In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding.

[13] In 1991, he moved to the French village of Lasserre in the Pyrenees, where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014.

[24] In the village of Le Chambon-sur-Lignon, he was sheltered and hidden in local boarding houses or pensions, although he occasionally had to seek refuge in the woods during Nazi raids, surviving at times without food or water for several days.

Many of the refugee children hidden in Le Chambon attended Collège Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics.

On the advice of Cartan and André Weil, he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, topological vector spaces: Jean Dieudonné and Laurent Schwartz.

Dieudonné and Schwartz showed the new student their latest paper La dualité dans les espaces (F) et (LF); it ended with a list of 14 open questions, relevant for locally convex spaces.

[37] In 1953 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he had refused to take French nationality (as that would have entailed military service against his convictions).

[38][39] It was in Lawrence that Grothendieck developed his theory of abelian categories and the reformulation of sheaf cohomology based on them, leading to the very influential "Tôhoku paper".

[41] Comparing Grothendieck during his Nancy years to the École Normale Supérieure-trained students at that time (Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, and Bernard Malgrange), Leila Schneps said: He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics.

Collaborators on the SGA projects also included Michael Artin (étale cohomology), Nick Katz (monodromy theory, and Lefschetz pencils).

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique (EGA) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

[45] The results of his work on these and other topics were published in the EGA and in less polished form in the notes of the Séminaire de géométrie algébrique (SGA) that he directed at the IHÉS.

While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from the IHÉS, those who knew him say that the causes of the rupture ran more deeply.

[48] In that publication, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and the downtrodden.

In 1983, stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University, Grothendieck wrote a 600-page manuscript entitled Pursuing Stacks.

"[63] La Clef des Songes,[64] a 315-page manuscript written in 1987, is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a deity exists.

[65] As part of the notes to this manuscript, Grothendieck described the life and the work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age.

[1] His growing preoccupation with spiritual matters was also evident in a letter entitled Lettre de la Bonne Nouvelle sent to 250 friends in January 1990.

From approximately 1955 he started to work on sheaf theory and homological algebra, producing the influential "Tôhoku paper" (Sur quelques points d'algèbre homologique, published in the Tohoku Mathematical Journal in 1957) where he introduced abelian categories and applied their theory to show that sheaf cohomology may be defined as certain derived functors in this context.

[17] Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre[81] and others, after sheaves had been defined by Jean Leray.

Grothendieck went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which at the time were in a state of flux and under discussion in Claude Chevalley's seminar.

[44] Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal.

The collection Fondements de la Géometrie Algébrique (FGA), which gathers together talks given in the Séminaire Bourbaki, also contains important material.

[17] Grothendieck's work includes the invention of the étale and l-adic cohomology theories, which explain an observation made by André Weil that argued for a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.

Topological tensor products had played the role of a tool rather than of a source of inspiration for further developments; but he expected that regular configurations could not be exhausted within the lifetime of a mathematician who devoted oneself to it.

[11] In an obituary David Mumford and John Tate wrote: Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend.

"[71] Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures.

Grothendieck laid a new foundation for algebraic geometry by making intrinsic spaces ("spectra") and associated rings the primary objects of study.

They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension.

[101] The Benjamín Labatut book When We Cease to Understand the World dedicates one chapter to the work and life of Grothendieck, introducing his story by reference to the Japanese mathematician Shinichi Mochizuki.