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Article contents
CORONA RIGIDITY
Published online by Cambridge University Press: 30 June 2025
Abstract
We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra 
$\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of 
${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and 
$\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and 
$\mathrm {C}^{*}$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.
Keywords
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