# Mathematical sincerity

SPRAGGETT ON CHESS

Masha Gessen

Moscow

What was needed was a small army of mathematicians to recalculate speeds and distances to let the air force hit its targets.

In the years following Stalin’s death in 1953, the Iron Curtain began to open a tiny crack—not quite enough to facilitate much-needed conversation with non-Soviet mathematicians but enough to show off some of Soviet mathematics’ proudest achievements.

–Charles Forelle

For example, the American model may not be able to produce a breakthrough like the proof of the Poincaré Conjecture, carried out by the St. Petersburg mathematician

A child of the Soviet math counterculture, he still held a singular truth to be self-evident: Math as it ought to be practiced, math as the ultimate flight of the imagination, is something money can’t buy.

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician and theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

**Russia’s Conquering Zeros**

**The strength of post-Soviet math stems from decades of lonely productivity**

Masha Gessen

Moscow

It may be no accident that, while some of the best American mathematical minds worked to solve one of the century’s hardest problems—the

**Poincaré Conjecture**—it was a Russian mathematician working in Russia who, early in this decade, finally triumphed.Decades before, in the Soviet Union, math placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand; and, worst of all, mathematics lay claim to singular and knowable truths—when the regime had staked its own legitimacy on its own singular truth. All this made mathematicians suspect. Still, math escaped the purges, show trials and rule by decree that decimated other Soviet sciences.

Three factors saved math. First, Russian math happened to be uncommonly strong right when it might have suffered the most, in the 1930s. Second, math proved too obscure for the sort of meddling Joseph Stalin most liked to exercise: It was simply too difficult to ignite a passionate debate about something as inaccessible as the objective nature of natural numbers (although just such a campaign was attempted). And third, at a critical moment math proved immensely useful to the state.

Three weeks after Nazi Germany invaded the Soviet Union in June 1941, the Soviet air force had been bombed out of existence. The Russian military set about retrofitting civilian airplanes for use as bombers. The problem was, the civilian airplanes were much slower than the military ones, rendering moot everything the military knew about aim.

What was needed was a small army of mathematicians to recalculate speeds and distances to let the air force hit its targets.

The greatest Russian mathematician of the 20th century, Andrei Kolmogorov, led a classroom of students, armed with adding machines, in recalculating the Red Army’s bombing and artillery tables. Then he set about creating a new system of statistical control and prediction for the Soviet military.

Following the war, the Soviets invested heavily in high-tech military research, building over 40 cities where scientists and mathematicians worked in secret. The urgency of the mobilization recalled the Manhattan Project—only much bigger and lasting much longer. Estimates of the number of people engaged in the Soviet arms effort in the second half of the century range up to 12 million people, with a couple million of them employed by military-research institutions.

These jobs spelled nearly total scientific isolation: For defense employees, any contact with foreigners would be considered treasonous rather than simply suspect. In addition, research towns provided comfortably cloistered social environments but no possibility for outside intellectual contact. The Soviet Union managed to hide some of its best mathematical minds away in plain sight.

In the years following Stalin’s death in 1953, the Iron Curtain began to open a tiny crack—not quite enough to facilitate much-needed conversation with non-Soviet mathematicians but enough to show off some of Soviet mathematics’ proudest achievements.

By the 1970s, a Soviet math establishment had taken shape. A totalitarian system within a totalitarian system, it provided its members not only with work and money but also with apartments, food, and transportation. It determined where they lived and when, where, and how they traveled for work or pleasure. To those in the fold, it was a controlling and strict but caring mother: Her children were undeniably privileged.

Even for members of the math establishment, though, there were always too few good apartments, too many people wanting to travel to a conference. So it was a vicious, back-stabbing little world, shaped by intrigue, denunciations and unfair competition.

Then there were those who could never join the establishment: those who happened to be born Jewish or female, those who had had the wrong advisers at university or those who could not force themselves to join the Party. For these people,

*“the most they could hope for was being able to defend their doctoral dissertation at some institute in Minsk, if they could secure connections there,”*says Sergei Gelfand, publisher of the American Mathematical Society—who also happens to be the son of one of Russia’s top 20th-century mathematicians, Israel Gelfand, a student of Mr. Kolmogorov. Some Western mathematicians, Sergei Gelfand adds,*“even came for an extended stay because they realized there were a lot of talented people. This was unofficial mathematics.”*The Riemann Hypothesis

*To the enduring befuddlement of mathematicians, prime numbers—numbers divisible only by themselves and 1—exhibit no pattern at all: 2, 3, 5, 7, 11, 13 are the first few. They aren’t evenly spaced but get scarcer the further out you go. No formula can tell you what the next one will be. In 1859, the German mathematician Bernhard Riemann discovered that a function—known now as the Riemann zeta function (expressed in the graphic above)—appeared to give signposts to where primes lie in the great field of numbers. It provided some order to the mystery. Riemann conjectured that these key signposts—”zeros” of the function—all lie on a single straight line out to infinity, that none are flung off in strange places. In the 150 years since, no one has proved his hypothesis. To a mathematician, the hypothesis looks like this: All non-trivial zeros of the Riemann zeta function have a real part equal to ½.*

One such visitor was Dusa McDuff, then a British algebraist and now a professor emerita at the State University of New York at Stony Brook. She studied with the older Mr. Gelfand for six months, and credits this experience to opening her eyes both to what mathematics really is:

*“It was a wonderful education… Gelfand amazed me by talking of mathematics as though it were poetry.”*In the mathematical counterculture, math

*“was almost a hobby,”*recalls Sergei Gelfand.*“So you could spend your time doing things that would not be useful to anyone for the nearest decade.”*Mathematicians called it*“math for math’s sake.”*There was no material reward in this—no tenure, no money, no apartments, no foreign travel; all they stood to gain was the respect of their peers.Math not only held out the promise of intellectual work without state interference (if also without its support) but also something found nowhere else in late-Soviet society: a knowable singular truth.

*“If I had been free to choose any profession, I would have become a literary critic,”*says Georgii Shabat, a well-known Moscow mathematician.*“But I wanted to work, not spend my life fighting the censors.”*The search for that truth could take long years—but in the late Soviet Union, time seemed to stand still.When it all collapsed, the state stopped investing in math and holding its mathematicians hostage. It’s hard to say which of these two factors did more to send Russian mathematicians to the West, primarily the U.S., but leave they did, in what was probably one of the biggest outflows of brainpower the world has ever known. Even the older Mr. Gelfand moved to the U.S. and taught at Rutgers University for nearly 20 years, almost until his death in October at the age of 96. The flow is probably unstoppable by now: A promising graduate student in Moscow or St. Petersburg, unable to find a suitable academic adviser at home, is most likely to follow the trail to the U.S.

But the math culture they find in America, while less back-stabbing than that of the Soviet math establishment, is far from the meritocratic ideal that Russia’s unofficial math world had taught themto expect. American math culture has intellectual rigor but also suffers from allegations of favoritism, small-time competitiveness, occasional plagiarism scandals, as well as the usual tenure battles, funding pressures and administrative chores that characterize American academic life. This culture offers the kinds of opportunities for professional communication that a Soviet mathematician could hardly have dreamed of, but it doesn’t foster the sort of luxurious, timeless creative work that was typical of the Soviet math counterculture.

For example, the American model may not be able to produce a breakthrough like the proof of the Poincaré Conjecture, carried out by the St. Petersburg mathematician

**Grigory Perelman**.Mr. Perelman came to the United States as a young postdoctoral student in the early 1990s and immediately decided that America was math heaven; he wrote home demanding that his mother and his younger sister, a budding mathematician, move here. But three years later, when his postdoc hiatus was over and he was faced with the pressures of securing an academic position, he returned home, disillusioned.

In St. Petersburg he went on the (admittedly modest) payroll of the math research institute, where he showed up infrequently and generally kept to himself for almost seven years, one of the greatest mathematical discoveries of at least the last hundred years. It’s all but impossible to imagine an American institution that could have provided Mr. Perelman with this kind of near-solitary existence, free of teaching and publishing obligations.

After posting his proof on the Web, Mr. Perelman traveled to the U.S. in the spring of 2003, to lecture at a couple of East Coast universities. He was immediately showered with offers of professorial appointments and research money, and, by all accounts,

**he found these offers gravely insulting**, as he believes the monetization of achievement is the ultimate insult to mathematics.So profound was his disappointment with the rewards he was offered that, I believe, it contributed a great deal to his subsequent decision to quit mathematics altogether, along with the people who practice it. (He now lives with his mother on the outskirts of St. Petersburg.)

*Masha Gessen’s latest book is*

**“Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century,”**a story of Grigory Perelman and the Poincaré Conjecture. She lives in Moscow and is the author of three previous books.In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal. The claim concerns a space that locally looks like ordinary three dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is just a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians,

**Grigori Perelman**sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have since verified the correctness of Perelman’s proof.**THE FIELDS MEDAL**

The Poincaré conjecture was, before being proven, one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a

**$1,000,000 prize for the first correct solution**. Perelman’s work survived review and was confirmed in 2006, leading to his being offered a Fields Medal,**which he declined**. The Poincaré conjecture remains the only solved Millennium problem.On December 22, 2006, the journal Science honored Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year,” the first time this had been bestowed in the area of mathematics.

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician and theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is considered to be one of the founders of the field of topology.

Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell’s equations, an important step in the formulation of the theory of special relativity.

The Poincaré group used in physics and mathematics was named after him.

Grigori Yakovlevich Perelman born 13 June 1966 in Leningrad, USSR (now St. Petersburg, Russia), sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology. In particular, he proved Thurston’s geometrization conjecture. This solves in the affirmative the famous Poincaré conjecture, posed in 1904 and regarded as one of the most important and difficult open problems in mathematics until it was solved.

In August 2006, Perelman was awarded the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” Perelman declined to accept the award or to appear at the congress.

On 22 December 2006, the journal Science recognized Perelman’s proof of the Poincaré Conjecture as the scientific “Breakthrough of the Year,” the first such recognition in the area of mathematics

Masha Gessen (born 1967) is a Russian journalist and author.

Born into an Ashkenazi Jewish family in Russia, in 1981 she moved with her family to the United States, returning to Moscow in 1991. She writes in both Russian and English, and has contributed to The New Republic, New Statesman, Granta and Slate. Gessen is the Russian correspondent for US News & World Report. She lives in Moscow.