Corona 刚性
Corona Rigidity
AI总结：
本文综述附加集合论公理对商结构刚性的影响，以布尔代数 $\mathcal{P}(\mathbb{N})/\text{Fin}$ 为模板，探讨力迫公理与连续统假设下自同构的平凡性差异，并推广至布尔代数、Čech-Stone 余紧化和 $\mathrm{C}^\ast$-代数等范畴。
摘要：
我们给出了关于附加集合论公理对商结构影响的统一综述。我们的焦点是刚性，以所讨论商的适当非平凡自同构的存在性（或更确切地说，不存在性）来衡量。研究这一主题的一个教科书式例子是布尔代数 $\mathcal{P}(\mathbb{N})/\text{Fin}$，其行为是本综述围绕的模板：力迫公理蕴含其所有自同构都是平凡的，即它们由 $\mathbb{N}$ 的几乎置换诱导，而在连续统假设下，这种刚性失效，$\mathcal{P}(\mathbb{N})/\text{Fin}$ 承认不可数多个非平凡自同构。我们考虑这一现象的深远推广，并展示大量类似模式持续存在的情形，主要（但不仅限于）关注布尔代数、Čech-Stone 余紧化和 $\mathrm{C}^\ast$-代数范畴。我们综述了这一领域的现状和未来前景，讨论了主要开放问题，并尽可能概述证明的主要思想。
英文摘要：
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 $\mathcal{P}(\mathbb{N})/\text{Fin}$, 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 $\mathcal{P}(\mathbb{N})/\text{Fin}$ 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 $\mathrm{C}^\ast$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
