Alexander Grothendieck (1928–2014) is, by near-consensus among algebraic geometers and number theorists, the most influential mathematician of the second half of the twentieth century. In a concentrated fifteen-year period between roughly 1955 and 1970, he rebuilt algebraic geometry from scratch on new foundations, invented the tools that would eventually prove the Weil conjectures, and reshaped what pure mathematics looked like for the generations that followed. Then, abruptly and completely, he walked away from mathematics and spent the last four decades of his life in self-imposed isolation.
His life reads like two different stories stitched together — the central figure of a scientific revolution and a philosophical exile.
Life
Grothendieck was born in Berlin in 1928 to stateless parents: his mother, Hanka Grothendieck, was a German journalist; his father, Sascha Schapiro, was a Russian-Jewish anarchist who had fought in the failed 1905 uprising and, later, in the Spanish Civil War. Sascha was murdered at Auschwitz in 1942. Alexander spent the war years in France, partly in an internment camp at Rieucros near Mende, where his mother was also held.
After the war, with no passport and no nationality, he began studying mathematics in Montpellier. He was entirely self-taught until he moved to Paris in 1948, where he quickly became a protégé of Laurent Schwartz and Jean Dieudonné. His doctoral work, completed in 1953, was on functional analysis — specifically, topological tensor products. Even within that narrow subject he was already reorganising the field.
In 1958 he was invited to the newly founded Institut des Hautes Études Scientifiques (IHÉS) outside Paris. The next twelve years there were arguably the most productive concentrated stretch of mathematics ever carried out by a single person.
Scientific contribution
Schemes
The central insight of Grothendieck’s programme can be summarised in one sentence: the right objects of algebraic geometry are not varieties sitting in space, but schemes — defined locally by commutative rings.
Before Grothendieck, an “algebraic variety” was a set of solutions to polynomial equations, studied by a mixture of classical geometric intuition and commutative algebra. The definitions were ad hoc, working over the complex numbers was different from working over finite fields, and many theorems had to be reproved case by case.
Grothendieck’s move was to define the basic object of study functorially. A scheme is, roughly, a space whose local models are spectra of commutative rings — every commutative ring corresponds to a geometric object, and morphisms of rings give morphisms of geometric objects. This was not merely an abstraction; it was a unification. Number theory and geometry became two specialisations of a single framework. The integers became a “geometric object” with “points” corresponding to prime numbers. A whole landscape opened up.
The construction is now entirely standard. Every graduate student in algebraic geometry or number theory learns Grothendieck’s foundations in their first year.
Étale cohomology
For decades the Weil conjectures — a set of statements connecting the number of solutions of polynomial equations over finite fields to the topology of complex varieties — had been the most tantalising open problem in arithmetic geometry. André Weil had proved a small case and conjectured that the general pattern followed from a cohomology theory with very specific properties. Nobody knew how to construct such a theory.
Grothendieck, with his students Michael Artin and Jean-Louis Verdier, invented the required theory: étale cohomology. Using schemes plus a new definition of “topology” that no longer required a topological space — the Grothendieck topology — they built a cohomology theory for varieties over any field that had exactly the properties Weil had predicted. Pierre Deligne, Grothendieck’s student, used this machinery to complete the proof of the last Weil conjecture in 1974, winning the Fields Medal.
Étale cohomology is now one of the most important tools in arithmetic geometry. Its consequences include Deligne’s theorem, much of the proof of Fermat’s Last Theorem, and significant parts of the Langlands program.
Topoi
Grothendieck realised that the Grothendieck topology he had introduced for étale cohomology was a far more general object than a cohomology tool. Any category with a notion of “covering” gives rise to a topos — a generalised space where the points themselves can be algebraic or logical objects. Topos theory has since spread well beyond algebraic geometry into mathematical logic, theoretical computer science, and the foundations of mathematics.
The same pattern appears again and again in his work: noticing that a technical construction invented for one purpose is actually the visible shadow of a much larger abstract object, and pursuing that abstract object until it reveals its own structure.
Other contributions
Even a partial list of Grothendieck’s lasting creations is long:
- Derived categories and triangulated categories, now the standard setting for homological algebra.
- The Grothendieck-Riemann-Roch theorem, generalising the classical result to arbitrary morphisms of schemes.
- The systematic use of functorial methods throughout algebraic geometry.
- Motives, a still-conjectural framework for unifying cohomology theories (the theory is not fully built even today).
- The foundational treatises EGA (Éléments de Géométrie Algébrique) and SGA (Séminaire de Géométrie Algébrique), co-authored with Dieudonné and seminar participants.
The break
In 1970, Grothendieck discovered that the IHÉS, where he had worked for twelve years, accepted partial funding from the French Ministry of Defence. He resigned in protest. He was 42.
He spent the next few years in political activism, founding the ecological group Survivre et Vivre and opposing the militarisation of science. He moved to Montpellier in 1973, took a teaching position there, and slowly withdrew from conventional academic life. By the early 1980s he had begun writing long, introspective manuscripts — most famously Récoltes et Semailles (“Reapings and Sowings”), an enormous unfinished autobiography-plus-philosophy that ran to over a thousand pages.
In 1988 he was jointly awarded the Crafoord Prize with his former student Pierre Deligne. He refused the money. That year he wrote his “Declaration of Intent,” formally breaking with the mathematical community.
In 1991 he disappeared. He moved to the small Pyrenean village of Lasserre and lived there in near-total isolation until his death in November 2014. He refused contact with nearly all former students and colleagues, including those who travelled to try to see him. The manuscripts he continued to write — tens of thousands of handwritten pages — were found after his death. Most have not yet been published.
Legacy
Grothendieck’s fifteen years of active mathematics transformed more than one field. Modern algebraic geometry, in every serious sense, is built on his framework. Number theory uses his tools at every level. The machinery of derived categories, sheaves, and stacks that appears throughout mathematical physics descends from his constructions. The word “functorial” appears so often in contemporary mathematics papers that it is easy to forget it was once a revolutionary thing to demand.
What sets him apart from other mathematicians of his rank is the degree to which his contributions were re-foundational. Most great mathematicians prove hard theorems. Grothendieck built the mathematical language in which those theorems could even be stated. It is a rarer kind of achievement and, in the view of many who knew him, a deeper one.
Grothendieck refused to attend the 1966 International Congress of Mathematicians in Moscow — where he was to receive the Fields Medal — as a protest against the Soviet treatment of dissidents. His Fields Medal is, to this day, one of the more conspicuous absences in the history of the award.
He left mathematics believing, with some reason, that mathematicians had failed to take responsibility for how their work was used. He spent the second half of his life on questions that mathematics could not answer. Whether his withdrawal was a tragedy or a principled stand is still debated. What is not debated is the magnitude of what he built before he left.
Known for
- Schemes — foundation of modern algebraic geometry
- Topos theory
- Fields Medal (1966, declined to attend the Moscow ceremony)
- Étale cohomology and the Weil conjectures
- EGA and SGA — the most ambitious foundational treatise of the century
Frequently asked
Why is Grothendieck considered so important?
Because he rewrote the foundations of a whole field. Before Grothendieck, algebraic geometry was a mix of classical geometry and commutative algebra with plenty of ad-hoc constructions. His theory of schemes unified everything under a single abstract framework, and the techniques he invented — étale cohomology, topoi, derived categories — now appear throughout number theory, representation theory, and mathematical physics.
Why did he leave mathematics?
In 1970, Grothendieck discovered that the IHÉS, the institute where he worked, accepted partial funding from the French defense ministry. He resigned in protest and never returned to mainstream mathematics. He spent the rest of his life in increasing isolation, eventually living as a hermit in a Pyrenean village until his death in 2014.
What are EGA and SGA?
Éléments de Géométrie Algébrique (EGA) and Séminaire de Géométrie Algébrique (SGA) are the two massive collective works Grothendieck co-authored at the IHÉS in the 1960s. Together they run to thousands of pages and contain the systematic foundation of scheme theory. Generations of algebraic geometers learn from them.
More on: en.wikipedia.org