In the relatively new subject of semidefinite programming,
positive definiteness plays the role of scalar positivity and
inequalities in the PSD partial order replace standard inequalities,
relative to linear programming. The subject has
had great success in solving or approximately solving hard problems, and
it has also presented interesting new
mathematical problems about positive definite matrices. One of these
deals with maintenance of positive semidefiniteness under 0-ing out some
off-diagonal entries ("gangster operators"). We discuss some recent results
in this area.