Grigory Perelman: My humble homage to Greatness

How I stumbled upon an interesting book

I cannot recall how I bought this book. Whatever the motivation for the purchase I can say two things for sure.  One, I did not do the customary due diligence I carry out before I buy a book.  Two, I did not know enough about the main protagonist to be persuaded by the fact that the book was about him. 

The book I am referring to is Perfect Rigour – A Genius and the Mathematical Breakthrough of the Century by Masha Gessen (Houghton Mifflin Harcourt Publishing Company).

By the time I picked it up to read, nearly two years later, and I realized that it was about Grigory Perelman I had heard about him twice.  I recalled reading an inconspicuous story in the newspaper and then I had heard his name in passing from a colleague who referred to him as an example of renunciation. 

Neither of it was sufficient reason to allocate an extreme scarce personal resource, namely reading time, to reading this obscure book.  But then here I was slowly plodding through it day after day.

A brief account of the life of Grigory Perelman

For those who don’t like a long drawn suspense the book is ostensibly the life story of Grigory Perelman who shot to fame by proving the Poincare Conjecture.  The Conjecture itself was one of the six unsolved problems in mathematics identified by the Clay Institute for which it announced a challenge prize of one million dollars. 

The Poincare Conjecture was an unresolved problem in topology put forth by French mathematician Henri Poincare.  The problem seems to have been so seductive that many a brilliant mathematical career seems to have been sacrificed at it.  It took seven years of Perelman’s single-minded brilliance to prove the Conjecture.  Gessen weaves the story of many other equally brilliant mathematicians, both inside and outside Russia, who float in and out of the narrative like minor characters in a grand musical opera.

The book ends with the sad story of how Perelman turned down many of the material payoffs that would have been his for his stupendous mathematical achievement.

The author is an unusual person too.  If you like me, enjoy knowing as much about the author as about the work, you can read about Masha Gessen here.   https://en.wikipedia.org/wiki/Masha_Gessen

After I started reading the book I did the usual rummaging of reviews that I normally do before I buy a book.   Most reviews on Amazon on this book were unflattering.  After reading the book I suspect that they were not unjustified.  A somewhat similar book by Sylvia Nassar, for example, A Beautiful Mind has many glowing reviews.  But the blurbs from the reviews in the Times and Sunday Times that show on the back jacket of the book are far more kind, perhaps fairer too.

What the Book is all about

Having pressed on with the book undeterred I am glad I did.  And that is so for what I learned, stuff of which I had no inkling until then. I am glad in spite of Gessen’s somewhat unusual style of English writing, the kind of which I have not come across so far.  I do not mean that as a compliment. The style of narration and the jerky flow of the narrative are two important peeves I have against the book. 

Gessen’s book is interesting because of the three broad themes that the book deals with:  The institutions of science and scientific education in Soviet Russia, the world of mathematics and the rather unusual social beings that mathematicians are and finally the life of one remarkable mathematician who epitomizes it.  Her account of the training in mathematics in Russia through its Math Olympiad schools is quite fascinating.

I would not have known anything about any of these if I had not read this book.  Having read it I think I am better off personally, even though it adds nothing to my academic welfare.

The other aspect that makes Gessen’s book remarkable is the fact that she wrote about Perelman without speaking to him even once.  Now that does raise question marks about whether the account can be authentic enough.  Viewed differently it is an interesting experiment.  Having read about him I believe that while he is important enough to be written about. Equally, I suspect that there is no way anyone could write a book on Perelman with his cooperation.

Kolmogorov – The Life of a Mathematician in Soviet Russia

Gessen’s account of life in academe is represented by her account of Andrey Kolmogorov.  Kolmogorov was a child prodigy of sorts.  His childhood reminded me of that of von Neumann’s.

Kolmogorov’s is a name that crops up so often in the lives of those of who work with applied statistics as a tool of research in social sciences.  Gessen presents him as a quintessential figure in the Soviet mathematical establishment.  He was a Renaissance man with deep involvement in the classics in music and literature and who openly carried on a same-sex relationship with Pavlov Alexandrov. 

When I read about the plan that Kolmogorov wrote up “of how to become a great man should I have sufficient desire and diligence my heart” warmed and my eyes lit up – until I reread that point and noted that Kolmogorov was all of forty when he wrote that plan.” (p-40). I sadly realized that I had missed that opportunity by well over sixteen years.  More importantly those sixteen years also happen to be the period when the brain is in radioactive decay mode.  Admittedly I was willing to assume for that brief while that I might have hidden deep inside me an intellect as formidable as that of Kolmogorov, waiting to be unleashed.

Life for Kolmogorov was not easy in the Soviet establishment.  One of the strange things I read was that the state would not allow, let alone encourage, the teaching of set theory since it was antithetical to the ideals of the state.  I have not understood the basis for this ideological antipathy though. 

Thus mathematicians like Kolmogorov were almost tolerated only when their utility in the space and military programmes of the Soviet state became evident.  Once that was realized mathematicians were even beginning to be pampered, except when it came to travelling overseas to attend conferences, especially to the USA.  International travel and collaborations were a nearly paranoia-inducing proposition to that totalitarian establishment.

The degree of tolerance for mathematicians comes out of this observation by Gessen:  “That Kolmogorov’s marked social problems did not impair his career is a measure of the degree to which a sort of Apsergian culture was built into the larger Russian culture of mathematics.” (p-177)

Kolmogorov’s story eventually ends in a tragedy.  “Kolmogorov never recovered from a scandal in which he was implicated as “an agent of western cultural influence in the Soviet Union….He died at eighty four, speechless, blind and motionless, but surrounded by his students, who for the preceding couple of years had taken turns providing round the clock care at his house.” 

I had no idea of how politicized the Russian scientific and mathematical establishments were until I read this book.  And that politicization was in more ways than one. 

At one predictable level the entire scientific establishment was just as much hostage to the propaganda blitzkrieg of the state.  At another level, narrow bigoted considerations such as anti Semitism seemed to govern the running of the establishment.  That anti-Semitism continued to haunt Perelman’s career and would perhaps have paid to his academic aspirations but for the numerous godfathers he had as we will soon see. 

Coming as it did from a state that officially did not subscribe to the idea of God it surprised me that the state would actively discriminate scientists on the basis of their religion. 

Grigory Perelman himself was a product of the emerging mathematics training phenomenon in the Soviet Union.  Gessen’s book provides an interesting peep – but just a peep – into the institutional and social dynamics of being a mathematician.  It talks about the scramble for good schools, the limited funding and resources the schools provided for international work and the privileges mathematicians enjoyed in a resource starved society. 

What is striking of course is the phenomenon of training schools for the International Math Olympiads (IOM) to which Gessen devotes a great deal of real estate in the book.  The book gives the impression that the IOM pervaded social life to a great extent.  That could however be a flawed conclusion given that these narratives tend to create a larger than life impression of the phenomenon they deal with.  I suspect that is partly the magic that every writer tends to create.

Perelman, the Man and the Mathematician

Perelman the mathematician was an unmistakable outcome of the IOM phenomenon in Russia.  So much so when he sets out to prove the Poincare conjecture Gessan describes it as a well-defined but challenging IOM type problem.

Perelman would have been a sad casualty but for the efforts of so many people.  Gessen sums this up in an interesting fashion:  “From the moment Perelman entered Rukshin’s math club at the age of ten – or perhaps from a much earlier point, when his mother told her professor she was leaving mathematics to have a baby – Perleman had been a human math project.  He was raised by his mother, reared by Rukshin, coddled by Ryzhik, coached by Abramov, directed by Zalgaller, protected by Alexandrov, tended by Burago, and promoted by Bromov so that he could do pure mathematics in a world of pure mathematics.”

But with all that Perelman turned out to what the average man or woman would consider a quirky individual.  At the age of fourteen “Perelman turned out to be Perelman, which is to say, rigid, demanding and hypercritical; these qualities would only intensify with age, ultimately making it impossible for him to be any kind of teacher or, indeed, communicator.” (p-96)

Perelman’s behaviour did not change much with age.  “The seventeen year old Perelman – university student, Olympiad champion, and universal problem-solving machine – did not and could not imagine that these math club teenagers, who had two years’ fewer problem solving and competition experience and who simply lacked his problem crunching skills could not do what he could if he they really, really put their minds to it. “ (p-97)

When Perelman arrived in the United States he was twenty six. . By that time his transformation into being what Jeff Cheeger, a mathematician at the Courant Institute at BYU, would describe as “eccentric” appears to have been complete.  Gessen pieces together evidences of this transformation.    “He did not believe in cutting hair or finger nails – some people thought they remembered his saying something about the unnaturalness of such trimming…( C) hances  are at least as good that Perelman found the conventions of personal hygiene and appearance both taxing and unreasonable…. He wore the same clothes every day – most notably a brown corduroy jacket – and his holding forth on the virtues of a particular kind of black bread that could be procured only from a Russian store in Brooklyn Beach, where Perelman walked from Manhattan.” (p—114).

Quirks aside, to say that Perelman was a gifted mathematician would be stating the obvious.  Gessen provides some interesting reference points to bring out the scale and nature of his brilliance.

“What the world had given Perelman was the habit of honing the power of his incomparable mind on a single problem.  In the world of top mathematicians, the intellectual elite are people who open new horizons by posing questions no one else has thought to ask.  A step down are the people who devise ways to answer those questions; often these are members of the elite at earlier stages in their career – few years after obtaining their PhDs, for example when they are proving other people’s theorems before they start formulating their own.  And finally there are the rare birds, those who take the last steps in completing proofs.  These are the persistent, exacting, patient mathematicians who finally lay down the path others have dreamed up and marked out.  In our story Poincare and Thurston represent the first group, Hamilton the second group and Perelman the one who finished the job. Indeed, it was a problem that perhaps could not be solved in any amount of time by anyone – except Perelman.  And Perelman was a man in search of just such a problem, one that would finally utilize the full capacity of the supercompactor that was his mind.” (p-146).

The Hamilton Gessen refers to is Richard Hamilton, one of the brilliant mathematicians who had worked on the Poincare Conjecture.  Hamilton’s work led to “Ricci flow”, Hamilton’s approach to proving the conjecture.  To give an idea of the prominence of his work, a whole group of other mathematicians which Gessen refers to as the “Ricci flow community”, had started work on and around Ricci flow.

Gessen’s detailed account of how Perelman eventually got to work on the proof and the dynamics of cutting edge research in mathematics is quite engaging and is worth a read.  You get interesting vignettes of interesting mathematicians like James Thurston who could “visualize” four dimensional space. 

Gessen also makes a quick detour into the world of modern psychology to explain the behaviour of these brilliant individuals.  She cites research by Simon Baron-Cohen on a disorder known as Asperger Syndrome, named after the Austrian pediatrician, Hans Asperger who is credited with identified the disorder.

Gessen believes that much of the behaviour of these brilliant mathematicians could be attributed to or explained by this syndrome.  “The correlation between math and autism and / or Asperger’s was proved again: mathematicians scored higher than other scientists, who scored higher than students in the humanities who scored roughly the same as the random controls.” (p-176)

But the extension that Gessen makes of this explanation sounds like a stretch.  “So it is perhaps no accident that the founders of the dissident movement in the Soviet Union were mathematicians and physicists.” (p-178)

In short I got to read about a world that sounds like fantasy in that I will never get to be a part of real.  Yet one knows that it is all very real.

The Not so Happy Denouement

For the next seven years Perelman toiled away, cut off from nearly all of mankind, working out the proof that was to set the world of maths agog with not so hushed excitement.  The beast at whose altar many a brilliant career in mathematics had been sacrificed had been finally tamed it would appear.

Perelman’s engagement with the world of maths, with academe and with the world of publishing – all distinct yet closely intertwined worlds - however does not end on a happy note.  Perelman declined all the recognition and rewards that would have been his if only he had chosen to accept them.  But in his world view they did not appear to be adequate honour for the seminal nature of what he had accomplished.

Perelman chose not to publish his work though the conventional channels for academic publishing. Gessen explains that “(f)inally, his decision to post his proof on the arXiv had been an intentional revolt against the very idea of scientific journals distributed by paid subscription.”  (p-157)  “Perelman’s revolt against the conventions of scientific publication was not based on an ideology; he simply had no use and therefore no regard for them.” (p-164)

He also turned down an offer to join the faculty at Princeton.  Gessen notes that he “abhorred the idea of being some department’s prized possession.” (p-164)

Having posted the third part of his proof Perelman managed to disappear from the world of mathematics and the world at large, leaving the world to just imagine that Perelman had back to inhabit the small world comprising just him and his mother.  Gessen notes that “(h)e cancelled his email account at the Steklov and left mathematics by walking out through the heavy oak double doors that led on to the embankment of the Fontanka River and into the oppressing grayness that masqueraded as daylight in St. Petersburg in winter.” (p-185)

Even Rukshin, the man behind the mathematician that Perelman is, who was a part of Perelman’s small world does not appeared to be a part of any more.

Gessen goes to great length to explain this extraordinary behaviour of Perelman.  She appears to proceed with the premise that this was all the response from an unhappy man.

Perelman’s “script also contained rules, obvious ones….Great mathematical achievement should be rewarded with professional recognition, which can take only one form: the form of studying and understanding the work that the person has done.  Money is no substitute for work.  In fact, money is insulting.  If you think it is natural for a university to offer money to someone who has solved a huge problem even though no one at this university understands the solution…(t)his is a caricature. There was no place for caricatures in Perelman’s script.” (p-174)

“He had given mathematics something great, something truly valuable.  Mathematics had responded feebly, trying to convince him to accept substitutes for true recognition.  No wonder he was disappointed in mathematics.” (p-181)

Perelman’s view of the world was best summed up in the comments of Rukshin who perhaps knew him better than anyone else in the world, including his own mother:  “ The world of science – the science that Perelman had considered the most honest of the sciences – had turned its other side to him.  It had been soiled and turned into market goods.” (p-165)

A Beautiful Mind vs Perfect Rigour

As I read the story of Perelman's life I was reminded of the life of John Nash, in Sylvia Nassar's book A Beautiful Mind.  I do not know enough maths to draw a comparison of the greatness of the two mathematicians.  I do get the sense that even if I could make that comparison it would be meaningless, given the sheer greatness of the two characters.  It would be like trying to establish who is a greater rishi between between Viswamitra and Vasishtha, I suspect.

Nash's story is a sad one too.  But in some ways it offers a sense of hope at the end.  Although the disease pretty much finished off his career in maths, he lived to harvest the recognition that his early work offered him.  He had a little more of a normal family and social life than Perelman did.  And possibly because of the close proximity of his work to the world of applications he has become a "household name" among economists and a whole lot of other social science disciplines.

Perelman on the contrary walked away from all of it, in his endeavor not to compromise what he seemed to see as the pristineness of maths and the life of an academician.  For all we know he has found his peace in this small world of his.

I am also led to wonder if Perelman and Nash are both products of the different societies they were raised in.  One very consumerist and grounded in this world and the other one of austerity and severity. 

As  I was reading this book, coincidentally I came across an anecdote involving a Soviet indologist who was told by an Indian seer that Russia was the home of the vedas and that its original name was Rishi Varsha and that explained the presence of a significant level of Sanskritic expressions in the Russian dialect spoken in Northern Russia.

OK, OK.  I know I am pushing it too far.  But I assure I have not been smoking anything...

But let me say this.  For many days after I read the book Perelman’s life coming back to my thoughts like very few others have from those I have read about.  And every time I thought about him I was reminded of a Sanskrit sloka I read in school:

उदारस्य तृणं वित्तं शूरस्य मरणं तृणं
विरक्तस्य तृणं भार्या निस्पृहस्य तृणं जगत् ;

Udaarasya trunam vittam shoorasya maranam trunam
Viraktasya trunam bhaaryaa nispruhasya trunam jagat.

i.e.   For a generous person money or wealth is insignificant (like a blade of grass), for a  warrior the prospect of facing death is immaterial.  Likewise, a person unattached to family life has no interest in his wife, and for a person having no desires this living Earth is immaterial.

Verse and Translation accessed Aug 10, 2016 at http://mcjoshi21.blogspot.in/2012/08/to-daus-subhashit.html]

Nanni... Namaskaaram...

 

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