Grothendieck the Mathematician

A Time for Reflections 1973-1991

 

Preface: Invitation to Travel

Heureux celui qui peut d’une aile vigoureuse
S’élancer vers les champs lumineux et sereins;

Charles Baudelaire, Les Fleurs du mal, “Élévation”

There was a mathematician who, at the height of his fame, abandoned scientific research to advocate radical ecology, became a professor at the University of Montpellier, devoted himself to meditation, was regarded by some as a prophet and by others as hotheaded, and then disappeared. One could use these episodes to construct the biography of a “frightening genius” and write stories embellished with anecdotes and colorful portraits likely to entertain amateurs of famous scientists.

Our approach is different. In this exhibition, we wanted to compose a portrait of Alexandre Grothendieck, the mathematician, placing ourselves as far as possible from the storms of his biography.

However, traces of his unceasing mathematical research between 1973 and 1991 are both visible and numerous, as evidenced by the thousands of pages written by Grothendieck between his arrival in Montpellier and his departure for the village of Lasserre in Ariège. These pages, entrusted to Jean Malgoire and now available on the website of the Grothendieck Archives at the University of Montpellier, provided most of the documentation and images for this exhibition.

These texts provide the constantly renewed material for “mathematical reflections,” of which La Longue Marche à travers la théorie de Galois (1981), Pursuing Stacks (1983) and Esquisse d’un Programme (1984) are examples that we have examined. As for Récoltes et Semailles (1984-1986), in which Grothendieck once again offers “reflections” but also a “testimony about a mathematician’s past,” it came after these samples and ended up replacing them and those announced but never published.

It has been argued that the baroque and provincial mathematics written by Grothendieck in a “renewed style” between 1973 and 1991 formed the tail of a comet observed in all its glory between 1955 and 1970: the comet of algebraic geometry of schemes and topos, classical and sophisticated, written in the formal style of a member of Bourbaki.

This judgement is contradicted by the mathematical texts of this period and their outcome, but also by Grothendieck himself in his retrospective reflection on “arithmetic geometry” and “topological algebra.” Indeed, these new sciences are developed, according to a necessary balance, over the two periods. As for his style of mathematical expression, while he concedes that it changed and conformed in Pursuing Stacks to that of a “logbook,” Grothendieck did not reject “formal writing” subject to the venerable canon of the axiomatic-deductive method.

We are aware of the difficulties of reading an exhibition that focuses on Grothendieck as a mathematician. It is true that it is aimed more at people with a university education in mathematics. We do not claim to solve this problem inherent in the dissemination of mathematics. Despite the defects of this first attempt, we still believe that a complex analysis of facts worthy of interest is more respectful to the reader than simplified and therefore misleading formulas.

We hope that, despite these difficulties, readers will enjoy following with us some fragments of the mathematical exploration formerly undertaken by Grothendieck. The territories of this new continent are the mathematical theories he discovered, and the stages of this journey are the texts he wrote.

Sébastien Maronne, Bertrand Toën
Toulouse, November 2025 (English version)