Birth of a Theorem: A Mathematical Adventure

by Cédric Villani

  Once the kids are safely on their way I rush back home and settle down in an armchair to try out the idea that came to me when I woke up, as if by magic. Mumbling to myself all the while: “I stay in Fourier, as Sigal recommended, completely avoiding the Laplace transform. But before inverting, I begin by separating thus. Next, in a second step…”

  I go on scribbling, then pause for a moment to reflect.

  “It works! I think.…”

  “YES! It works!!! Of course that’s the way you’ve got to do it, no question. We can elaborate later, add more details. But now the framework’s finally in place.”

  From here on it’s only a matter of patience. Developing the idea will undoubtedly lead to schemes that I know what to do with. I write out the details carefully, taking my time. The moment has finally come to bring to bear all the ingenuity I can muster from eighteen years of doing mathematics!

  “Hmmmm, now that resembles a Young inequality … and then it’s like proving Minkowski’s inequality … you change the variables, separate the integrals.…”

  * * *

  I went into semi-automatic pilot, drawing on the whole of my accumulated experience … but in order to be able to do this, first you’ve got to tap into a certain line—the famous direct line, the one that connects you to God, or at least the god of mathematics. Suddenly you hear a voice echoing in your head. It’s not the sort of thing that happens every day, I grant you. But it does happen.

  I’d tapped into the direct line once before. In the winter of 2001, when I was teaching in Lyon, I lectured every Wednesday at the Institut Henri Poincaré in Paris. On one of those Wednesdays I was explaining my quasi-solution to Cercignani’s conjecture when Thierry Bodineau interrupted me and asked whether a certain part of the statement couldn’t be improved. Thinking about it on the high-speed train back to Lyon, the TGV, divine inspiration or something very much like it showed me the way to a much more powerful proof scheme, which made it possible to give a complete demonstration of the conjecture. Then, over the next few days, I was able to broaden the conjecture by extending the argument to cover a more general case.

  But come the following Tuesday, just as I was about to proudly present my new results, I discovered a fatal error in the proof of the second theorem! I worked until three or four in the morning trying to fix it. No success.

  Scarcely awake after only a few hours’ sleep, I began mulling over the problem again. I couldn’t bear the thought of not being able to announce my results. When I left home to go to the station, my head was filled with dead ends. But no sooner had I settled into my seat on the TGV than inspiration suddenly struck again: I knew what I had to do to fix the proof.

  I spent the rest of the trip from Lyon to Paris putting everything in order, and when I entered the lecture hall I was able to keep my promise to myself after all. This made-in-TGV proof furnished the basis for one of my best articles only a few months later.

  Now, once again, on this morning, the morning of April 9, 2009, another bit of inspiration came knocking at my brain’s door and illuminated everything!

  * * *

  Probably no one who read the article that finally appeared in Acta Mathematica had the least inkling of the euphoria I experienced that morning. Technique is the only thing that matters in a proof. It’s a pity there’s no place for the most important thing of all: illumination.

  * * *

  * * *

  7.4. Growth control. To state the main result of this section we shall write and if a sequence of functions is given, then We shall use K(s)Φ(t) as a shorthand for etc.

  Theorem 7.7 (Growth control via integral inequalities). Let f 0 = f 0(v) and W = W(x) satisfy condition (L) from Subsection 2.2 with constants C0, λ 0, κ; in particular Let further

  Let with 0 < λ* < λ 0. Let be a continuous function of t ≥ 0, valued in such that

  (7.22)

  where c0 ≥ 0, m > 1, and K 0(t,τ), K1(t,τ) are nonnegative kernels. Let Then

  (i) Assume γ > 1 and K1 = cK (α),γ for some c > 0, where K(α),γ is defined by

  and appears in Proposition 7.1. Then there are positive constants C and χ, depending only on γ, λ*, λ 0, κ, c0, CW, m, uniform as γ → 1, such that if

  (7.23)

  and

  (7.24)

  then for any

  (7.25)

  where

  (7.26)

  (ii) Assume for some , where appears in Proposition 7.1; then there is a numeric constant Γ > 0 such that whenever

  one has, with the same notation as in (i),

  (7.27)

  where

  Proof of Theorem 7.7. We only treat (i), since the reasoning for (ii) is rather similar; and we only establish the conclusion as an a priori estimate, skipping the continuity/approximation argument needed to turn it into a rigorous estimate. Then the proof is done in three steps.

  Step 1: Crude pointwise bounds. From (7.22) we have

  (7.28)

  We note that for any and t ≥ 0,

  where (here as below) C stands for a numeric constant which may change from line to line. Assuming ∫ K 0(t, τ)dτ ≤ 1/2, we deduce from (7.28)

  and by Gronwall’s lemma

  (7.29)

  where

  Step 2: L2 bound. This is the step where the smallness assumption (7.23) will be the most important. For all we define

  (7.30)

  (7.31)

  (7.32)

  and we extend all these functions by 0 for negative values of t. Taking the Fourier transform in the time variable yields since condition (L) implies we deduce i.e.,

  (7.33)

  Plugging (7.33) into (7.32), we deduce

  (7.34)

  Then

  (7.35)

  (Note: We bounded by which seems very crude; but the decay of as a function of k will save us.) Next, we note that

  so

  Plugging this into (7.35) and using (7.22) again, we obtain

  (7.36)

  We separate this (by Minkowski’s inequality) into various contributions which we estimate separately. First, of course,

  (7.37)

  Next, for any T ≥ 1, by Step 1 and

  (7.38)

  Invoking Jensen and Fubini, we also have

  (7.39)

  (Basically we copied the proof of Young’s inequality.) Similarly,

  (7.40)

  The last term is also split, this time according to τ ≤ T or τ > T:

  (7.41)

  and

  (7.42)

  Gathering estimates (7.37) to (7.42), we deduce from (7.36)

  (7.43)

  where

  Using Propositions 7.1 (case γ > 1) and 7.5, as well as assumptions (7.23) and (7.24), we see that a ≤ 1/2 for χ small enough and T satisfying (7.26). Then from (7.43) it follows that

  Step 3: Refined pointwise bounds. Let us use (7.22) a third time, now for t ≥ T:

  (7.44)

  We note that for any

  so

  Then the conclusion follows from (7.44), Corollary 7.4, conditions (7.26) and (7.24), and Step 2.

  TWENTY-EIGHT

  Princeton

  April 14, 2009

  Today I officially accepted the directorship of the IHP.

  And we’re right on track with the theorem. Twice in the last few days I’ve worked until four in the morning, my resolve undiminished.

  This evening I was getting ready for another long private session with the Problem. The first step is always to boil some water.

  Then suddenly I realized there wasn’t any more tea in the house—panic! Without the stimulating leaves of Camellia sinensis, I couldn’t possibly face the hours of calculation that lay in store.

  Night had already fallen, futile to imagine finding a store open in town. Untroubled by the thought of crime, I hopped on my bike and set off to steal some tea bags from the common room of the School of Mathematics.

  Made it to Simonyi Hall in no time flat, typed in the entry code, and furtive
ly climbed the stairs to the second floor. All was darkness, except for a ray of light glimmering beneath Jean Bourgain’s door. I wasn’t the least bit surprised: even though Jean has won the highest honors and is universally regarded as one of the most powerful analysts of recent decades, he has kept the working hours of an ambitious young up-and-comer—in part, too, because he regularly visits the West Coast and likes to stay on Pacific Standard Time. It’s always a good bet you’ll find him working late into the night.

  Slipped into the common room and noiselessly pocketed the precious packets. André Weil staring down at me in disapproval. I hurried back downstairs.

  But there, standing right in my way, was Tom Spencer, a big name in statistical physics and one of my best friends at the Institute. I had no choice but to confess to my burglary.

  “Oh, tea! Keeps you going, eh?”

  Back home in a flash. The time had come at last to perform my private tea ceremony.

  And some music, please—or I shall die!

  I’ve been listening to a lot of singers lately. Catherine Ribeiro, whose songs play in a continuous loop on my computer. The tragic Danielle Messia, the forsaken one. Ribeiro, la pasionaria, hopelessly committed to her cause, come what may. The hypersensitive Mama Béa Tekielski, with her magnificent shrieking and screeching. Ribeiro, Ribeiro, Ribeiro. Music, my indispensable companion in solitary research.

  Nothing brings back long-forgotten moments in our lives quite the way music does. I remember the shocked look on my grandfather’s face the first time he heard me play a piece by Francis Poulenc: instantly he was transported back sixty years in time, back to the modest apartment whose wallpaper-thin walls were powerless to stifle the sounds produced by his next-door neighbor, a classical composer inspired by the same aesthetic ideas as Poulenc.

  For me it’s no different. When I hear Gundula Janowitz launch into Schubert’s “Gretchen am Spinnrade,” I become once more the young man hospitalized for pneumothorax in the intensive care unit of the Hôpital Cochin in Paris who spent part of his days devouring Carmen Cru and part of his nights talking about music with the interns, sleeping with an Irish teddy bear a girl had given him.

  Hearing Tom Waits rasp his way through “Cemetery Polka” takes me back to the time of my second pneumothorax, in a large hospital in Lyon where I shared a room with a guy whose ribald banter made the nurses roar with laughter.

  John Lennon’s metamorphosis into the Walrus ushers me into a hallway at the École Polytechnique in Paris after the first of my two oral examinations. Eighteen years old—the future drawing an elegant question mark above my head.

  Three years later, the dramatic opening chords of Brahms’s First Piano Concerto sounded at exactly the right moment in my little dormitory room at the École Normale Supérieure when a girl knocked on my door, distraught, looking for answers.

  Going back further, to my early childhood, several pieces of music bring back especially vivid memories: Jeannette’s insistent “Porque te vas,” the song that made her famous; Steve Waring’s gently sarcastic “Baleine bleue”; Henri Tachan’s caustic “Grand Méchant Loup”; also (no idea why) a theme from Beethoven’s Violin Concerto that my mother liked to hum.

  From when I was twelve, some of my parents’ favorite songs: Jean Ferrat’s “Les Poètes,” Maxime Le Forestier’s “Éducation sentimentale,” Leonard Cohen’s “Seems So Long Ago, Nancy,” Beau Dommage’s “La Plainte du phoque,” two songs by Les Enfants Terribles, “Horloge du fond de l’eau” and “Sur un fil blanc,” Jean-Michel Jarre’s “Oxygène,” and Graeme Allwright’s “Jusqu’à la ceinture,” in which an army captain orders his men to keep going forward even though the water is waist-high and rising.

  And from when I was a teenager, choosing among the many music videos I watched on M6 and the many cassettes I picked up here and there along the way, a medley of my own favorite songs: “Airport,” “Envole-moi,” “Tombé du ciel,” “Poulailler’s Song,” “Le Jerk,” “King Kong Five,” “Marcia Baïla,” “Lætitia,” “Barbara,” “L’Aigle noir,” “L’Oiseau de nuit,” “Les Nuits sans soleil,” “Madame Rêve,” “Sweet Dreams (Are Made of This),” “Les Mots bleus,” “The Sounds of Silence,” “The Boxer,” “Still Loving You,” “L’Étrange comédie,” “Sans contrefaçon,” “Maldòn’,” “Changer la vie,” “Le Bagad de Lann-Bihoué,” “Aux sombres héros de l’amer,” “La Ligne Holworth,” “Armstrong,” “Mississippi River,” “Le Connemara,” “Sidi H’Bibi,” “Sunday Bloody Sunday,” “Wind of Change,” “Les Murs de poussière,” “Mon Copain Bismarck,” “Hexagone,” “Le France,” “Russians,” “J’ai vu,” “Oncle Archibald,” “Sentimental Bourreau” …

  * * *

  So many times I’ve been utterly captivated by a piece of music, classical or pop or rock; so many times I’ve listened to a piece of music over and over again, in some cases hundreds of times, marveling at the state of grace that must have presided over its creation. My entry into the new world of so-called classical music began with Dvořák’s New World Symphony. After that came Bach’s Fifth Brandenburg Concerto, Beethoven’s Seventh Symphony, Rachmaninov’s Third Concerto, Mahler’s Second Symphony, Brahms’s Fourth Symphony, Prokofiev’s Sixth Sonata, Berg’s First Sonata … Liszt’s Sonata, Ligeti’s Études for Piano, Shostakovich’s ambiguous Fifth Symphony, Schubert’s Sonata D.784, Chopin’s Sixth Prelude (with a suitably dramatic interpretation, thank you very much). Boëllmann’s Toccata, Britten’s War Requiem, John Adams’s fabulous Nixon in China, the Beatles’ “A Day in the Life,” the Zombies’ “Butcher’s Tale,” the Beach Boys’ “Here Today,” Divine Comedy’s “Three Sisters,” the Têtes Raides’ “Gino,” Anne Sylvestre’s “Lisa la Goélette,” William Sheller’s “Excalibur,” Thomas Fersen’s “Monsieur.” Étienne Roda-Gil’s faux-lighthearted “Ce n’est rien,” his faux-faux-serious “Makhnovchina,” his “Le Maître du Palais” (with its palace of tartaric columns), and his “Patineur” (to the north, or possibly the south, of July). François Hadji-Lazaro singing of dikes, barges, and Paris at the barricades. Mort Shuman’s love letter to Brooklyn by the Sea and Herbert Pagani’s ode to Venice, now in danger of being drowned by a different sea. Léo Ferré’s mysterious, reorchestrated version of “Inconnue de Londres,” and the rabid dog of his “Chien,” the only one that will be left when “Il n’y a plus rien.” Bob Dylan in his watchtower recounting the terrible fate of John Brown; Pink Floyd waxing nostalgic over the green grass of yesteryear; Ástor Piazzola evoking Buenos Aires at zero hour. Two film soundtracks, Prokofiev’s “Romance” and Ennio Morricone’s “Romanzo.” Salvatore Adamo’s moving “Manuel,” whose words I once transcribed for friends in Moscow, lovers of French music and the French language, back in the days when song lyrics couldn’t be found on the Internet. Fabrizio De André weeping for Geordie, hung by a golden rope; Giorgio Gaber, taking himself for God; Paolo Conte, inviting his sweetheart to follow him. Little René Simard, making mothers in Quebec cry with his crystalline “Oiseau,” as well as young girls in Japan with his breathtaking “Non ne pleure pas” / “Midori iro no yane.” Les Frères Jacques buying a five-star general in Francis Blanche’s song “Général à vendre.” Weepers Circus offering love to vixens, Olivia Ruiz repairing broken hearts and windows, Mes Aïeux’s moronic crooks adulterating marijuana in “Ton père est un croche.” Boris Vian waxing lyrical about a nuclear thrashing, Gilbert Bécaud about a diabolical auction. Renaud reciting the saga of Gérard Lambert, François Corbier the tale of the unfortunate elephant lover. Hubert-Félix Thiéfaine’s strange world, populated by “weed-reaper” girls dispensing joints, by coffins on wheels and atomic Alligators and mucous Diogenes that got all the girls and boys dancing at the all-night parties I used to go to when I was in my twenties. Moments of riveting drama: Jacques Brel crying out, nailed to the Big Dipper; Serge Utgé-Royo reviving Jacques Debronckart’s banned song “Mutins de 1917”; Jean Ferrat hailing two children who fell in battle, to their mother Maria’s undying sorrow
; Henri Tachan howling that he doesn’t want to have a child! And the elves who take you by surprise: Kate Bush and her “Army Dreamer,” France Gall and her “Petit Soldat,” Loreena McKennitt and her “Highwayman,” Tori Amos imagining herself as the “Happy Phantom,” Jeanne Cherhal shouting “Un trait danger,” Amélie Morin playfully blaspheming “Rien ne va plus.” And my favorites, the tigresses who give you goose bumps: Melanie reproaching all the ones around her, Danielle Messia demanding to know why she has been abandoned, Patti Smith taking refuge in the night that belongs to lovers, Ute Lemper bemoaning the fate of Marie Saunders, Francesca Solleville bringing the Commune back to life, Juliette playing the little boy manqué, Nina Hagen growling Kurt Weill, Gribouille roaring her Ravens’ song, the sublime duo of Patrice Moullet and Catherine Ribeiro singing of peace, death, and the bird in front of the door!