Perfect Rigor in NY Review of Books

Earlier I recommended Masha Gessen's book Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. In case you haven't read it, there is a nice review and summary at the NY Review of Books.

... Up until March of this year, there remained one more chapter to the Perelman saga. Would he accept the one-million-dollar prize promised by the Clay Mathematics Institute for solving one of the seven so-called Millennium Problems? While the rules say that a proof must appear in a peer-reviewed mathematics journal (not just in an Internet posting), the mathematicians mentioned above have published papers in such journals expounding and amplifying the proof. Surely Perelman deserves the prize, which he was finally and officially offered on March 18.

Five days later, on March 23, Perelman rejected the Clay prize. He reportedly said through the closed door to his spartan apartment, "I have all I want." The comments he made after rejecting the Fields Medal probably reflect his present state of mind as well:

I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful. That is why I don't want to have everybody looking at me.

Some might argue that monetary awards for mathematical work are inappropriate, or that the Poincaré Conjecture is of little practical value and not worth the one-million-dollar prize. The aesthetic and epistemic value of the proof is priceless, however, and it may eventually yield more earthly consequences as well. As for the size of the award—how many no-name hacks are there on Wall Street who make a million dollars or more not just once but every year, and contribute exactly what? Whether Perelman has practical need for the money or not, he could use it to help support his mother or mathematicians of his liking, or to advance the kind of education conceived by Andrei Kolmogorov, or for some purpose only he could imagine. Reconsider your decision, Grisha.

Two book recommendations

Very high recommendations for the following books, which I stockpiled in advance of my trip to China, but ended up reading most of before departure...

The Lost Books of the Odyssey by Zachary Mason, a Borges-ian retelling of the Odyssey in 44 short stories.

Perfect Rigor by Masha Gessen, on Grisha Perelman and the proof of the Poincare / Geometrization Conjecture. I already knew a bit about the completion of Hamilton's program thanks to colleagues who work on Ricci flow. What I found most interesting in the book is the insider description of the Soviet math system, including the nurturing of young talents and the training for competitions like the IMO.

Galison: Poincare and Einstein

A colleague and I recently discussed Peter Galison and his book Einstein's Clocks, Poincare's Maps: Empires of Time . (See also here and here.) The book explores how practical concerns of the era (in particular, clock synchronization -- important for longitudinal navigation as well as for the European train system) influenced the discovery of special relativity.

Both my friend and I are great admirers of Galison. After earning his doctorate in the history of science, he wrote a second dissertation in particle theory under Howard Georgi while a Junior Fellow at Harvard. Other than particle theorists turned science historians like Sam Schweber or Abraham Pais (see here and here), I can't think of anyone more qualified to work on the (underdeveloped) history of modern physics.

When I learned about special relativity as a kid, I first went through a phase of suspicion about Einstein's operational approach -- how could one be sure, I wondered, that light beams were the best primitive for the operation and synchronization of clocks? After I accepted this idea, I was shocked that someone could be so imaginative as to come up with his clever gedanken experiments, involving moving trains, light beams, lattices of clocks. I thought to myself -- I could have never invented that! It was only much later that I learned about his patent office work on clocks and how synchronization of time between distant rail stations was an important practical problem of the day. I agree completely with Galison that practical concerns had a strong influence on both Einstein's and Poincare's thinking.

NYTimes: ...Einstein's relativity has long been regarded by scholars as a monument to the power of abstract thought. But if Dr. Peter Galison, 48 -- a Harvard professor of the history of science and of physics, a pilot, art lover and nascent filmmaker -- is right, physics and Einstein have flourished more in their connections to the world than in any ivory tower aloofness. And one clue to the origin of relativity can be found in something as mundane and practical as a 19th-century train schedule. ''It's in as plain sight as it could possibly be,'' he said.

As Dr. Galison relates, before the advent of factories began to standardize life, and railroad systems with crisscrossing tracks made it imperative to know which train was where and when, there were too many times, one for every village.

In the last part of the 19th century, the coordination of clocks and the standardization of time had engaged the passions of nations, business leaders, astronomers and philosophers. The patent office in Bern, Switzerland, where Einstein worked, was a clearinghouse for patents on the synchronization of clocks.

In New England, the Harvard and Yale observatories were competing to sell time signals to the public, and in Paris pneumatic tubes snaked under the streets to synchronize the city's clocks with blasts of air. Far from being a bit of abstraction by a loner genius, the clocks that Einstein used as examples in his papers were as familiar then as computers are today.

...In addition to all his high-flown academic activities, Poincaré was immersed in practical work. He was a mining inspector, for example. Most important, he was deeply involved with the French Board of Longitude, even serving as president, sending teams of soldiers and surveyors across the oceans to map the far-flung empire.

Coordinated clocks were central to this enterprise. To measure the longitude of some mountain or port or gold mine in the New World, it was necessary to measure the difference between the time some star crossed the meridian there and the time it did back in Paris. The leaders and rivals in filling in this ''electric world map,'' as Dr. Galison calls it, were England and France, even though for several years they were embarrassingly unable to agree on the distance between their own principal observatories, Greenwich and Paris. Paris lost out to Greenwich as the locus of zero longitude, but in 1909 Poincaré used the Eiffel Tower to broadcast time signals to the world.

...In his papers Einstein was always using modern machines to illustrate his ideas, Dr. Galison noted. ''There is something wonderful about Einstein invoking trains and telegraphs to get a transformation of space-time, Poincaré turning the Eiffel Tower into a radio,'' Dr. Galison said.

''In the long run I think what's happened to them is that we, partly through our own doing and partly through our doing to them, removed these physicists from the concrete situations that they were involved in. And I think in a way lose some of the fascination that these ideas had for them and still could have for us in a way.''

It's our loss, he said.

Galison: "My question is not how different scientific communities pass like ships in the night,'' he wrote in Image and Logic. ''It is rather how, given the extraordinary diversity of the participants in physics -- cryogenic engineers, radio chemists, algebraic topologists, prototype tinkerers, computer wizards, quantum field theorists -- they speak to each other at all."

Perelmania

The New Yorker has a nice article on Perelman, his proof of the geometrization and Poincare conjectures, the politics surrounding the proof, and S.T. Yau's antics. If the article is correct (see end of excerpts below), Yau's maneuvering is directly related to Perelman's declining the Fields medal. Note he has not said whether he would decline the $1 million Clay Foundation prize ;-)

One of the authors is Sylvia Nasar, who wrote A Beautiful Mind. The writers actually went to St. Petersberg to track down the elusive genius, and have reconstructed the history behind his decade of work to produce the proof.

...In 1982, the year that Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and the gold medal at the International Mathematical Olympiad, in Budapest.

...In 1993, he began a two-year fellowship at Berkeley. While he was there, Hamilton gave several talks on campus, and in one he mentioned that he was working on the Poincaré. Hamilton’s Ricci-flow strategy was extremely technical and tricky to execute. After one of his talks at Berkeley, he told Perelman about his biggest obstacle. As a space is smoothed under the Ricci flow, some regions deform into what mathematicians refer to as “singularities.” Some regions, called “necks,” become attenuated areas of infinite density. More troubling to Hamilton was a kind of singularity he called the “cigar.” If cigars formed, Hamilton worried, it might be impossible to achieve uniform geometry. Perelman realized that a paper he had written on Alexandrov spaces might help Hamilton prove Thurston’s conjecture—and the Poincaré—once Hamilton solved the cigar problem. “At some point, I asked Hamilton if he knew a certain collapsing result that I had proved but not published—which turned out to be very useful,” Perelman said. “Later, I realized that he didn’t understand what I was talking about.” Dan Stroock, of M.I.T., said, “Perelman may have learned stuff from Yau and Hamilton, but, at the time, they were not learning from him.”

By the end of his first year at Berkeley, Perelman had written several strikingly original papers. He was asked to give a lecture at the 1994 I.M.U. congress, in Zurich, and invited to apply for jobs at Stanford, Princeton, the Institute for Advanced Study, and the University of Tel Aviv. Like Yau, Perelman was a formidable problem solver. Instead of spending years constructing an intricate theoretical framework, or defining new areas of research, he focussed on obtaining particular results. According to Mikhail Gromov, a renowned Russian geometer who has collaborated with Perelman, he had been trying to overcome a technical difficulty relating to Alexandrov spaces and had apparently been stumped. “He couldn’t do it,” Gromov said. “It was hopeless.”

Perelman told us that he liked to work on several problems at once. At Berkeley, however, he found himself returning again and again to Hamilton’s Ricci-flow equation and the problem that Hamilton thought he could solve with it. Some of Perelman’s friends noticed that he was becoming more and more ascetic. Visitors from St. Petersburg who stayed in his apartment were struck by how sparsely furnished it was. Others worried that he seemed to want to reduce life to a set of rigid axioms. When a member of a hiring committee at Stanford asked him for a C.V. to include with requests for letters of recommendation, Perelman balked. “If they know my work, they don’t need my C.V.,” he said. “If they need my C.V., they don’t know my work.”

...Perelman had posted a thirty-nine-page paper entitled “The Entropy Formula for the Ricci Flow and Its Geometric Applications,” on arXiv.org, a Web site used by mathematicians to post preprints—articles awaiting publication in refereed journals. He then e-mailed an abstract of his paper to a dozen mathematicians in the United States—including Hamilton, Tian, and Yau—none of whom had heard from him for years. In the abstract, he explained that he had written “a sketch of an eclectic proof” of the geometrization conjecture.

Perelman had not mentioned the proof or shown it to anyone. “I didn’t have any friends with whom I could discuss this,” he said in St. Petersburg. “I didn’t want to discuss my work with someone I didn’t trust.” Andrew Wiles had also kept the fact that he was working on Fermat’s last theorem a secret, but he had had a colleague vet the proof before making it public. Perelman, by casually posting a proof on the Internet of one of the most famous problems in mathematics, was not just flouting academic convention but taking a considerable risk. If the proof was flawed, he would be publicly humiliated, and there would be no way to prevent another mathematician from fixing any errors and claiming victory. But Perelman said he was not particularly concerned. “My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased,” he said. “I never set out to be the sole solver of the Poincaré.”

Gang Tian was in his office at M.I.T. when he received Perelman’s e-mail. He and Perelman had been friendly in 1992, when they were both at N.Y.U. and had attended the same weekly math seminar in Princeton. “I immediately realized its importance,” Tian said of Perelman’s paper. Tian began to read the paper and discuss it with colleagues, who were equally enthusiastic.

On November 19th, Vitali Kapovitch, a geometer, sent Perelman an e-mail:

Hi Grisha, Sorry to bother you but a lot of people are asking me about your preprint “The entropy formula for the Ricci . . .” Do I understand it correctly that while you cannot yet do all the steps in the Hamilton program you can do enough so that using some collapsing results you can prove geometrization? Vitali.

Perelman’s response, the next day, was terse: “That’s correct. Grisha.”

...Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. ... “It is not people who break ethical standards who are regarded as aliens,” he said. “It is people like me who are isolated.” We asked him whether he had read Cao and Zhu’s paper. “It is not clear to me what new contribution did they make,” he said. “Apparently, Zhu did not quite understand the argument and reworked it.” As for Yau, Perelman said, “I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.”

The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. “As long as I was not conspicuous, I had a choice,” Perelman explained. “Either to make some ugly thing”—a fuss about the math community’s lack of integrity—“or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit.” We asked Perelman whether, by refusing the Fields and withdrawing from his profession, he was eliminating any possibility of influencing the discipline. “I am not a politician!” he replied, angrily. Perelman would not say whether his objection to awards extended to the Clay Institute’s million-dollar prize. “I’m not going to decide whether to accept the prize until it is offered,” he said.

Mikhail Gromov, the Russian geometer, said that he understood Perelman’s logic: “To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness.” Others might view Perelman’s refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. “The ideal scientist does science and cares about nothing else,” he said. “He wants to live this ideal. Now, I don’t think he really lives on this ideal plane. But he wants to.”