Hitler’s ascent to power in 1933 marked a profound change in the balance of power in mathematics. Not only did German-Jewish mathematicians escape to Britain and America, but a revival of French mathematics began under the influence of young scholars who had gone to Germany to study. Lamenting the lack of suitable modern textbooks, they created their own using the pseudonym Nicolas Bourbaki — the name of a French general in the Franco-Prussian War.
The Greek origin of the name suited the intended style of a multi-volume work under the heading Éléments de mathématique, inspired by the title of Euclid’s great work Elements. Following their first meeting in 1934, the Bourbaki group’s influence went international, with new volumes of Éléments published over the next 50 years. After the Second World War, they produced a series of advanced monographs by individual authors based on public lectures in Paris in the Séminaire Bourbaki. Paris was rapidly taking over from Germany as the centre of mathematics.
During this time, a brilliant young boy named Alexander, born in Berlin in 1928, was hidden on a farm in northern Germany. His father, Alexander Schapiro, a Russian anarchist, had taken part in the failed 1905 revolution. He lost an arm escaping from prison ten years later; leaving Russia on forged papers, he reached Berlin in 1922. When the Nazis took power in 1933 Schapiro escaped to France and was later joined by Alexander’s mother, Hanka Grothendieck. The two of them joined in the Spanish Civil War. After returning to France in 1939, Schapiro was interned in Camp Vernet in the French Pyrenees, before the Vichy regime handed him over to the Nazis and he disappeared into Auschwitz. That year, Alexander’s surrogate parents decided Germany was too dangerous and put him on a train for France where he rejoined his mother, living in various camps for displaced persons.
After the war, Alexander Grothendieck (he took his mother’s surname) became a student at Montpellier. There his professor told him that Henri Lebesgue, a French mathematician who died in 1941, had already solved all the problems of mathematics but his ideas and methods would be too difficult to teach. Undeterred, Grothendieck rediscovered much of Lebesgue’s great work. He achieved his first mathematical success in isolation, honestly believing that he was the only mathematician in the world.
Grothendieck was a phenomenon, unique in modern mathematics. After writing a dense and profound article that fuelled the research of a whole school for the next 40 years, he abandoned that area. He later said of himself that he was destined to be the builder of houses he would never inhabit. Aged 27, he started a new chapter in his work, turning the subject of algebraic geometry into something far grander. Roughly speaking, algebraic geometry studies curves and surfaces representing solutions to algebraic equations. For instance, solutions to the equation xn+yn=zn determine a surface in three-dimensional space framed by the coordinates x, y, and z. This particular equation inspired Fermat’s Last Theorem, which states that when n is greater than 2 there are no non-trivial solutions for which x, y and z are whole numbers. In other words, the surface defined by that equation contains no such points.
Fermat’s famous conjecture (he was only able to prove some special cases) remained unresolved for more than 350 years until the last decade of the 20th century. Its resolution was extraordinarily difficult: direct number-theoretic approaches never succeeded, nor did direct geometric approaches. Its solution finally came from a viewpoint at a far higher level, facilitated by Grothendieck — though he never went near such concrete questions, preferring to live in a world of higher abstraction.
His experiences, hidden in Germany for years, then escaping to France, losing a father who had battled Tsarist Russia and the Communists and who was finally killed by the Nazis, gave him a yearning for extreme abstraction. He would have nothing to do with physics, nor with any kind of military support for mathematics: when as a professor at the Institut des Hautes Études Scientifiques (IHÉS) he discovered that some of its funding was defence-related he abandoned that brilliant research centre. In 1970, he went to Montpellier, where he had once been a student. It was the beginning of the end for his mathematical work, and after retirement from academia he went to live in the French Pyrenees, not far from the internment camp where his father had lived before deportation to Auschwitz. Grothendieck’s address and telephone number were known only to a select few, sworn to secrecy.
In happier days, the Bourbaki group had been ready and willing to help him. In particular, Jean Dieudonné and another mathematician from the circle took him on, encouraging his visions and helping restrain his most extreme tendencies towards abstraction. Working at the IHÉS, he turned out mathematics at such a rate that it needed all Dieudonné’s God-given talents as expositor to knock them into shape, writing from five until eight every morning before doing his day job. With help from his “12 disciples”, Grothendieck’s magnum opus on algebraic geometry spanned more than 10,000 pages.
Like his mathematical predecessors, Gauss and Riemann, and the physicist Einstein, Grothendieck was fascinated by the concept of space. For him a key ingredient was the concept of a point, to which he attached far more meaning than Euclid’s notion of something having no dimension. He drew algebraic geometry into a broader context embracing not just curves and surfaces but much of number theory too, creating highly sophisticated mathematical tools to handle this new abstract terrain.
It departed from the usual more concrete concerns about equations and their geometric representations. Eventually Grothendieck himself departed from mundane concerns altogether. He abandoned his disciples, his five children by three different mothers, and reached for ever deeper abstraction. This man who eschewed war, yet gave mathematical seminars in North Vietnam under US bombing raids, who fought a losing custody battle for his eldest son and a later court case involving the phalanstery he founded in his Montpellier home, retreated to the Pyrenees to engage with the religious aspect of his life and study the greatest battle of all, between God and the Devil.
Grothendieck died in November aged 86. If in 2,000 years’ time contemporary accounts by friends and colleagues are lost, and later commentaries are all that survive, future historians may wonder whether, like Pythagoras or Euclid, such an extraordinary person ever existed.