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.