Are Mathematicians Weird?

Grigory Perelman and the “Perelman Stick” 

In the fall of 1976, a strange, pudgy, and socially awkward 10-year-old was enrolled by his mother in a Soviet school math club. Like thousands of talented young people across the Soviet Union, Grigory Perelman was entering an apprenticeship in the service of the Soviet Union as a future mathematician. The after-school math club met twice a week for two hours of immersion in mathematical problem solving. In each session, the coach Sergei Rukshin, who would later attain recognition for his prolific output of mathematical superstars, assigned students the types of problems they would encounter in competitions. By the time Grigory Perelman entered the All-Soviet Olympiad in 1980, he had four years of Rukshin’s math club training behind him, and he came to the competition expecting nothing less than first place. When the Olympiad results were announced, Perelman came second. This was, for him, a devastating blow that prompted him to double down on his efforts. His biographer, Masha Gessen observed:

From then on he practiced ceaselessly. While for other kids, life was divided into school and leisure, for Perelman it was split into time devoted to solving problems without disruption and the rest of the time.

On July 9 and 10, 1982, the International Mathematics Olympiad (IMO) was held in Budapest, Hungary. Grigory Perelman, was just 16 years old and had logged 6 years of intense problem solving under the tutelage of coach Rukshin. He came ready for the challenge. In two 4½-hour sessions spread over those two days, he grappled with the full set of 6 challenging problems. When the marking of the solutions was completed 3 days later, the name Grigory Perelman stood at the top of the list as the winner of the gold medal at the IMO. This gold medal earned him admission to Mathmech–Leningrad University’s Mathematics and Mechanics department.

Five years later in the fall of 1987, Grisha (as Grigory was known to his small circle of friends) Perelman had finished his undergraduate work and was embarking on his graduate studies at the Leningrad site of the prestigious Steklov Mathematics Institute. His assigned dissertation advisor was the famous Russian topologist Alexander Alexandrov. It was a time of historic change in the USSR. Mikhail Gorbachev was preparing to introduce reforms known as perestroika that would lighten travel restrictions on Soviet citizens, allowing them to connect with the outside world. Andrei Sakharov, the Nobel Laureate we met in chapter 3, who had been held under house arrest for years, was released and a thaw in the Cold War was becoming apparent. In the next four years, Grisha was able to immerse himself in the study of topology, defend his dissertation, and travel to France and the US to share ideas with the international mathematics community. 

In 1994, he published a short, concise proof of the Soul Conjecture, a conjecture in topology that had been proposed 22 years earlier, but unproven until yielding to what his colleagues called “Perelman’s stick.” This moniker referred to the imagined weapon that Perelman was believed to wield in dealing the death blow to a hard problem. Colleagues reported that Grisha would become absorbed in a problem until he could reduce it to its essence and then deliver the killer blow. The Perelman stick was respected, but it had not yet been wielded with its full power. 

The Poincaré Conjecture

In the year 2000, the Clay Foundation of Cambridge, Massachusetts launched the new millennium by posting a one-million-dollar prize for the solution of any one of seven so-called millennium problems in mathematics. Each of these problems is considered a “Mount Everest” of mathematics problems because each has eluded proof by the greatest mathematicians and may remain beyond the reach of the human mind for centuries to come.

In 1904, Henri Poincaré was studying a special topological property of the n-dimensional sphere, (in an n + 1 dimensional space) that we’ll call the “shrink-to-a-point” property. For example, if we place an elastic band on the surface of a ball, then no matter where the band is placed, we can shrink the elastic band to a circle of zero radius and remain on the surface of the ball. However, a donut-shaped surface, that mathematicians call a torus, does not have this “shrink-to-a-point” property, because an elastic band placed as shown, cannot be contracted to a point without leaving the surface. Poincaré conjectured that any n-dimensional surface with the “shrink-to-a-point” property is an n-dimensional sphere, or can be transformed into such a sphere by bending or stretching. However, Poincaré was unable to prove his conjecture and it remained unproved for a century in spite of a multitude of failed attempts by some of the world’s greatest mathematicians. Consequently, it became one of the seven millennium problems called the Poincaré Conjecture. Stephen Smale had proved it in 1960 for n ≥ 5 and then in 1991 another American mathematician, Michael Freedman had proved the conjecture for n = 4 and won the Fields Medal. However, the Poincaré Conjecture remained unproved for n = 3. 

By the time the Millennium prize was posted, Grisha Perelman had returned to Russia and disappeared from the world mathematics community. He stopped answering emails from friends and colleagues and became a recluse. In the words of his biographer Masha Gessen:

Even Gromov [his mentor] heard nothing from him and assumed he was still stuck on Alexandrov spaces–in other words, that he had joined the sizable ranks of talented mathematicians who did brilliant early work and then disappeared into the black hole of some impossible problem.

Then, on November 12, 2002 at 5:09 EST an email sent to a dozen US mathematicians read: May I bring to your attention my paper in arXiv math.DG 0211159, followed by an abstract of the paper he posted the day before on the arXiv.org website hosted by Cornell University Library in Ithaca, NY. The email was signed, “Best regards, Grisha.” In this email, Grisha Perelman didn’t indicate that he had solved the Poincaré Conjecture; he merely let the mathematics speak for itself. 

The email launched a tsunami of cyberspace communication that swept through the mathematics community in a giant wave of speculation. A dumbfounded gaggle of the world’s greatest intellects were engaged in frantic efforts to make sense of what it all meant. Ensuring that it was, indeed, a proof with no fundamental flaws would take months or possibly years. Mathematicians who were working on the Poincaré Conjecture had mixed feelings about this epic event. If in fact, Grisha Perelman had beaten them to the finish line, their life’s work would evaporate instantly. This was the high-stakes risk that haunted all mathematicians who dedicated substantial chunks of their lives to the search for the proof of a difficult conjecture. As a minor consolation, these mathematicians could play a supportive role in checking the proof and in helping people understand the implications of this new theorem.

Weird and Weirder

Non-mathematicians might think that the monetary reward offered by the Clay Institute was the incentive for proving the Poincaré Conjecture, but mathematicians understand the real incentive. If you’re a mathematician, solving a problem of this magnitude catapults you into the pantheon of the gods. You become an immediate celebrity in the community of all the people who matter to you. Brilliant people come to your lectures just to see what a “genius” looks like. And all the respect and attention that you wanted from early childhood comes in gushing waves. Universities flood you with offers to join their faculties as a professor who is unencumbered by teaching or research duties. They seek only the prestige of your name. So it was with Andrew Wiles when he solved Fermat’s Last Theorem and it would be with Grigory Perelman when his proof was ultimately validated. 

However, Grisha was somewhat different. Though it is generally accepted that mathematicians have a “weirdness” relative to the rest of the population, Grisha Perelman was perceived as weird by those with a high “weirdness coefficient.” Though he may have enjoyed the respect of colleagues, he was self-contained. His eight or nine-year struggle was between himself and the problem; that was the battlefield and that was all that mattered. In this sense, he was the epitome of a pure mathematician. Knowing he had solved the Poincaré Conjecture was the ultimate reward and validation. The external trappings were irrelevant. Solving hard math problems was his shtick.

Following the email in November 2002, Perelman was flooded with invitations to speak in the US at conferences, in colloquia, and at informal meetings of his colleagues. He agreed to come for only a month to speak on the Poincaré Conjecture, and he followed through on his promise, arriving in the United States in early April 2003, and returning to Russia at the end of that month. Grisha was described as cordial and friendly with his colleagues but contemptuous of the press and the American focus on extrinsic rewards. He perceived the process of submitting publications to journals supported by paid subscriptions and the posting of monetary rewards as incentives to mathematical scholarship as vulgar. As a purist, he seemed to believe that the mathematics community in America had prostituted itself by promoting its ambitions in the media. 

By the middle of 2005, the mathematics community had reached consensus that Perelman’s proof was, indeed, valid. To celebrate his triumph, the organizers of the International Congress of Mathematicians (ICM) selected him to receive a Fields Medal–the highest honor that a mathematician can receive–to be announced and presented at the ICM in Madrid on August 22, 2006. Traditionally, the winners of the Fields Medal (presented every four years) are not informed of their award until the Conference. However, the governing board of the International Mathematical Union wanted to ensure that Perelman would attend the Congress and accept his medal, so they sent their President, Sir John Ball to St. Petersburg to inform Grisha of his award and to persuade him to attend. The unyielding Perelman refused to attend or even accept delivery of the Fields Medal. It was the first time anyone had ever refused this prestigious award. 

In March 2010, the Millennium Prize of $1,000,000 was awarded to Perelman in absentia. As expected, he refused the money and the invitation to attend any ceremony in his honor stating, “the main reason is my disagreement with the organized mathematical community. I don’t like their decisions, I consider them unjust.”¹⁷ Grigory Perelman not only turned his back on the mathematics community, but walked away from virtually all former friends and associates, declaring that he was abandoning mathematics forever. However, there had been some speculation that he may be working on another Millennium problem–a solution to the Navier-Stokes fluid flow equations. But this is now in doubt. A picture of Grisha with his mother (he never married) accessible at: https://intelligenceandiq.quora.com  suggests that he is semi-retired.