Note: For a Romanian translation of this post, see here.
Misha Shifman has edited a wonderful book about the mathematician Felix Berezin, which recently appeared with the title Felix Berezin: Life and Death of the Mastermind of Supermathematics. Berezin was a Soviet mathematician largely responsible for many new ideas about “supermathematics”, working out the analog for anticommuting variables of many standard concepts in analysis. Path integrals for fermions crucially use an analog of the standard integral that is now known as the Berezin integral.
Berezin began his mathematical career working with Gelfand on representation theory. While Gelfand thought very highly of him, at some point the two of them had a falling out, which is alluded to without any details in several of the contributions to this book. Since Berezin’s mother was Jewish, his professional life was often difficult due to the anti-semitism that was prevalent in the Soviet mathematical establishment. Between this and being on the outs with Gelfand, he had continual problems with things like getting his papers published, as well as being able to travel or effectively communicate with people in the West.
Tragically, Berezin died at the age of 49, under somewhat unclear circumstances on a trip to Siberia he took with a geological team. The largest segment of the book is a wonderful and touching piece by Elena Karpel, who lived with him for many years (they had a daughter together, Natasha). Karpel describes their life together in detail, as well as the circumstances following his death. It is a moving portrayal of a complex relationship of two highly intelligent and cultured people, with one of them, Berezin, extremely seriously devoted to his work, one cause of stress in his relations with Karpel. Together with contributions from his colleagues, the book gives a fascinating portrayal of the mathematical culture that Berezin was an important part of.
With his interest in quantum mechanics, quantum field theory, path integrals, and anticommuting variables, Berezin helped to transform the field of mathematical physics into something much more modern. His book written during the sixties, The Method of Second Quantization remains one of the classics of quantum field theory. I remember being especially impressed by his paper with Marinov Particle spin dynamics as the Grassmann variant of classical mechanics, which gives an amazing interpretation of the physics of a spin-1/2 particle by invoking anti-commuting variables in a very simple way. The book contains a summary of some of Berezin’s scientific work by Andrei Losev, and this article is available on-line.
The Mathematician’s Brain
Princeton University Press seems to be trying to corner the market on popular books about mathematics, bringing out in quick succession a novel about mathematics (A Certain Ambiguity), a book about The Pythagorean Theorem, and two books trying to explain what it is that mathematicians do: How Mathematicians Think by William Byers, and The Mathematician’s Brain by mathematical physicist David Ruelle. The Ruelle book is the only one of the four that I’ve had a chance to read.
The New York Sun recently published a review of The Mathematician’s Brain by David Berlinski. It’s one of the great mysteries of the popular science book business why anybody publishes the writings of Berlinski. His recent claim to fame is as an affiliate of the Discovery Institute, critic of Darwinism and proponent of Intelligent design, but he has also authored various popular books, including some on mathematics. Some web-sites claim that he has a Ph.D. in mathematics from Princeton, but it appears that the truth of the matter is that he was in the philosophy department there, writing a doctoral thesis on Wittgenstein. His writings on math and science that I’ve seen over the years have always struck me as singularly incoherent and confused.
Berlinski actually doesn’t do that bad a job with the Ruelle review, picking up on one of the things that might interest mathematicians and physicists about the book, the part about Alexandre Grothendieck (I confess to skimming some of the material explaining what mathematicians do, since I spend far too much of my life watching them do it). Ruelle has some interesting stories to tell about Grothendieck and the IHES, where they both worked for many years. The IHES was founded in the late 1950s by Leon Motchane, who had studied mathematics before going into business. Ruelle describes well the IHES during the 1960s, including the various conflicts which existed between Motchane and the IHES members, one of which ended up leading to Grothendieck’s resignation.
Ruelle also has quite a lot to say about the structure of power in mathematics, and how the desire for recognition and honors motivates people. His portrayal of mathematicians is a very well-rounded one, examining not just how they do mathematics, but how they live their lives, noting that:
But one should not forget that, besides beautiful mathematical ideas, there are many more obscure things that crawl in the mind of a mathematician.
Many of the footnotes in the back are well worth reading, such as one that tells us:
As my wife puts it, there are fewer bastards and fewer frauds among mathematicians than in the general population, but maybe also fewer amusing people!
Ruelle also tells a favorite anecdote I’ve heard from several mathematicians. The version I’ve heard is somewhat different than Ruelle’s, and goes:
At the Institute for Advanced Study in Princeton, a visitor once came up to Armand Borel and asked him
“Do you know about algebraic groups?”
Borel answered that, yes, he did. The visitor then went on
“Good. Can I ask a stupid question then?”
to which Borel responded:
“That’s two already.”
La Theorie des Cordes
A colleague brought me back from France a science fiction novel written by the Spanish writer Jose Carlos Somoza. In French the book is called La Theorie des Cordes (String Theory), but the Spanish and English versions have the title Zig Zag. The plot revolves around a discovery about string theory that allows physicists to look back into the past. It begins with some promise, describing the world of theoretical physics as seen from Spain, with references to Witten and other theorists. But it soon degenerates into a long tale revolving around a threatened attractive young female scientist. The string is somehow responsible for forcing her into sexual depravity and the prospect of nearly infinitely long and horrific bloody torture, with time suspended and no end in sight. OK, I guess maybe this does have to do with present-day particle theory, except for the sexual depravity part…
Reviews by Atiyah in the Notices
The October Notices of the AMS contains very interesting reviews by Michael Atiyah of two books about Bourbaki: Bourbaki: A Secret Society of Mathematicians by Maurice Mashaal, and The Artist and the Mathematician by Amir Aczel. Atiyah speaks from personal experience, knowing many of the members of Bourbaki and their work well, and having attended one of the Bourbaki gatherings where they hashed out the text of one of their books. He gives an excellent summary of the Bourbaki story and its place in recent mathematical history, finding the Mashaal book to be both highly readable and reliable on the facts and personalities involved. As for the Aczel book, he’s much more dubious. Aczel tries to claim an important impact of Bourbaki on sociology and structuralism via Claude Levi-Strauss, but Atiyah is not convinced by this, and takes issue with what Aczel has to say about Grothendieck, someone Atiyah knew well. Atiyah’s characterization of Grothendieck goes as follows:
I greatly admired his mathematics, his prodigious energy and drive, and his generosity with ideas, which attracted a horde of disciples. But his main characteristic, both in his mathematics and in social life, was his uncompromising nature. This was, at the same time, the cause both of his success and of his downfall. No one but Grothendieck could have taken on algebraic geometry in the full generality he adopted and seen it through to success. It required courage, even daring, total selfconfidence and immense powers of concentration and hard work. Grothendieck was a phenomenon.
But he had his weaknesses. He could navigate like no one else in the stratosphere, but he was not sure of his ground on earth—examples did not appeal to him and had to be supplied by his colleagues.
He ends with the following critical remarks
Aczel’s total endorsement of Grothendieck leads him to make such fatuous statements as: “Weil was a somewhat jealous person who clearly saw that Grothendieck was a far better mathematician than he was.” Subtle balanced judgement is clearly not Aczel’s forte, and it hardly encourages the reader to take seriously his confident and sweeping assertions in the social sciences.