From perturbations of operators to noncommutative condensers

A numerical invariant, the quasicentral modulus underlies the multivariable generalizations of the classical Weyl-von Neumann-Kuroda and Kato-Rosenblum theorems. There are also connections to the Kolmogorov-Sinai dynamical entropy. I will also point out some of the open problems. Recently a noncommutative analogy with condenser capacity in nonlinear potential theory is emerging, that provides a new perspective on the subject.