Cocycles de groupe pour $\mathrm {GL}_n$ et arrangements d'hyperplans

Ce livre constitue un expose detaille de la serie de cours donnes en 2020 par le Prof. Nicolas Bergeron, titulaire de la Chaire Aisenstadt au CRM de Montreal. L'objet de ce texte est une ample generalisation d'une famille d'identites classiques, notamment la formule d'addition de la fonction cotangente ou celle des series d'Eisenstein. Le livre relie ces identites a la cohomologie de certains sous-groupes arithmetiques du groupe lineaire general. Il rend explicite ces relations au moyen de la theorie des symboles modulaires de rang superieur, devoilant finalement un lien concret entre des objets de nature topologique et algebrique. This book provides a detailed exposition of the material presented in a series of lectures given in 2020 by Prof. Nicolas Bergeron while he held the Aisenstadt Chair at the CRM in Montreal. The topic is a broad generalization of certain classical identities such as the addition formulas for the cotangent function and for Eisenstein series. The book relates these identities to the cohomology of arithmetic subgroups of the general linear group. It shows that the relations can be made explicit using the theory of higher rank modular symbols, ultimately unveiling a concrete link between topological and algebraic objects. I think that the text ``Cocycles de groupe pour $\mathrm{GL}_n$ et arrangements d'hyperplans'' is terrific. I like how it begins in a leisurely, enticing way with an elementary example that neatly gets to the topic. The construction of these ``meromorphic function''-valued modular symbols are fundamental objects, and play (and will continue to play) an important role. --Barry Mazur, Harvard University