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The Ultimate Space Explorer
January/February 2015

An extraordinary life: Grothendieck escaped Nazi Germany and changed the world of algebraic equations (photo: IHÉS)

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.

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