This book pays homage to the mathematical life and achievements of one of the greatest mathematicians ever, a unique genius: Alexandre Grothendieck. Surely he changed Mathematics, and Mathematics would not be the same without him. EGA, SGA, FGA, schemes, flatness, strict localization, fundamental group of a scheme, Picard scheme, change of base point, descent, infinitesimal lifting, relative geometry, completeness theorem, standard conjectures, sites, topos, derived functors, étale cohomology, higher categories, stacks, motives… The book consists of 13 papers by mathematicians who knew him, as his students and/or collaborators. These papers discuss various aspects of Grothendieck’s work: his approach to mathematical problems, his goals and visions of Mathematics, his devotion and generosity in research for the sake of science, and also biographical recollections. In different degrees each paper presents technical concepts and results, discloses metamathematics behind the scenes and comments personal remembrances. Surely some parts will be somehow out of reach for a non-specialist, but even then they are a highly profitable reading. One finds here an enthralling presentation by Schneps of the Grothendieck-Serre correspondence, some quite private considerations by Mumford and Cartier, what Diestel explains about the transformation of Banach spaces theory under Grothendieck’s spell, the track of Grothendieck’s inspiration sources by Oort, the discussions by Simpson of the phylosophy of descent and then its higher and higher in crescendo, Kleiman historical account of the Picard group back to Clebsch, Murre on the fundamental group, Illusie on cohomology étale, Karoubi on K-theory, Raynaud, Manin, Hartshorne! And everywhere the impressive depth and transcendency of Grothendieck visions of Mathematics. Even in one does not understand completely the music, the noise is wonderful. This reviewer can only say: of course, anyone must read this, and close with three quotations:
[Cartier] When one follows Grothendieck’s work throughout its development, one has exactly this impression of rising step by step toward perfection.
[Schneps] Grothendieck devoted his life to the pursue of the absolute.
[Grothendieck] If it is not obvious, it is probably false.