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Topological interpretations of provability logic

Provability logic concerns the study of modality $\Box$ as provability in formal systems such as Peano arithmetic. Natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970's by Harold Simmons and Leo Esakia. They have observed that the dual $\Diamond$ modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere. More recently, a new impetus came from the study of polymodal provability logic GLP that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that GLP was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged. We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also included a few results that have not been published so far, most notably the results of Section 6 (due the second author) and Sections 10, 11 (due to the first author).