Alexander Grothendieck: Mathematician Extraordinaire

                                    The lion in winter: Grothendieck in old age 

Mathematicians and theoretical physicists fascinate me. This is because, I can imagine myself, if I work hard enough, sometime solving a biological conundrum, or evaluating a chemical compound. But the idea of sitting with a piece of paper and pencil and calculating away for days or months or years and then coming up with a proof of an esoteric mathematical problem, or a theory to explain an observation made in deep space seems to me to be more like magic. Anybody who can do this, in my opinion is made of stuff that I cannot even imagine, let alone ever to hope to emulate.

There are several such mathematicians. One of them passed away last November. I refer to Alexander Grothendieck who passed away on the 13^th of last month. Grothendieck was the mathematician par excellence. He embodied all that is mysterious and unknowable about these “beautiful minds” as somebody picturesquely described another mathematician.. He was perhaps the greatest mathematician of the twentieth century and his passing is, though it may sound clichéd, really the passing of an era.

Born of anarchist parents in the Germany of the inter war years, Grothendieck was a Jew at a time and place when it was the most dangerous to be one. They escaped to France but that did not prevent the death of his father in Auschwitz. He studied in the University of Montpellier and then in Paris during the Second World War and just after it. Even during his school days he was dissatisfied with the state of Geometry as he found it. He wrote  “What was least satisfying to me in our high school mathematics books was the absence of any serious definition of the notion of length of a curve, of area of a surface, of volume of a solid. I promised myself I would fill this gap when I had the chance” During his university days he came up with a general version of the Lebesgue integral, working all alone with no supervision. Lebescgue himself described this integral as follows in a letter written to one of his colleagues:  “I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.”

I do not pretend to really understand all this, but the crux is that Grothendieck immediately showed his genius and went on to redefine the frontiers of algebra and geometry. It is said that he was an immensely charismatic person. In 1958 he set up a team of mathematicians who met 10 hours a day, 5 days a week for more than a decade to create a new foundation for geometry. It said that he talked; others took notes, filled in the details and returned the next day for more ideas. One of his colleagues, Jean Dieudonne, a prominent mathematician in his own right, immersed himself totally in the project, arriving at his desk at 5 AM every morning so that he could do three hours of editing before the seminars started at 8 AM.

Grothendieck was genius of generalizing a problem. Whenever he was confronted with a problem that seemed unsolvable, he simply “zoomed away” from the problem and attempted to generalize it . This led to some amazing insights which, as I said earlier, made him the doyen of mathematicians and got him the Field medal in 1966. He, however, refused it.

Grothendieck , as one mathematician has put it  “rewrote the foundations of geometry not just once, but three times, first replacing classical geometry with his “theory of schemes” —and then going on to the “theory of toposes”, and finally the great unfinished “theory of motives”. One of the great goals of contemporary mathematics is to complete that final theory, which, if our expectations are correct, would give us the tools to settle many of the hardest outstanding problems in several areas of mathematics.”

 “When he was just in his early twenties, he wrote a doctoral thesis in the subject of “functional analysis” that provided new tools so powerful that they left almost no problems in that field left to solve. He then went off searching for a new field vast enough to contain his talent, and found it in algebraic geometry. In the days of the Seminaire de Geometrie Algebrique, where he spent most of his working life, he reportedly worked 18 hours a day, 7 days a week, 10 years per decade.”

Unfortunately as the sixties ended, he withdrew from the world of mathematics to embroil himself in politics and self-examination. It is said that he was scarcely sane during the last years of his life. He died on the 13 November 2014, aged 86, in Saint-Lizier, Ariège, France.