It is proved that the relation of isomorphism between separable Banach
spaces is a complete analytic equivalence relation, i.e., that any
analytic equivalence relation Borel reduces to it. Thus, separable Banach
spaces up to isomorphism provide complete invariants for a great number of
mathematical structures up to their corresponding notion of isomorphism.

Forcing axioms are natural combinatorial statements which decide many
of the questions undecided by the usual axioms ZFC of set theory. The
study of these axioms was initiated in the late 1960s by Martin and
Solovay who introduced Martin's Axiom, followed by the formulation of
the Proper Forcing Axiom by Baumgartener and Shelah in the early 1980s
and Martin's Maximum by Foreman, Magidor and Shelah in the mid-1980s.
In the mid 1990s Woodin's work on Pmax extensions established deep
connections between forcing axioms and the theory of large cardinals
and determinacy. Nevertheless, some of the key problems remained open.
In 2003 Moore formulated the Mapping Reflection Principle (MRP) which
seems to be the missing ingredient needed in order to resolve many of
the remaining open problems in the subject and a number of important
developments followed.

In this lecture I will survey some recent results on forcing axioms:
Moore's work on MRP, my work with A. Caicedo on definable well-orderings
of the reals, Viale's result that the Proper Forcing Axiom implies the
Singular Cardinal Hypothesis, etc.