In the early years of the XX century, Weyl initiated study of compact perturbations of pseudo-differential operators. The Weyl-von Neumann theorem asserts that two self-adjoint operators on a complex Hilbert space are unitarily equivalent modulo compact perturbations if and only if their essential spectra coincide. Berg and Sikonia (independently) extended this result to normal operators. New impetus to the subject was given in 1970s by Brown, Douglas, and Fillmore, who replaced single operators with (separable) C*-algebras and realized that compact perturbations can be considered as extensions by the ideal of compact operators. After passing to the quotient (the Calkin algebra, Q) and identifying an extension with a *-homomorhism into Q, analytic methods had been supplemented with methods from algebraic topology, homological algebra, and (most recently) logic.  Around the same time, Shelah proved one of his many influential results, by showing that the assertion `all automorphisms of $\ell_\infty/c_0$ are trivial' is relatively consistent with ZFC. Surprisingly, these two directions of research are intimately connected. 

This talk will be about rigidity of quotient structures and it is  partially based on the preprint Farah, I., Ghasemi, S., Vaccaro, A., and Vignati, A. (2022). Corona rigidity. arXiv preprint arXiv:2201.11618

https://arxiv.org/abs/2201.11618 and some more recent results.