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Alexandre Grothendieck, the greatest mathematician for the twenties century, passed away on November 13, 2014 at the Saint-Girons hospital (Ariège) near the village Lasserre.
Alexandre Grothendieck spent about the last 24 years of his life in this village in Pyrenees range of mountains in a self-imposed retirement avoiding all contacts with the outside world and the mathematical community.
He had good reasons for this, but till now the mathematical community does not want to listen, or, rather, to read his extensive partially autobiographical, partially philosophical texts.
Alexandre Grothendieck, with help of his pupils, collaborators, and admires, completely transformed mathematics. His best known contribution is the proof of most of the Andre Weil conjectures (with the last step done by his pupil Pierre Deligne). Much more important is his transformation of the algebraic geometry from relatively obscure branch of mathematics to its central part. Even more important is his most intangible contribution, the concept known as th "rising sea", the idea that every mathematical problem should be immersed in a sufficiently abstract theory, which will made the solution trivial. This theory should be, in a sense, trivial too - it should not involve any tricks or convoluted arguments. This was a drastic departure from the mathematical analysis, the central branch of mathematics at the time, which was dominated by proofs demonstrating not so much the vision, but the "executive power" of the authors (the concept introduced by G. Hardy, who valued the executive power most). These ideas are still far from being internalized or even understood by the mathematical community.
Despite his tremendous influence, surpassing by a large margin the influence of any mathematician after David Hilbert, Alexandre Grothendieck was at least about 100 years ahead of his time.
His integrity and his concern about the perils people put each other into are hardly matched by any other contemporary scientist. He did not succeed much in this respect, apparently because his concerns only appeared to be left wing politics, but in fact were not of political nature.
With Alexandre Grothendieck passing away we lost the last living giant in mathematics.
Here is a link to a memorial article Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort in Le Monde, France (in French).
Next post: Where one can find an autobiography of Alexander Grothendieck? Part 1.
When I read some excerpts from his autobiography, the way he talks about life is as if he lives in a different world. His soul must have been connected to a place where most of us can not go. I don't think he was 100 years ahead of his time, but from another stage of evolution that humans are not guaranteed to reach even in a 1000 years. He was an outlier, but his unique nature was in control of him and not the other way around. Otherwise, he could have been more patient with other people, realizing that they also are not really in control of their nature.
ReplyDeleteWhere can I find his autobiography?
ReplyDeleteThe link to the post
DeleteWhere one can find an autobiography of Alexander Grothendieck? Part 1.
disappeared in Google servers.
michal2602: While the short answer is "I have no idea", I posted a long reply as the next two posts.
DeleteThe second one:
Where one can find an autobiography of Alexander Grothendieck? Part 2.
Google turned it into not working javascript. I will delete my original attempt to reply.
Dear Sowa,
ReplyDeleteWhat do you think mathematics would look like today without Grothendieck? What parts of it would still be missing, and what important results would still be open? Would all of his ideas inevitably have been discovered by someone else by now? If young Grothendieck would enter mathematics right now, what would he be doing? I know it's all a fantasy, but I am interested to hear your opinion.
Jamie Vitara: Such questions are impossible to answer. If not Grothendieck, may be somebody else would do the same five years later? How can we know?
ReplyDeleteA more accessible version of the first question is: how mathematics would look today without Grothendieck's algebraic geometry?
Since it occupies now the central place in mathematics, one can expect that it would look very differently. Before Grothendieck algebraic geometry was a fairly isolated branch of mathematics, more or less like the riemannian geometry (not differential geometry). Most likely, it will keep this position. The theory of complex (not necessarily algebraic) manifolds, continuing the work of Kodaira, would be much more important. But, most likely, it will meet the same resistance it actually met in this world. The theory of complex manifolds is based on complex analysis, and, in a sense, a part of it. Analysts hate topological methods in general and the sheaf theory in particular (W. Rudin wrote about this most openly). Without if not technical, but definite moral support from algebraic geometry, Kodaira's methods would die out no later than in the 1970-ies.
Etc.
One of the Weil conjectures still would be proven. In fact, B. Dwork proved it before Grothendieck. Dwork's methods are very beautiful, but are as alien to the "classical" mathematics as Grothendieck's ones. It is hard to speculate what would happen in Dwork's direction.
The Mordell conjecture (G. Faltings) and the Fermat's Last Theorem (A. Wiles) would be not proven.
The Atiyah-Singer index theorem would be proven, but probably, not in its most useful form: the index theorem for families (its proof is based on Grothendieck's ideas). In addition, there would be no K-theory, and there would be no K-theoretic form of the index theorem, which was (and is) very important for the development of the index theory.
The modern interaction of mathematics with the theoretical physics, probably, would be much much narrow in scope.
Jamie Vitara: In your second question, do you assume that there was no Grothendieck in the past, or that the situation is the same as today, and a new genius of his caliber appears?
ReplyDeleteIf there was no Grothendieck in the past, for many people mathematics would be much less interesting. A new Grothendieck would begin studying it, prove some remarkable theorems in analysis (as Grothendieck did), and the mathematical community will be unable to absorb them very much like the community of analysts wasn't able to absorb Grothendieck's ideas in functional analysis. And, perhaps, he would quit mathematics and start to write some innovative fiction.
This may happen even in the our world with Grothendieck in the past. May be, this already happened. By the very definition of the notion "a Grothendieck" (well, it is my definition), a Grothendieck cannot work in the Bourgain-Gowers-Tao style mathematics.
Some people leave mathematics, and this is rarely noticed. Somebody just stopped publishing papers. Who knows why? Sometimes one occasionally learns that he or she left academia and does something different. But one learns this only about people who worked in your field, and one learns this by the word of mouth. There is no publicity.
More optimistically, a new Grothendieck would create a new algebraic geometry for us. If there is no Grothendieck's algebraic geometry, she or he will create it. And if we are in our current world, she will create something new absorbing the Grothendieck's algebraic geometry, algebraic topology, and the conceptual part of analysis (and replacing PDE estimates by a categorical formalism).