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.
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.
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