COVER
ABOUT THE BOOK
ABOUT THE AUTHOR
TITLE PAGE
PREFACE
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
EIGHT
NINE
TEN
ELEVEN
TWELVE
THIRTEEN
FOURTEEN
FIFTEEN
SIXTEEN
SEVENTEEN
EIGHTEEN
NINETEEN
TWENTY
TWENTY-ONE
TWENTY-TWO
TWENTY-THREE
TWENTY-FOUR
TWENTY-FIVE
TWENTY-SIX
TWENTY-SEVEN
TWENTY-EIGHT
TWENTY-NINE
THIRTY
THIRTY-ONE
THIRTY-TWO
THIRTY-THREE
THIRTY-FOUR
THIRTY-FIVE
THIRTY-SIX
THIRTY-SEVEN
THIRTY-EIGHT
THIRTY-NINE
FORTY
FORTY-ONE
FORTY-TWO
FORTY-THREE
FORTY-FOUR
EPILOGUE
A NOTE ABOUT THE TRANSLATION
A NOTE ABOUT THE TRANSLATOR
NOTES
COPYRIGHT
ABOUT THE BOOK
What goes on inside the mind of a rock-star mathematician? Where does inspiration come from?
With a storyteller’s gift, Cédric Villani takes us on a mesmerising journey as he wrestles with a new theorem that will win him the most coveted prize in mathematics. Along the way he encounters obstacles and setbacks, losses of faith and even brushes with madness. His story is one of courage and partnership, doubt and anxiety, elation and despair.
We discover how it feels to be obsessed by a theorem during your child’s cello practise and throughout your dreams, why appreciating maths is a bit like watching an episode of Columbo, and how sometimes inspiration only comes from locking yourself away in a dark room to think. Blending science with history, biography with myth, Villani conjures up an inimitable cast of characters including the omnipresent Einstein, mad genius Kurt Godel, and Villani’s personal hero, John Nash.
Birth of a Theorem combines passion and imagination to take us on a fantastical adventure through the beautiful, mysterious world of mathematics.
Maths has never seemed so magical or so exciting.
ABOUT THE AUTHOR
Cédric Villani is a French mathematician who has received many international awards for his work including the Jacques Herbrand Prize, the Prize of the European Mathematical Society, the Fermat Prize and the Henri Poincaré Prize.
In 2010 he was awarded the Fields Medal, the International Medal for Outstanding Discoveries in Mathematics, for his work on Landau damping and the Boltzmann equation. Often called ‘the mathematicians’ Nobel Prize’, it is awarded every four years and is viewed by some as the highest honour a mathematician can achieve.
He is a professor at Lyon University and Director of the Institut Henri Poincaré in Paris, working primarily on partial differential equations and mathematical physics.
I am often asked what it’s like to be a mathematician—what a mathematician’s daily life is like, how a mathematician’s work gets done. In the pages that follow I try to answer these questions.
This book tells the story of a mathematical journey, a quest, from the moment when the decision is made to venture forth into the unknown until the moment when the article announcing a new result—a new theorem—is accepted for publication in an international journal.
Far from moving swiftly between these two points, in a straight line, the mathematician moves forward haltingly, along a long and winding road. He meets with obstacles, suffers setbacks, sometimes loses his way. As we all do from time to time.
Apart from a few insignificant details, the story I have told here is in agreement with reality, or at least with reality as I experienced it.
My thanks to Olivier Nora for having encouraged me, on the occasion of a chance encounter, to write this book; thanks to Claire for her careful reading and many helpful suggestions; thanks to Claude for his fine illustrations; thanks to Ariane Fasquelle and the staff at Grasset for grasping at once my purpose in writing this book and for their care in preparing the final manuscript for the typesetter; thanks, finally, to Clément for an unforgettable collaboration, without which this book wouldn’t exist.
Cédric Villani
Paris, December 2011
Lyon
March 23, 2008
One o’clock on a Sunday afternoon. Normally the laboratory would be deserted, were it not for two busy mathematicians in need of a quiet place to talk—the office that I’ve occupied for eight years now on the third floor of a building on the campus of the École Normale Supérieure in Lyon.
I’m seated in a comfortable armchair, insistently tapping my fingers on the large desk in front of me. My fingers are spread apart like the legs of a spider. Just as my piano teacher trained me to do, years ago.
To my left, on a separate table, a computer workstation. To my right a cabinet containing several hundred works of mathematics and physics. Behind me, neatly arranged on long shelves, thousands and thousands of pages of articles, lawfully photocopied back in the days when scientific journals were still printed on paper, and a great many mathematical monographs, unlawfully photocopied back in the days when I didn’t make enough money to buy all of the books I wanted. There are also a good three feet of rough drafts of my own work, meticulously archived over many years, and quite as many feet of handwritten notes, the legacy of hours and hours spent listening to research talks. In front of me, Gaspard, my laptop computer, named in honor of Gaspard Monge, the great mathematician and revolutionary. And a stack of pages covered with mathematical symbols—more notes from every one of the eight corners of the world, assembled especially for this occasion.
My partner, Clément Mouhot, stands to one side of the great whiteboard that takes up the entire wall in front of me, marker in hand, eyes sparkling.
“So what’s up? Your message was pretty vague.”
“My old demon’s back again—regularity for the inhomogeneous Boltzmann.”
“Conditional regularity? You mean, modulo minimal regularity bounds?”
“No, unconditional.”
“Completely? Not even in a perturbative framework? You really think it’s possible?”
“Yes, I do. I’ve been working on it again for a while now and I’ve made pretty good progress. I have some ideas. But now I’m stuck. I broke the problem down using a series of scale models, but even the simplest one baffles me. I thought I’d gotten a handle on it with a maximum principle argument, but everything fell apart. I need to talk.”
“Go on, I’m listening. . . .”
I went on for a long time. About the result I have in mind, the attempts I’ve made so far, the various pieces I can’t fit together, the logical puzzle that so far has defeated me. The Boltzmann equation remains intractable.
Ah, the Boltzmann! The most beautiful equation in the world, as I once described it to a journalist. I fell under its spell when I was young—when I was writing my doctoral thesis. Since then I’ve studied every aspect of it. It’s all there in Boltzmann’s equation: statistical physics, time’s arrow, fluid mechanics, probability theory, information theory, Fourier analysis, and more. Some people say that I understand the mathematical world of this equation better than anyone alive.
Seven years ago I initiated Clément into this mysterious world when he began his own thesis under my direction. He was eager to learn. Certainly he’s the only person who has read everything I’ve written on Boltzmann’s equation. Now Clément is a respected member of the profession, a mathematician in his own right, brilliant, eager to get on with his own research.
Seven years ago I helped him get started; today I’m the one who needs help. The problem I’ve chosen to work on is exceedingly difficult. I’ll never solve it by myself. I’ve got to be able to explain what I’ve done so far to someone who knows the theory inside out.
“Let’s assume grazing collisions, okay? A model without cutoff. Then the equation behaves like a fractional diffusion, degenerate, of course, but a diffusion just the same, and as soon as you’ve got bounds on density and temperature you can apply a Moser-style iteration scheme, modified to take nonlocality into account.”
“A Moser scheme? Hmmmm … hold on a moment, I need to write this down.”
“Yes, a Moser-style scheme. The key is that the Boltzmann operator … true, the operator is bilinear, it’s not local, but even so it’s basically in divergence form—that’s what makes the Moser scheme work. You make a nonlinear function change, you raise the power. . . . You need a little more than temperature, of course, there’s a matrix of moments of order 2 that have to be controlled. But the positivity is the main thing.”
“Sorry, I don’t follow—why isn’t temperature enough?”
I paused to explain why, at some length. We discussed. We argued. Before long the board was flooded with symbols. Clément was still unsure about the positivity. How can strict positivity be proved without any regularity bound? Is such a thing even imaginable?
“It’s not so shocking, when you think about it: collisions produce lower bounds; so does transport, in a confined system. So it makes sense. Unless we’re completely missing something, the two effects ought to reinforce each other. Bernt tried a while ago, he gave up. A whole bunch of people have tried, but no one’s had any luck so far. Still, it’s plausible.”
“You’re sure that the transport is going to turn out to be positive without regularity? And yet without collisions, you bring over the same density value, it doesn’t become more positive—”
“I know, but when you average the velocities, it strengthens the positivity—a little like what happens with the averaging lemmas for kinetic equations. But here we’re dealing with positivity, not regularity. No one’s really looked at it from this angle before. Which reminds me … when was it? That’s it! Two years ago, at Princeton, a Chinese postdoc asked me a somewhat similar question. You take a transport equation, in the torus, say. Assuming zero regularity, you want to show that the spatial density becomes strictly positive. Without regularity! He could do it for free transport, and for something more general on small time scales, but for larger times he was stymied. . . . I remember asking other people about it at the time, but no one had a convincing answer.”
“Back up. How did he handle the simple free transport case?”
“Free transport” is a piece of jargon that refers to an ideal gas in which the particles do not interact. The model is too simplified to be at all realistic, but you can still learn a lot from it.
“Not sure—but it should be obvious from an explicit solution. Let’s try to figure it out, right now. . . .”
Each of us set about reconstructing the argument that this postdoc, Dong Li, must have developed. No big deal, more like a minor exercise in problem solving. But maybe it will help us resolve the great enigma, who knows? And besides, it’s a contest—who can come up with the answer first? We scribbled away in silence for a few minutes. I won.
“I think I’ve got it.”
I got up and went over to the board, just like in school when the teacher shows the class how to solve a problem.
“You break down the solution in terms of the replicas of the torus … you change the variables in each piece … a Jacobian drops out, you use the Lipschitz regularity … and finally you end up with convergence in 1/t. Slow, but it looks about right.”
“But then you don’t have regularization … you get convergence by averaging … by averaging. . . .”
Clément was thinking out loud, staring at my calculation. Suddenly his face lit up. In a state of great excitement, he jabbed at the board with his index finger: “But then you’d have to check to see whether that helps with Landau damping!”
I was at a loss for words. Three seconds of silence. A vague feeling this could be important.
Now it was my turn to ask Clément to explain. He didn’t know what to say either. He hemmed and hawed, shifting his weight from one foot to the other. Then he said that my solution reminded him of a conversation he’d had three years ago with a Chinese-born mathematician in the United States, Yan Guo, at Brown.
“In Landau damping you want to have relaxation for a reversible equation—”
“Yes, yes, I know. But doesn’t interaction play a role? We’re not dealing with the Vlasov here, it’s just free transport!”
“Okay, maybe you’re right, interaction must play a role—in which case … the convergence should be exponential. Do you think 1/t is optimal?”
“Sounds right to me. What do you think?”
“But what if the regularity was stronger? Wouldn’t it be better if it was?”
I groaned. Doubt mixed with concentration, interest with frustration.
We stood in silence, staring at each other, wondering where to go from here. After a while conversation resumed. As fascinating as it is, the weird (and possibly mythical) phenomenon of Landau damping has nothing to do with what we’ve set out to accomplish. A few more minutes passed and we'd moved on to something else. We talked for a long time. One topic led to another. We took notes, we argued, we got annoyed with each other, we reached agreement about a few things, we prepared a plan of attack. When we left my office a few hours later, Landau damping was nevertheless on our long list of homework assignments.
•
The Boltzmann equation,
discovered around 1870, models the evolution of a rarefied gas made of billions and billions of particles that collide with one another. The statistical distribution of the positions and velocities of these particles is represented by a function f(t, x, v), which at time t indicates the density of particles whose position is (roughly) x and whose velocity is (roughly) v.
Ludwig Boltzmann was the first to express the statistical notion of entropy, or disorder, in a gas:
By means of this equation he was able to prove that, moving from an initial arbitrarily fixed state, entropy can only increase over time, never decrease. Left to its own devices, in other words, the gas spontaneously becomes more and more disordered. He also proved that this process is irreversible.
In stating the principle of entropy increase, Boltzmann reformulated a law that had been discovered a few decades earlier, the second law of thermodynamics. But he did several things that enriched it immeasurably from the conceptual point of view. First, by providing a rigorous proof, he placed an experimentally observed regularity that had been elevated to the status of a natural law on a secure theoretical foundation; next, he introduced an extraordinarily fruitful mathematical interpretation of a mysterious phenomenon; finally, he reconciled microscopic physics—unpredictable, chaotic, and reversible—with macroscopic physics—predictable, stable, and irreversible. These achievements earned Boltzmann a place of honor in the pantheon of theoretical physicists and stimulated an enduring interest in his work among epistemologists and philosophers of science.
Additionally, Boltzmann defined the equilibrium state of a statistical system as the state of maximum entropy, thus founding a vast field of research known as equilibrium statistical physics. In so doing, he demonstrated that the most disordered state is the most natural state of all.
The triumphant young Boltzmann turned into a tormented old man who took his own life, in 1906. His treatise on the theory of gases appears in retrospect to have been one of the most important scientific works of the nineteenth century. And yet its predictions, though they have been repeatedly confirmed by experiment, still await a satisfactory mathematical explanation. One of the missing pieces of the puzzle is an understanding of the regularity of solutions to the Boltzmann equation. Despite this persistent uncertainty, or perhaps because of it, the Boltzmann equation is now the object of intensive theoretical investigation by an international community of mathematicians, physicists, and engineers who gather by the hundreds at conferences on rarefied gas dynamics and many other meetings every year.
Lyon
Last week of March 2008
Landau damping!
In the days following our working session, a confused series of recollections came to me—snatches of conversation, discussions begun but never finished. . . . Plasma physicists have long been used to the idea of Landau damping. But as far as mathematicians are concerned, the phenomenon remains a mystery.
In December 2006 I was visiting Oberwolfach, the legendary institute for mathematical research deep in the heart of the Black Forest, a retreat where mathematicians come and go in an unending ballet of the mind, giving talks on every subject imaginable. No locks on the doors, an open bar, cakes and pastries galore, small wooden cash boxes in which you put payment for food and drinks, tables at which your seat is determined by drawing lots.
One day chance placed me at the same table with two Americans, Robert Glassey and Eric Carlen, both of them authorities on the kinetic theory of gases. The evening before, at the opening of that week’s seminar, I had proudly presented a whole batch of new results, and that same morning Eric had given a truly memorable performance, bursting with energy and jam-packed with ideas. The two events, coming one right after the other, were a bit overwhelming for Robert, who confessed to feeling old and worn out. “Time to retire,” he sighed. “Retire?” Eric exclaimed in disbelief. There’s never been a more exciting time in the theory of gases! “Retire?” I cried. Just when we are so urgently in need of the wisdom this man has accumulated in his thirty-five years as a professional mathematician!
“Robert, what can you tell me about the mysterious Landau damping effect? Do you think it’s real?”
The words “weird” and “strange” stood out in Robert’s reply. Yes, Maslov worked on it; yes, there is a paradox of reversibility that seems incompatible with Landau damping; no, it isn’t at all clear what’s going on. Eric suggested that the effect was chimerical—a product of physicists’ fertile imaginations that had no hope of being rigorously formulated in mathematical terms. None of this meant much to me at the time, but I did manage to make a mental note and file it away in a corner of my brain.
Now here we are in 2008, and I don’t know anything more about Landau damping than I did two years ago. Clément, on the other hand, had a chance to discuss the matter at length with Yan Guo, one of Robert’s younger brothers in mathematics (they both had the same thesis director, twenty years apart). The heart of the difficulty, according to Yan, is that Landau didn’t work on Vlasov’s original model but on a simplified, linearized version. No one knows if what he found also applies to the “true” nonlinear model. Yan is fascinated by this problem—and he’s not alone.
Could Clément and I tackle it? Sure, we could try. But in order to solve a problem, you’ve got to know at the outset exactly what the problem is! In mathematical research, clearly identifying what it is you are trying to do is a crucial, and often very tricky, first step.
And no matter what our objective might turn out to be, the only thing we’d be sure of to begin with is the Vlasov equation,
which determines the statistical properties of plasmas with exquisite precision. Mathematicians, like the poor Lady of Shalott in Tennyson’s Arthurian ballad, cannot look at the world directly, only at its reflection—a mathematical reflection. It is therefore in the world of mathematical ideas, governed by logic alone, that we will have to track down Landau. . . .
Neither Clément nor I have ever worked on this equation. But equations belong to everybody. We’re going to roll up our sleeves and give it our best shot.
•
Lev Davidovich Landau, a Russian Jew born in 1908, winner of the Nobel Prize in 1962, was one of the greatest theoretical physicists of the twentieth century. Persecuted by the Soviet regime and finally freed from prison through the devoted efforts of his colleagues, he survived to become a towering, almost tyrannical figure in the world of science. With Evgeny Lifshitz he wrote the magisterial ten-volume Course of Theoretical Physics, still a standard reference today, and made two fundamental contributions to the study of plasma physics in particular: the Landau equation, a sort of little sister to the Boltzmann equation (I studied both in preparing my thesis), and Landau damping, a spontaneous phenomenon of stabilization in plasmas—that is, a return to equilibrium without any increase in entropy, in contrast to the mechanisms described by the Boltzmann.
With the physics of gases we are in the realm of Boltzmann: entropy increases, information is lost, the arrow of time points toward the future, the initial state is forgotten; gradually the statistical distribution of neutral particles approaches a state of maximum entropy, the most disordered state possible.
With plasma physics, on the other hand, we are in the realm of Vlasov: entropy is constant, information is conserved, there is no arrow of time, the initial state is always remembered; disorder does not increase, and there is no reason for the system to approach one state rather than another.
Landau had a low opinion of Vlasov, even going so far as to say that almost all of Vlasov’s results were wrong. And yet he adopted Vlasov’s model. Landau drew from it a conclusion that Vlasov had completely overlooked, namely, that the electrical forces weakened spontaneously over time without any corresponding increase in entropy or any friction whatsoever. Heresy?
Landau’s ingeniously complex mathematical calculation satisfied most physicists, and the so-called damping phenomenon soon came to be named after him. But not everyone was convinced.
Lyon
April 2, 2008
In the hallway, a low table strewn with pages of hastily scribbled notes and a blackboard covered with little drawings. Through the great picture window, a view of a gigantic long-legged black cubist spider, the famous P4 laboratory where experiments are conducted on the most dangerous viruses in the world.
My guest, Freddy Bouchet, gathered up his notes and put them in his bag. We’d spent a good hour talking about his research on the numerical simulation of galaxy formation and the mysterious power of stars to spontaneously organize themselves in stable clusters.
This phenomenon is not contemplated by Isaac Newton’s law of universal gravitation, discovered more than three hundred years ago. And yet when one observes a cluster of stars governed by Newton’s law, it does indeed seem that the entire cloud settles into a stable state after a rather long time—an impression that has been confirmed by a great many calculations performed on very powerful computers.
Is it possible, then, to deduce this property from the law of universal gravitation? The English astrophysicist Donald Lynden-Bell had no doubt whatsoever about the reality of dynamic stabilization in star clusters. He thought it was a “hard” phenomenon—as hard as, well, an iron meteorite—and gave it the name violent relaxation. A splendid oxymoron!
“Violent relaxation, Cédric, is like Landau damping. Except that Landau damping is a perturbative regime and violent relaxation is a highly nonlinear regime.”
Freddy was trained in both mathematics and physics, and he has devoted a good part of his professional life to studying such problems. Today he had come to talk to me about one question in particular.
“When you model galaxies, you treat the stars as a fluid—as a gas of stars, in effect. You go from the discrete to the continuous. But how great an error does this approximation entail? Does it depend on the number of stars? In a gas there are a billion billion particles, but in a galaxy there are only a hundred billion stars. How much of a difference does that make?”
Freddy went on in this vein for a long while, raising further questions, explaining recent results, drawing figures and diagrams on the board, noting references. I pointed out the connection between his research and one of my hobby horses, the theory of optimal transport inaugurated by Monge. Freddy seemed pleased; it was a profitable conversation for him. For my part, I was thrilled to see Landau damping suddenly make another appearance, scarcely more than a week after my discussion with Clément.
Just as I was coming back to my office after saying goodbye to Freddy, my neighbor Étienne, who until then had been bustling about, noiselessly filing papers, popped his head into the hallway. With his long gray hair cut in a bob, he looks like an elderly teenager, anticonformist but hardly threatening.
“I didn’t really want to say anything, Cédric, but those figures there on the board—I’ve seen them before.”
A plenary speaker at the last International Congress of Mathematicians, member of the French Academy of Sciences, often (and probably rightly) described as the world’s best lecturer on mathematics, Étienne Ghys is an institution unto himself. As a staunch advocate of promoting research outside the Paris region, he has spent the past twenty years developing the mathematics laboratory at ENS-Lyon. More than anyone else, he is responsible for turning it into one of the leading centers for geometry in the world. Étienne’s charisma is matched only by his grumpiness: he has something to say about everything—and nothing will stop him from saying it.
“You’ve seen these figures?”
“Yes, that one’s from KAM theory. And this one, I know it from somewhere. . . .”
“Where should I look?”
“Well, KAM is found almost everywhere. You start from a completely integrable, quasi-periodic dynamic system and you introduce a small perturbation. There’s a problem with small divisors that eliminate certain trajectories, but even so, probabilistically speaking, you’ve got long-term stability.”
“Yes, I know. But what about the figures?”
“Hold on, I’m going to find a good book on the subject for you. But a lot of the figures you see in works on cosmology are usually found in dynamical systems theory.”
Very interesting, I’ll have to take a look. Maybe it will help me figure out what stabilization is really all about.
That’s what I love most of all about our small but very productive laboratory—the way conversation moves from one topic to another, especially when you’re talking with someone whose mathematical interests are different from yours. With no disciplinary barriers to get in the way, there are so many new paths to explore!
I didn’t have the patience to wait for Étienne to rummage through his vast collection of books, so I rooted around in my own library and came up with a monograph by Alinhac and Gérard on the Nash–Moser theorem. As it happens, I’d made a careful study of this work a few years ago, so I was well aware that the method developed by John Nash and Jürgen Moser is one of the pillars of the Kolmogorov–Arnold–Moser (KAM) theory that Étienne had mentioned. I also knew that Nash–Moser relies on Newton’s extraordinary iteration scheme for finding successively better approximations to the roots of real-valued equations—a method that converges unimaginably fast, exponentially exponentially fast!—and that Kolmogorov was able to exploit it with remarkable ingenuity. Frankly, I couldn’t see any connection whatever between these things and Landau damping. But who knows, I muttered to myself, perhaps Étienne’s intuition will turn out to be correct. . . .
Enough daydreaming! I wedged the book into my backpack and rushed off to pick up my kids from school, got on the métro and immediately took out a manga from my coat pocket. For a few brief and precious moments life around me disappeared, giving way to a world of supernaturally skilled physicians with surgically reconstructed faces, hardened yakuza who lay down their lives for their children, little girls with huge doe eyes, cruel monsters who suddenly turn into tragic heroes, little boys with blond curls who gradually turn into cruel monsters. . . . A skeptical and tender world, passionate, disillusioned, devoid of either prejudice or Manichaean certainties; a world of emotions that strike deep down in the soul and bring tears to the eyes of anyone innocent enough to surrender himself to them—
Hôtel de Ville! My stop! During the time it took to get here the story had flowed through my brain and through my veins, a small torrent of ink and paper. I felt cleansed through and through.
While I’m reading manga all thoughts of mathematics are suspended. It’s like hitting a pause button: manga and mathematics don’t mix. But what about later, when I’m dreaming at night? What if Landau, after the terrible accident that should have cost him his life, had been operated on by Black Jack? Surely the fiendishly gifted surgeon would have fully restored his powers, and Landau would have resumed his superhuman labors. . . .
For at least a brief time anyway, I was able to forget Étienne’s remark and this business about KAM theory. What connection could there possibly be between Kolmogorov and Landau? The moment I got off the métro, the question echoed through my mind over and over again. If there really is a connection, I’ll find it.
At the time I had no way of knowing that it would take me more than a year to find the link between the two. Nor could I have suspected the fantastic irony that would finally emerge: the figure that caught Étienne’s attention, that put him in mind of Kolmogorov, actually illustrates a situation where Landau damping and KAM theory have nothing to do with each other! Étienne’s intuition was right, but for the wrong reason—as though Darwin had guessed correctly about the evolution of species by comparing bats and pterodactyls, mistakenly supposing that the two were closely related.
Ten days after the unexpected turn taken by my working session with Clément, a second miraculous coincidence had occurred—and on the same subject! The timing could not have been more fortuitous.
Now to take advantage of it.
•
What was the name of that Russian physicist? Just like what happened to me, everyone thought he was dead when they pulled him out from the wreckage. Medically, he was dead. An extraordinary story. The Soviet authorities mobilized every resource in order to save an irreplaceable scientist. An appeal for help was even issued to physicians in other countries. The dead man was revived. For weeks the greatest surgeons in the world took turns at his bedside. Four times the man died. Four times life was artificially breathed into him. I’ve forgotten the details, but I do remember how fascinating it was to read about this struggle against an inadmissible fatality. His tomb was opened up and he was forcibly removed. He resumed his post at the university in Moscow.
[Paul Guimard, Les choses de la vie]
•
Newton’s law of universal gravitation states that any two bodies are attracted to each other by a force proportional to the product of their masses and inversely proportional to the square of the distance between them:
In its classical form, this law does a good job of accounting for the motion of stars in galaxies. But even if Newton’s law is simple, the immense number of stars in a galaxy makes it difficult to apply. After all, just because we understand the behavior of individual atoms doesn’t mean that we understand the behavior of a human being. . . .
A few years after formulating the law of gravitation, Newton made another extraordinary discovery: an iterative method for calculating the solutions of any equation of the form
Starting from an approximate solution x0 , you replace the function F by its tangent Tx0 at the point (x0 , F(x0)) (more precisely, the equation is linearized around x0 ) and solve the approximate equation Tx0(x) = 0. This gives a new approximate equation x1, and you now repeat the same procedure: replace F by its tangent Tx1 at x1 , obtain x2 as the solution of Tx1(x2)= 0, and so on. In exact mathematical notation, the relation that associates xn with xn + 1 is
The successive approximations x1 , x2 , x3 , … obtained in this fashion are incredibly good: they approach the “true” solution with phenomenal swiftness. It is often the case that four or five tries are all that is needed to achieve a precision greater than that of any modern pocket calculator. The Babylonians are said to have used this method four thousand years ago to extract square roots; Newton discovered that it can be used to find not only square roots but the roots of any real-valued equation.
Much later, the same preternaturally rapid convergence made it possible to demonstrate some of the most striking theoretical results of the twentieth century, among them Kolmogorov’s stability theorem and Nash’s isometric embedding theorem. Single-handedly, Newton’s diabolical scheme transcends the artificial distinction between pure and applied mathematics.
•
The Russian mathematician Andrei Kolmogorov is a legendary figure in the history of twentieth-century science. In the 1930s, Kolmogorov founded modern probability theory. His theory of turbulence in fluid dynamics, worked out in 1941, remains the starting point for research on this subject today, both for those who seek to corroborate it and for those who seek to disconfirm it. His theory of complexity prefigured the development of artificial intelligence.
Henri Poincaré had convinced his fellow mathematicians that the solar system is intrinsically unstable, and that uncertainty about the position of the planets, however small, makes any prediction of the position of the planets in the distant future impossible. But some seventy years later, in 1954, at the International Congress of Mathematicians in Amsterdam, Kolmogorov presented an astonishing result. Harnessing probabilities and the deterministic equations of mechanics with breath-taking audacity, he argued that the solar system probably is stable. Instability is possible, as Poincaré correctly saw—but if it occurs, it should occur only rarely.
Kolmogorov’s theorem asserts that if one assumes an exactly soluble mechanical system (in this case, the solar system as Kepler imagined it to be, with the planets endlessly revolving around the sun in regular and unchanging elliptical orbits), and if one then disturbs it ever so slightly (taking into account the gravitational force of attraction, neglected by Kepler), the resulting system remains stable for the great majority of initial conditions.
Kolmogorov’s argument was not widely accepted at first. This was mainly because of its complexity, but Kolmogorov’s own elliptical style of exposition didn’t help matters. Less than a decade later, however, the Russian mathematician Vladimir Arnold and the German mathematician Jürgen Moser, using different approaches, succeeded in providing a complete demonstration, Arnold proving Kolmogorov’s original statement of the theorem and Moser a more general variant of it. Thus was born KAM theory, which in its turn has given birth to some of the most powerful and surprising results in classical mechanics.
The singular beauty of this theory silenced skeptics, and for the next twenty-five years the solar system was believed to be stable, even if the technical constraints of Kolmogorov’s theorem did not correspond exactly to reality. With the work of the French astrophysicist Jacques Laskar in the late 1980s, however, opinion reversed itself once more. But that’s another story. . . .
Chaillol
April 15, 2008
The audience holds its breath, the teacher gives the cue, and the young musicians all at once make their bows dance across the strings. . . .
The Suzuki method requires parents to attend their children’s group lessons. Here, high in the French Alps, the lessons are given in a grand ski chalet. The main floor is entirely taken up by a stage and rows of seats. What else is there to do but watch—and listen?
We try not to grimace at the most grating noises. Those of us who volunteered yesterday to make fools of ourselves (to our children’s great delight) by playing their instruments know full well how difficult it is to make these diabolical contraptions produce the right sound! Today the atmosphere’s just right: the adults are in a good mood, the children are happy.
Suzuki method or no Suzuki method, what matters most of all is the teacher, and the one who is helping my son learn to play the cello is really, really talented.
Sitting toward the front, I find myself in almost the same position as my son—devouring Binney and Tremaine’s classic work, Galactic Dynamics, with the enthusiasm of a small child discovering a new world. I had no idea that the Vlasov equation was so important in astrophysics. Boltzmann’s is still the most beautiful equation in the world, but Vlasov’s isn’t too shabby!