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Alexander Grothendieck (1928–2014)

Grothendieck refounded algebraic geometry on categorical language and, in doing so, transformed the practice of mathematics. His programme — developed at the Institut des Hautes Études Scientifiques (IHÉS) from 1958 to 1970 — replaced the classical objects of algebraic geometry (varieties defined by polynomial equations) with a far more general framework (schemes, defined over arbitrary commutative rings) that unified number theory, algebraic geometry, and topology under a single conceptual roof. The tools he built — schemes, sheaves, toposes, étale cohomology, motives — are now the standard infrastructure of algebraic geometry and large parts of number theory. The framework depended on categorical thinking at every level: Grothendieck’s objects are defined by their relationships (functors, natural transformations, representability), not by their internal construction. He was, by wide acknowledgment, the most consequential mathematician of the second half of the twentieth century.

Life

Born 28 March 1928 in Berlin, Germany. His father Alexander Schapiro was a Russian anarchist; his mother Hanka Grothendieck was a German journalist. Both parents were politically active; his father had participated in the Russian Revolution and fought in the Spanish Civil War. After the Nazi rise to power, the family fled to France (1933). During the German occupation, Grothendieck’s father was interned at Le Vernet and then deported to Auschwitz, where he was killed in 1942. Grothendieck and his mother survived the war in southern France; he was interned briefly at the Rieucros camp as a child.

After the war, Grothendieck studied mathematics at the University of Montpellier (1945–48), then moved to Paris, where he attended Henri Cartan’s seminar at the École Normale Supérieure. PhD at the University of Nancy (1953), under Laurent Schwartz and Jean Dieudonné, on topological vector spaces — a subject unrelated to his later work but already displaying his characteristic approach: rebuilding a subject from its foundations rather than solving problems within an existing framework.

Appointed professor at the IHÉS in Bures-sur-Yvette (1958). The IHÉS period (1958–70) was the most productive mathematical decade of the twentieth century by any measure: Grothendieck, with a group of collaborators (the “Grothendieck school”), produced the Éléments de géométrie algébrique (EGA) and the Séminaire de Géométrie Algébrique du Bois Marie (SGA) — thousands of pages refounding algebraic geometry. Fields Medal (1966) — though Grothendieck boycotted the Moscow ICM rather than accept the medal in person, in protest against Soviet military intervention in Eastern Europe. The boycott prefigured the pattern of political refusal that marked his later career.

Grothendieck left the IHÉS in 1970 over the institute’s acceptance of military funding — a departure driven by political and moral conviction. He became increasingly involved in anti-war and ecological activism (founding the group Survivre et Vivre). Held a position at the University of Montpellier (1973–88) but published little mathematics. In 1988, he publicly declined the Crafoord Prize in a letter citing his break with the scientific establishment and his conviction that the prize system corrupts the values of science — completing the arc from the 1966 boycott through the 1970 departure. In 1991, he retreated to a village in the Pyrenees (Lasserre), where he lived in near-total isolation for the last twenty-three years of his life, writing voluminously but publishing nothing. Récoltes et Semailles (“Reaping and Sowing”), a long, autobiographical, and often bitter reflection on mathematics and the mathematical community, circulated in manuscript from the late 1980s and was published posthumously in 2022. Died 13 November 2014 in Saint-Girons, Ariège.

The refounding of algebraic geometry

The method that would define the programme was already visible before the IHÉS period. Grothendieck-Riemann-Roch (c. 1957, published by Armand Borel and Jean-Pierre Serre in 1958) generalised the classical Hirzebruch-Riemann-Roch theorem from a statement about a single variety to a statement about morphisms between varieties — proving a theorem about relationships rather than about objects considered in isolation. The relativising move — prove the general (morphism) case and recover the classical (object) case as a special instance — became the signature of Grothendieck’s method. GRR also introduced the Grothendieck group (K-group) — the “K” is from the German Klasse — which seeded algebraic K-theory, an entire field of mathematics connecting algebra, topology, and number theory.

Grothendieck’s programme can be understood through three further constructions that display the method at full scale.

Schemes. Classical algebraic geometry studies algebraic varieties — the solution sets of polynomial equations in some number of variables. Grothendieck generalised varieties to schemes: geometric objects defined not over a specific number field (like the rational or complex numbers) but over an arbitrary commutative ring. The generalisation unified algebraic geometry over the complex numbers (where the geometric intuition lives) with algebraic number theory (where the arithmetic lives). A scheme is defined by its functor of points — its relationships to all other schemes — rather than by its internal construction. The approach is categorical at its core: objects are characterised by morphisms, not by elements.

Toposes. Grothendieck introduced the concept of a topos (plural: toposes or topoi) as a generalisation of the notion of a topological space. A topos is a category that behaves like the category of sheaves on a topological space — it supports a notion of “local” and “global” without requiring an underlying point-set topology. The generalisation made it possible to do geometry in settings where no classical topological space exists — particularly in algebraic geometry over finite fields. Lawvere and Myles Tierney later abstracted the topos concept into elementary topos theory, connecting it to logic and foundations.

Étale cohomology. The technical achievement that made the Weil conjectures accessible. André Weil had conjectured (1949) deep connections between the number of solutions of equations over finite fields and the topology of the corresponding variety over the complex numbers. Proving the conjectures required a cohomology theory for varieties over finite fields — a topological invariant in a setting where classical topology does not apply. Grothendieck constructed étale cohomology for this purpose, using the étale topology (a generalisation of the Zariski topology that captures more geometric information). The Weil conjectures were eventually proved by Pierre Deligne (1974), Grothendieck’s most distinguished student, using the tools Grothendieck had built.

Where Grothendieck stops

Grothendieck’s programme is the most consequential mathematical achievement of the twentieth century’s second half, but its relationship to the wider mathematical community is complicated. The EGA and SGA are forbiddingly technical — thousands of pages of dense, foundational material that require years of study to absorb. The “Grothendieck school” produced a generation of extraordinary mathematicians (Deligne, Luc Illusie, Jean-Pierre Serre, Michel Raynaud), but the programme’s demands on its practitioners were extreme, and the community outside the school often found the material inaccessible. Whether the inaccessibility is an inherent feature of the mathematics or a consequence of the expository style is debated.

The unfinished projects are as significant as the completed ones. Grothendieck outlined a programme of “motives” — a universal cohomology theory that would unify all existing cohomology theories (étale, de Rham, crystalline, Hodge) into a single framework. The motive programme remains unfinished and is one of the central open problems in algebraic geometry and number theory. The “anabelian geometry” programme, outlined in Esquisse d’un Programme (1984), proposed connections between algebraic geometry and the absolute Galois group that have been partially developed by others but remain largely unrealised.

Grothendieck’s withdrawal from mathematics after 1970, and his increasingly isolated and bitter later years, is part of his story. Récoltes et Semailles contains accusations of plagiarism and intellectual theft against former students and colleagues — accusations that the mathematical community has largely rejected but that reflect Grothendieck’s sense of betrayal. The relationship between the mathematical vision (the most generous and unifying programme in modern mathematics) and the personal trajectory (withdrawal, isolation, bitterness) is a question in the history of mathematics that does not have a simple answer.

Key works

See also: Category theory · Eilenberg · Mac Lane · Lawvere