A Quick Glance at Grothendieck's Work

Alexander (Alexandre) Grothendieck (Berlin 1928–Saint-Girons 2014) must be considered, without doubt, as the major mathematical genius of the last sixty years, and, next to David Hilbert, as one of the two fundamental mathematicians of the last century. With a published work of nearly ten thousand pages (covering geometry, topology, number theory, complex variables, to only quote the heart of mathematics), with more than one thousand new definitions (when an ordinary mathematician might be happy if she or he has introduced one new definition in the field) and with completely revolutionary understandings of the notions of number (‘schemes’), space (‘topos’) and form (‘motives’), Grothendieck has opened all kinds of new roads for the development of mathematics in the coming centuries.

Nevertheless, Grothendieck's work is almost completely unknown outside a small group of specialists. Unlike Einstein, whose ideas (simplified or deformed) have reached the public domain, Grothendieck still lies in the shadow, even if his conceptual revolution goes much further. In fact, if Einstein studies the relativisation of space-time and discovers its local invariants, Grothendieck studies the relativisation of spacenumber and discovers its global universal invariants. After his recent death, a sudden interest has arisen in a life worthy of fiction (there are three partial biographies and an inspiring novel), but time is still needed to fully appreciate the magnitude of his mathematical work.

Grothendieck's inexhaustible universe may be divided (wrongly, as in any compartmentalisation) into four main periods.

• 1) From 1949 to 1957, along certain geographical margins (Nancy, São Paulo, Kansas), he produces fundamental contributions in topological vector spaces, homology, category theory and complex variables, obtaining profound theorems and leaving rich seeds to be developed in the following decades.

• 2) Between 1958 and 1970, in the very centre of mathematics (Paris), Grothendieck advances his research at the Institut des Hautes Études Scientifiques (IHES), specially constructed to shelter him. After a scintillating, condensed vision of his future programmes, he writes with the formidable Jean Dieudonné the Éléments de Géométrie Algébrique (EGA) (the appearance of schemes), and directs his famous Séminaire de Géométrie Algébrique (SGA) (the appearance of topoi).