Alexandre Grothendieck passed away yesterday, November 13, 2014

Previous post: And who actually got Fields medals?


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.

William P. Thurston, 1946-2012

Previous post: The twist ending. 4


William P. Thurston passed away yesterday (August 21) at 8:00 p.m. at a hospital in Rochester, NY, surrounded by the members of his family.

From the announcement of the Cornell Department of Mathematics (there is no permalink for this):

"All those who knew Bill, especially his many students and collaborators, know that nothing can replace his insight and personality. We are all terribly saddened by this loss."

American Mathematical Society posted a short obituary: William P. Thurston, 1946-2012.


William Thurston was the greatest geometer of the last century. The word "geometry" is very fashionable since about fifty years ago, and this phrase now calls for a clarification. William P. Thurston was able to see unexpected, remarkable, beautiful pictures hidden from all other mathematicians. After he showed them to other mathematicians, they saw and appreciated them also. His thinking was predominantly visual. Perhaps, in this respect he was the greatest geometer of all times.

It is extremely difficult to convey any visual concept by the means of a conventional mathematical text, even with a lot of illustrations. Some visual concepts are too complicated or too many dimensional (here the usual 3 dimensions are often already too many) to be adequately explained by a 2-dimensional picture. Apparently, mathematics lacks a proper language to efficiently describe visions of Thurston's level of complexity and originality. Perhaps, this is the main reason why Thurston did published only sketches or partial expositions of his results (his own published explanation is different, but compatible with this one). Some of his ideas were successfully translated into the conventional language and written down by other mathematicians. But some others are not, and the results themselves are reproved using different means. Of course, Thurston's visions are more important than his theorems, and I am afraid that some were lost completely already in the last century. I hope that his students and collaborators will write down and publish everything they learned from Thurston.


William P. Thurston was looking for alternative ways to convey visual concepts; his prize-winning book ''Three-Dimensional Geometry and Topology'' is an attempt to deal with this problem, but covers mostly pre-Thurston ideas.

Creating a proper language to describe visual mathematical concepts is, may be, the most important problem for the future generations. The layman language has the same deficiency; every description of a tree or a landscape relies on the previous experience of the readers with trees and landscapes in the real world. This recourse is not available to mathematicians creating new visual concepts, and by this reason the problem is so difficult that it is extremely rarely even acknowledged as a problem.


The passing away of William Thurston created a hole in the mathematical community which cannot be filled. Everybody who was happy enough to talk with him at least ten minutes knows how remarkable person he was. We lost not only an exceptional mathematician; we lost also an exceptional person. This deepens the feelings of loss, emptiness, sadness, and sorrow.


Next post: William Thurston about the humanity and mathematics.