"FO Isomorphism Theorems and Descriptive Complexity".

For all important complexity classes, all complete problems via first-order projections are
first-order isomorphic. Thus, up to first-order isomorphism, there is only one complete problem for
each important complexity class.  This means that a restricted version of the Berman-Hartmanis
Conjecture holds even though the original conjecture -- which implies that P is not equal to NP ---
remains open.  I will explain this result in the context of Descriptive Complexity.  I will also
explain, in light of some new results on the number of quantifiers needed to express some
computational properties, what such results say about the relationships between complexity classes.