi and myself on this by means of Ramsey theory
and dichotomy theorems for the structure of Banach spaces. This
simultaneously allows us to answer some related questions of Gowers
concerning the quasiorder of subspaces of a Banach space under the
relation of isomorphic embeddability.

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.