Some new partial answers to a 52 year old interpolation question

It is now more than 52 years since Studia Mathematica received Alberto
Calder\'on's very remarkable paper about his theory of complex
interpolation spaces. And one of the questions which Calder\'on
implicitly asked in that paper, by solving it in a significant special
case, is apparently still open today:

DOES COMPLEX INTERPOLATION PRESERVE THE COMPACTNESS OF AN OPERATOR?

After briefly surveying attempts to solve this question over several
decades, I will also report on a few new partial answers obtained
recently, some of them (arXiv:1411.0171) jointly with Richard
Rochberg. Among other things there is an interplay with Jaak Peetre's
"plus-minus" interpolation method, (arXiv:1502.00986) a method which
probably deserves to be better known. Banach lattices and UMD spaces
also have some roles to play.

Several distinguished mathematicians have expressed the belief that
that the general answer to this question will ultimately turn out to be
negative. Among other things, I will try to hint at where a counterexample
might perhaps be hiding. You are all warmly invited to seek it out,
or prove that it does not exist.

A fairly recent survey which discusses this question is available at
arXiv:1410.4527.