The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. — George Bernard Shaw (attributed)
Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century
Boston/New York, 2009
The Poincare Conjecture: In Search of the Shape of the Universe
Walker and Company
New York, 2007
On November 11, 2002, Grigory Perelman, a Russian mathematician known to his friends as “Grisha”, posted a research paper to the www.arXiv.org preprint server containing, amongst other things, the outline of a proof of the Poincaré Conjecture, a famous conjecture in topology first articulated in 1904 by the great mathematician Henri Poincaré. Dr. Perelman also e-mailed a few selected mathematicians directly, drawing attention to his somewhat curious paper. This rapidly created a stir as the mathematicians realized that he might well have proven the Poincaré Conjecture, an extremely difficult problem that had eluded the talents of many top mathematicians including Poincaré. Perelman went on to post two more papers to arXiv.org elaborating his proof. The Clay Institute, which had offered a prize of $1 million for the proof (or disproof) of the Poincaré Conjecture, funded two teams of mathematicians to verify Perelman’s proof. The National Science Foundation also funded efforts to verify and expand upon the proof. By 2006, the “consensus” in the mathematical community was that Dr. Perelman had proved the Poincaré Conjecture. Dr. Perelman was offered the prestigious Fields Medal, close to the Nobel Prize of mathematics. He became the first mathematician to decline the Fields for reasons that remain somewhat unclear.
Two recent books attempt to tell the story of Grigory Perelman and the Poincaré Conjecture. Masha Gessen’s Perfect Rigor is the first biography of the elusive and enigmatic Perelman. It gives a great deal of information about the world of Soviet mathematics in which Perelman grew up and Perelman’s life to date. The author was unable to interview Perelman who has declined nearly all interviews; he has given an interview to Sylvia Nasar and David Gruber for their New Yorker article “Manifold Destiny“, about which more later. The book suffers from an unremittingly hostile, perhaps jealous, view of the unusual Dr. Perelman, who is variously portrayed as extremely naive, weird, and possibly mentally ill.
Dr. Perelman’s father was an electrical engineer and his mother a mathematics teacher at a Soviet trade school. His mother apparently had a strong interest in mathematics and almost pursued a doctorate before marrying his father. Perelman appears to have been involved in mathematics at an early age and joined a competitive math club. He competed and won a gold medal at the International Math Olympiad in Budapest, Hungary in 1982 at the age of 16. He attended a special math and physics school, Leningrad Secondary School #239, usually identified as “School 239” in Perfect Rigor. He then became a student at Leningrad State University. In 1987, he became a graduate student at the Leningrad (subsequently the St. Petersburg) branch of the Steklov Mathematical Institute, the mathematics division of the Soviet (now Russian) Academy of Sciences. The mathematician Yuri Burago was his adviser. Perelman defended his dissertation in 1990. He continued to work at the Steklov Institute until 1992, publishing a number of papers in Russian and American mathematical journals.
In the fall of 1992, Perelman came to the United States for a semester at the Courant Institute at New York University and then another semester at the State University of New York Stony Brook in early 1993. At New York University, he met and may have become friends with the mathematician Gang Tian. Perelman and Gang Tian traveled together from NYU to the Institute for Advanced Study at Princeton to listen to mathematics lectures. Then, Perelman became a prestigious Miller Fellow at Berkeley. During this time he proved the Soul Conjecture, a difficult problem in topology. His Miller Fellowship ended in 1995. He received several job offers from a number of top universities. However, he wanted a tenured position. His job offers appear to have been untenured, tenure-track positions. He returned to Russia and the Steklov Institute in 1995 where he was part of the Mathematical Physics group, dropping almost entirely out of sight, publishing nothing. He appears to have spent the next seven years working on the Poincaré conjecture. In 2002, he stunned the mathematical world by posting his proof to the Internet, flouting tradition by declining to submit the proof to a peer reviewed mathematics journal. The Clay Institute would fund mathematicians John Morgan and Gang Tian (Perelman’s friend or acquaintance at NYU) as well as a separate team at the University of Michigan to verify Perelman’s work in the form of a peer reviewed academic book.
In 2006, the prominent mathematician Shing-Tung Yau and two of his former students argued that Perelman had published an incomplete proof which they “fixed” in a lengthy paper published in the Asian Journal of Mathematics. At this point the elusive Dr. Perelman appears to have struck back with a vengeance, possibly exhibiting something other than the naivete imputed in Pefect Rigor. Perelman granted a rare interview to Sylvia Nasar, best known as author of A Beautiful Mind about the mathematician John Forbes Nash, and David Gruber for an article in the New Yorker magazine, “Manifold Destiny,” which all but openly accused Yau and his former students of blatant plagiarism.
The article quotes Perelman attributing his decision to decline the Fields medal and withdraw from the mathematics profession to the low ethical standards of the profession (in his opinion). The article also discusses the alleged rivalry between Yau and his former student Gang Tian, Perelman’s acquaintance from NYU and co-author with John Morgan of the book on Perelman’s proof. Yau threatened legal action against the New Yorker which stood by its story. Yau soon appears to have retreated under a storm of negative publicity and criticism within the mathematics “community”.
By most accounts, Perelman is an unusual person. He left his job at the Steklov Institute and apparently resides with his aging mother in her apartment in St. Petersburg. He has reportedly indicated that he is no longer interested in mathematics and generally refuses interviews, prizes, and so forth. It is not unlikely that many prominent research universities and institutions would fall over themselves to offer him a tenured professorship or something similar if he expressed any interest. It remains to be seen whether he will decline the Clay Institute’s $1 million prize if offered. Without knowing more about Perelman and his adventures in mathematics than can be found in Perfect Rigor or other accounts to date, it is difficult to draw firm conclusions about the man or even his discovery.
Notwithstanding, a few thoughts come to mind. Perfect Rigor and some other accounts implicitly criticize Perelman for his decision to turn down the job offers in 1995 and return to the Steklov Institute, imputing arrogance or just plain nuttiness. Some mathematicians and scientists would kill for some of the offers that Perelman turned down. Most major breakthroughs take a long time, usually five years or more. Perelman spent at least seven years on the Poincaré conjecture and he probably was working on it while in the United States. Most tenure track positions involve a seven year period. The assistant professor is up for review typically in six years; he or she usually must produce allegedly ground breaking work within six years. If he or she is denied tenure, he or she has one year, the seventh year, to find another job. Most assistant professors have acquired a spouse and small children by this time. There is considerable pressure to produce research papers, write grant proposals and raise money. Perelman apparently published nothing from 1995 until 2002. He most likely would not have gotten tenure had he tried to do this at any of the jobs that he turned down in 1995.
There appears to be a long history of mathematicians developing serious psychological problems. The aforementioned John Forbes Nash succumbed to mental illness, diagnosed as paranoid schizophrenia, and was well known to Princeton students for wandering around campus scribbling incomprehensible formulas on blackboards. Kurt Gödel developed psychological problems and allegedly starved himself to death. Georg Cantor became increasingly erratic as he got older. There are many anecdotal accounts of high levels of concentration and mental efforts sustained over months or years resulting in a kind of mental exhaustion and other problems. Both the western and eastern literature of meditation, which often involves prolonged concentration, contain warnings about various adverse psychological effects including anxiety attacks and hallucinations. Disillusioned former adherents of various meditation movements or “cults” have alleged serious adverse effects of heavy meditation, meaning many hours per day every day, similar to those recounted in ancient traditional sources on meditation. Although computer programming can be exhilarating, many programmers appear to experience mental exhaustion and “burnout” after lengthy programming projects involving high levels of sustained concentration.
In engineering there is an adage: “if you are one step ahead, you are a genius; if you are two steps ahead, you are an idiot!” Perfect Rigor portrays Perelman as astonishingly naive, protected from the “real world” by the bizarre Soviet mathematical system. While this may have some truth, a number of Perelman’s actions may exhibit much foresight, like a champion chess player sacrificing a piece for subsequent gain. Is pretending not to notice the alleged anti-Semitism (Perelman is a Russian Jew) in the Soviet mathematical system naive or politically astute? Declining the Fields medal, as some have noted, attracted enormous attention to Perelman. He is now one of the best known recipients (or non-recipients in this case) of the Fields Medal. It also gave him a great deal of moral authority which he seems to have used effectively to fend off Shing-Tung Yau’s alleged attempt to steal credit for proving the Poincaré Conjecture. Refusing to grant interviews also means that Perelman probably has a great deal of leverage with journalists in the rare cases when he grants an interview, as he did with such great effect in The New Yorker in 2006.
Perelman was a math prodigy, returning home with a gold medal and a perfect score from the 1982 International Math Olympiad. Prodigies are often not as successful as one might expect. Math and physics prodigies often flame out, sometimes catastrophically. While prodigies are more common among people who make major inventions and scientific discoveries than in the general population, they are not nearly as common as most people probably think. Perfect Rigor portrays Perelman’s success in proving the Poincaré Conjecture as a logical consequence of his youthful training and competition in the sometimes bizarre Soviet mathematical system. Since Perelman has revealed little about the process of his discovery, this is difficult to evaluate.
Prodigies often run into problems and don’t realize their seeming potential later in life. This has been observed in math, physics, and other fields for many generations. There are probably several causes. Some prodigies are probably frauds, manufactured by ambitious parents; that such people fail to make major breakthroughs is not surprising. Some prodigies are probably the product of a hothouse environment, driven or manipulated by parents or others to practice heavily and perform at an unusually high level that is difficult to sustain. As they get older and establish their own lives, other interests or needs intervene. Some prodigies undoubtedly fall afoul of politics that they are ill-prepared to deal with.
Academic homework, exams, competitions like the International Math Olympiad, admissions exams such as the SAT or GRE exams in the United States, specialized exams and competitions such as the famous Putnam math examinations, and so forth do not necessarily either teach or measure some of the skills required in actual invention or discovery. Exams and homework in math and physics tend to test the ability to accurately and quickly perform certain calculations or apply certain known mathematical methods to a problem. Some people either through heavy practice or rare natural ability can learn to perform these calculations rapidly with negligible error. This does not translate directly into the ability to handle unsolved research problems which often seem to require large amounts of frustrating trial and error and often deeper understanding of concepts, mental visualization, and so forth.
Many topics taught at a high school, college, and even beginning graduate school level are quite mature. Logical and technical flaws that abound in original research papers have been cleaned up and eliminated. Teachers and textbook writers have learned how to present the material clearly so that a bright or highly motivated student may be able to easily master the material quickly. Prodigies can sometimes read a textbook and immediately start performing the methods described in the textbook very accurately. This becomes more difficult as one reaches the “bleeding edge” where the available learning materials are original research papers or badly written textbooks that may contain errors, impenetrable jargon, opaque language, and even deliberate obfuscation of logical or technical flaws. Prodigies may encounter a sudden drop off of their remarkable abilities which they may inaccurately attribute to a lack of the magic “ability” required for the field rather than the immature state of the bleeding edge knowledge. Perelman presumably navigated these difficulties as he progressed in mathematical research.
One is reminded of the old sayings “actions speak louder than words” and “talk is cheap”. If Perelman’s proof stands the test of time, he has done much. If he is sincere in declining prizes, honors, and adulation, he sets an example by his actions. In reading Perelman’s story, one also cannot shake the impression that he may have had some unhappy experiences during his stay in the United States and went home silently vowing “I’ll show them,” which he apparently has.
The Poincare Conjecture
Donal O’Shea’s The Poincare Conjecture is a more pleasant book to read than Perfect Rigor, lacking the hostile tone of Perfect Rigor and sugar coating a number of topics. Perelman is “eccentric”. Little is said about “Manifold Destiny” or the ugly priority dispute. O’Shea focuses on the history of geometry, the Poincaré Conjecture, mostly inspiring stories about great mathematicians, and tries to explain the mathematics of the Poincaré Conjecture to a general audience.
On the whole, The Poincare Conjecture is an enjoyable and informative book to read. O’Shea carefully debunks the myth that scholars in the Middle Ages and the ancient world believed the Earth was flat. He gives an interesting account of Columbus, the slow discovery of the exact shape and geography of the Earth, confirming the ancient theory of the spherical Earth. He slowly and deftly leads the reader through the history of mathematics and geometry to the Poincaré Conjecture, the many failed attempts to prove it, and the seeming final solution by Perelman.
Some of the illustrations leave a bit to be desired. In discussing mathematics in the ancient world, O’Shea uses modern CIA maps of the modern world to show the ancient Greek kingdom of Ionia where Pythagoras was born and to show the Middle East. One map, for example, shows modern Bulgaria which did not exist in the time of Pythagoras. Similarly, O’Shea is discussing ancient Babylonia and Persia but the associated map shows modern Iraq and Iran. Hopefully, this will be fixed in a future edition.
Some of the discussion of hyperbolic geometry and most of the chapter on Poincaré’s topology papers, which presents the actual Poincaré conjecture, could be improved. The diagrams and explanation on page 27 in the chapter “Possible Worlds” showing how the surface of a two-holed torus can be mapped to an octagon is hard to follow. O’Shea returns to the two-holed torus and the octagon in Chapter 10, “Poincaré’s Topological Papers”. Probably many readers will have forgotten the discussion on page 27 by then. The term “natural geometry” is used in this chapter but not defined clearly. A number of diagrams in this chapter are small and difficult to follow. Interested readers can find a better explanation of some of the relevant aspects of hyperbolic geometry in the second chapter of Roger Penrose’sThe Road to Reality which features some entertaining Escher prints showing the so-called “Poincaré disc model” of hyperbolic geometry (first discovered not by Poincaré, but by Eugenio Beltrami, Penrose carefully points out).
One can only go so far with analogies to rubber sheets or cloth fabric in describing topology and especially differential geometry. This is a problem many popular mathematics and science books encounter. If we had a better way of explaining and introducing differential calculus to a general audience, this would improve the general public’s ability to follow issues in mathematics and science and also improve our educational system.
Pure mathematics today suffers from a particularly opaque and confusing language. It now typically takes several months for a skilled person to master the arcane language of modern pure mathematics. Abstraction has been taken to an extreme. Words and phrases such as “algebra”, “ring”, “module”, “field”, and so forth have meanings in pure mathematics that differ both from common usage and the language of applied mathematics used in most engineering and also much physics. The Poincare Conjecture suffers in places from terms like “natural geometry” that have a special meaning in pure mathematics.
Both books focus on the genius of Perelman and famous mathematicians such as Gauss, Riemann, Poincaré, and others. Indeed, the subtitle of Perfect Rigor is “A Genius and the Mathematical Breakthrough of the Century”. This superman theory of scientific progress and a strong focus on extreme intelligence is common in popular science and math books and articles.
The story of the Poincaré Conjecture, at least until Perelman, is a story of large amounts of trial and error (lots of error) as both books allude to. Henri Poincaré formulated the conjecture in 1904 and published an incorrect proof. Almost every year has seen publication or presentation of attempts to prove the Poincaré Conjecture. Numerous mathematicians, including very top mathematicians, have published incorrect proofs. Many different approaches to the problem have been developed. Most failed. Richard Hamilton developed the basic approach that Perelman built upon but apparently stopped making progress in the 1980’s or early 1990’s. It is common to find large amounts of trial and error in the detailed history of inventions and discoveries, including discoveries in pure and applied mathematics.
It is clear that Perelman spent at least seven years on the Poincaré conjecture. We have no idea how much trial and error and how much failure took place during those seven years. Perelman reportedly fixed two minor errors in his first paper in the subsequent two papers posted to www.arXiv.org in 2002 and 2003. Other inventors and discovers have frequently gone through long periods of trial and error and repeated failure before their “breakthrough”. While respecting Perelman’s accomplishments, we should also be interested in the precise process used to reach the answer and avoid attributing it to magical genius alone.
Both Perfect Rigor and The Poincare Conjecture are interesting and informative books for general audiences. Even practicing mathematicians may gain some insights and new information from Perfect Rigor. Yet, Grigory Perelman remains an enigma. A definitive biography remains to be written. The world might learn a lot from more details on how he discovered his proof of the Poincaré Conjecture.
(C) Copyright 2010, John F. McGowan, Ph.D.
About the Author
John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at [email protected].