The Weil Conjectures

by Karen Olsson

  “After this his fame grew great,” the ancient Roman philosopher Porphyry of Tyre wrote of Pythagoras, “and he won many followers from the city itself (not only men but women also, one of whom, Theano, became very well known too) and many princes and chieftains from the barbarian territory around. What he said to his associates, nobody can say for certain; for silence with them was of no ordinary kind.”

  Neugebauer, the scholar of ancient Babylonia, was an Austrian who fought in World War I and wound up in the same prisoner-of-war camp as Ludwig Wittgenstein. After Hitler rose to power, Neugebauer found a job in Denmark and over the course of several years published three volumes of his translations. Then, in 1939, he emigrated to America, where in collaboration with another scholar he published an English version of his magnum opus, as Mathematical Cuneiform Texts.

  He would also write and publish a journal article called “The Study of Wretched Subjects.”

  In his lectures, the Professor Gross of 2003 manages to impart something like suspense and drama to a subject without much inherent narrative tension. There’s an urgency to his presentation, a vigor born of logic itself. “Now I claim,” he says, raising his voice and drawing out the I and the claim, like a magician announcing his next trick—“Now I CLAIM . . . that r = 0.”

  He has a fondness for historical digressions (as do I, obviously), and during the first week of class, he presents one as though it were a theorem: “If you go to Paris and if you take the ligne de sud to the Parisian suburb of Bourg-la-Reine, an absolutely disgusting suburb, and you go to the main intersection of Bourg-la-Reine, an absolutely disgusting intersection, you will find a plaque that says, Ici est née Évariste Galois, mathematicien.”

  And if you give an introductory algebra course, surely you must mention Galois. A misunderstood, mournful prodigy, his short career a series of arguments and rebuffs, he bequeathed to the world not only the fundamentals of what is called group theory but also the romantic image of the mathematician in its purest, most distilled form. He died in 1832, when he was twenty, as the result of a duel: pistols at dawn, the whole shebang. His father, a provincial mayor slandered by enemies, had committed suicide. The papers he’d submitted to the Académie Française had been lost or rejected. He’d been jailed for making treasonous remarks while drunk at a dinner. The night before the duel, or so the legend goes, he tried to write down everything he knew but hadn’t yet committed to paper, all the math swirling around in his head, pausing to scribble in the margins, “I have not time, I have not time.”

  This tale circulated for many years, though it was steeped in exaggeration. (For instance, Galois wrote up his major mathematical ideas in those papers he submitted to the Académie Française, rather than on the eve of the duel.) Galois’s story aside, his insights helped transform the study of equations, which in the nineteenth century would be flipped on its head. He and others developed an inspired new approach: Instead of trying to understand an equation by finding the solutions, one could presume the solutions and examine the domain in which those solutions exist. To say it another way, given the polynomial equations we learned about in school (equations like x2 – 5x + 6 = 0) and were tasked with finding the roots of (here x = 2 and x = 3), you could take a step back into a more abstract realm: instead of solving for those roots, you could assume they existed, designate them by symbols (u and v rather than 2 and 3), and consider the smallest set containing u, v, and all the numbers you could obtain from them by adding, subtracting, multiplying, and/or dividing. By investigating the properties of such sets of numbers, rather than the original polynomials, you could cast the study of equations in a new light.

  “Gradually the attachment of the symbols to the world of numbers loosened, and they began to drift free, taking on lives of their own,” the historian and novelist John Derbyshire writes about this shift in algebraic thinking. What the x might represent mattered less than what the x might do in various circumstances. It was the sort of turn toward abstraction that Simone would come to hate but was hugely fruitful in a way that she would refuse to see.

  When the Pythagoreans claimed that the universe was number, they meant it, literally; for them creation had begun with a unit, a number somehow possessed of spatial magnitude. First there was a single unit, then it split into two units, and as it did so, the universe drew into itself a void that would keep the two units apart—as though taking in breath, they said. The universe inhaling the void. The two units begat the first line, and then, by more splitting, a triangle, then a tetrahedron. From a single point all of geometry spilled forth, and this geometry constituted the material world.

  Much of this belief system was noted by Aristotle, who wrote about Pythagorean thought and, more often than not, dismissed it as absurd. “All . . . suppose number to consist of abstract units, except the Pythagoreans; but they suppose the numbers to have magnitude.”

  Every day André is permitted a thirty-minute walk inside the circular prison yard. The yard is divided into sectors, and the prisoner is restricted to one of them, as the guards watch him from a tower in the center. Though André normally wears sandals even in winter, here he has only the ill-fitting, bulky work shoes that he was given in Finland. Nonetheless he tries to walk fast, clopping along in his brogans, around and around the sector’s edge. When he walks toward the guard tower he thinks about complex functions. When he walks away from it he cranes his neck to see the birds, the clouds beyond the fence.

  “My dear sister,” he writes, when he is back in his cell, “Telling nonspecialists of my research or of any other mathematical research, it seems to me, is like explaining a symphony to a deaf person. It could be attempted, you could talk of images and themes, of sad harmonies or triumphant dissonances, but in the end what would you have?

  “A kind of poem, good or bad, unrelated to the thing it pretends to describe.”

  You might compare mathematics to an art, he goes on, to a type of sculpture in a very hard, resistant material. The grains and countergrains of the material, its very essence, limit the mathematician in a manner that gives his work the aspect of objectivity. But just like any work of art, it is inexplicable: the work itself is its explanation.

  As for Babylonian algebra, it did in fact infiltrate Greek mathematics, but it was algebra translated into geometric terms, he writes. Take for example the work of Apollonius, in which algebraic equations became conic sections: parabolas, hyperbolas, ellipses. Yet the most original thing about Greek mathematics is that the Greeks didn’t deal in approximations; they killed number for the benefit of Logos. In other words the Greeks, rather than just using numbers to calculate, considered them as pure quantities, and this abstraction—that is, this conceiving of the whole numbers as such—constituted the leap forward from which everything else followed.

  So here we have André in prison, Simone at loose ends, all Europe going to hell—and they take to arguing about the nature of ancient mathematics.

  (But who’s to say, when the world is going to hell, that you shouldn’t argue about ancient mathematics?)

  When it was discovered that the whole numbers couldn’t fully account for even simple geometric relationships, the Greeks had to start over, he writes, at the foot of the hill. Since one could no longer be sure of anything.

  Another morsel from Eros the Bittersweet: our word symbol comes from the ancient Greek symbolon, which was half of a knucklebone, “carried as a token of identity to someone who has the other half,” like one of those cheap heart-shaped lockets that break into two pieces. I imagine André and Simone as having such half knucklebones. Each sibling carrying one, as a symbol.

  Although I’ve never been susceptible to phantom sense impressions, one day as I watched an algebra lecture on my laptop I began to smell chalk dust so strongly that I wondered whether my kids hadn’t stashed some sidewalk chalk in my office. (They hadn’t.) Thick in my nostrils, I could’ve sworn: vintage chalk dust from 2003.

  That year, the class reminds me, was a long tim
e ago. In one lecture, a phone starts to ring, and Professor Gross, with a very 2003 self-consciousness about the technology, pulls out a flip phone, opens it, and without preamble asks, “Is the governor of Puerto Rico here?”

  In 2003 he is also the dean of Harvard College, and as he explains after the call, he is supposed to receive the visiting governor of Puerto Rico later that day.

  He turns to face the class. “Moocho goosto!” he says with relish—mucho gusto, pleased to meet you. Then: “I’m practicing my Spanish.”

  In a postscript to that letter in which André tells his sister that it would be pointless to explain his work, he relates a story about the nineteenth-century Norwegian mathematician Sophus Lie—for whom I myself have a sympathetic fondness, because he was a tall Scandinavian who for a long time couldn’t make up his mind about what to study or which career to pursue. As a university student, Lie had first tried his hand at languages, then switched to science, after failing Greek. He would later say that he found the road to mathematics long and difficult. But he had a thing for long roads—one weekend he walked the sixty kilometers from the capital to the town where his parents lived, only to find that they were not at home, at which point he turned around and walked back—and when he was twenty-six, he encountered certain new ideas in geometry that led him to embrace math for good.

  “The powerful Northman with the frank open glance and the merry laugh, a hero in whose presence the common and the mean could not venture to show themselves,” wrote Anna Klein about Lie. Anna was the wife of Felix Klein, another renowned mathematician and a friend of Lie’s. As young men they both lived in Paris, in adjacent rooms. Felix Klein would recall that on one morning he had risen early and was headed out for the day when Lie, still in bed, called to him. During the night, Lie explained, he had discovered a connection between lines of curvature and minimal lines.

  I did not understand one word, Klein would write. Yet later on, after he had left and was going about his business, he perceived in a flash what Lie had meant and how to prove it geometrically. That afternoon he returned home and, finding that Lie had gone out, wrote a letter containing the proof.

  I imagine the excitement he must’ve felt as he slid it under Lie’s door.

  Dude, that is so rad.

  In another of his lectures, Gross pauses to consider a particular function (if you’re curious: the function f(t) = e2πit, along with its derivative f' = 2πif), and then he invokes Lie.

  “This is what Sophus Lie discovered,” Gross says, with feeling, “that the study of group homomorphisms in the context of continuous groups is intimately related to the solution of differential equations!”

  Leaving aside the details here, the gist of the matter is that Lie, by expanding a theory in algebra, namely the study of groups, found a way to shed light on an entirely separate area of math—or one that had seemed entirely separate, namely differential equations. It’s as though he located a wormhole from one mathematical realm to another.

  “Is that amazing?” Gross exclaims. A pause, and then: “But I digress.”

  The Franco-Prussian War forced Klein and Lie to flee Paris. Lie, naturally, left on foot. Near the town of Fontainebleau, he was arrested and (as André wrote to Simone) accused of spying for the Prussians. His mathematical notes were thought to contain coded military secrets, just as André’s papers would draw the attention of the Finnish police seventy years later.

  “Occupied unceasingly with the ideas which were fermenting in his head, he walked in the forest each day, stopping at the places furthest from the beaten track, taking notes, drawing diagrams in pencil,” recalled one of his French colleagues, Gaston Darboux. “At that time this was quite enough to awaken suspicions.” Darboux appealed to the imperial prosecutor, insisting that the notes had nothing to do with national security, and after a month in prison, Lie was released.

  He returned to Norway, though in his native country he eventually came to feel isolated, ignored by mathematicians working on the Continent. In a letter to Klein he complained that Darboux “plunders my work”—that is to say, the Frenchman who’d pleaded to save Lie’s neck was stealing his results and publishing them as his own.

  Simone imagines that a young paratrooper has landed on the terrace of the Weil family apartment, a bewildered German teenager for whom, in her reverie, she feels nothing but tenderness. At dinner she asks her parents whether they would offer shelter to a German soldier.

  Absolutely not, says her father. I would turn him in, of course.

  She refuses to eat another bite until he promises her that he would help the young man, this hypothetical soldier she saw in a daydream.

  Selma Weil, who has always fretted about her daughter, now worries about her son, too, not to mention the fate of France itself. She goes all around the neighborhood sharing gossip and advice, buying up more food than her kitchen can hold, huffing her way upstairs and back down again, and if she happens to sit still, she has the twitchy look of a hare. Her dark eyes are anxious, vigilant. She’s had her bangs cut very short, which has the contradictory effect of making her look girlish while exposing her lined forehead. One day she absently pulls a button off her sweater, unaware that she’s been tugging at it until she finds it in her hand. She stares at it in surprise, then mutters to herself in Russian, much as she claims to have forgotten every word of the language she spoke as a young girl. She writes to cousins in California who have invited the family to come live with them. She rereads André’s letters. She plays her beloved Beethoven sonatas, too loud and too fast.

  “What can I say about myself?” André writes to Eveline. “I am like the snail, I have withdrawn inside my shell; almost nothing can get through it, in either direction.”

  Only paper, that of his correspondence, his manuscripts, his books. Because he is allowed those things, it is a tolerable existence, that of a snail who executes the following actions: work, eat, write, read, sleep. While his life is stripped-down and constrained, he strides ahead in his mathematical investigations, making progress in the area of algebraic curves.

  His colleagues, who’d previously written him to offer their sympathy, now envy him instead, reminding him that not everyone is so lucky to sit and work undisturbed. “I’m beginning to think that nothing is more conducive to the abstract sciences than prison,” he tells Eveline. Her letters back to him are reassuringly full of activity; she says that she’s been cultivating pea plants, trying to teach Alain about fractions. In a sense this arrangement merely heightens a division of labor they’d already adopted, André off by himself, contemplating mathematics, and Eveline out in the world, with her child and later with their children.

  “A heavy, opaque, suffocating atmosphere has settled over the country,” Simone writes to André. “People are downhearted, discontent, but they also tend to swallow whatever they are served without protest or surprise. Characteristic situation of a period of tyranny. Unhappiness is joined to an absence of hope. France is and will be for a long time (unless there’s a social convulsion) in a state of torpor and resignation.”

  She sends a short letter while working on a draft of a longer one, funneling a fury of thoughts into her neat, upright script. She sets the finished pages next to her on the train-station bench and tucks the edges under her thigh while she prosecutes her argument about the ancients. Countering her brother, she insists that the Greek stance toward algebra can’t be explained merely by saying that they assimilated algebra into their geometry. It must have been taboo—algebra must have seemed impious, she writes. Mathematics, for the Greeks, was not just a mental exercise but a key to nature. It illuminated a structural identity between the human mind and the universe.

  As for André’s comparison of mathematics to art in a hard material, Simone has her doubts: What material? The proper arts work on material that exists physically. Even poetry has the material of language regarded as an ensemble of sounds. The material of mathematical art is a metaphor, and to what does it correspond?
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  The material of Greek geometry was space, but space in three dimensions, actual space, a constraint imposed on all men. It is no longer this way. Your material, she says to André, isn’t it just the ensemble of previous mathematical work, a system of signs? Rather than a point of contact between man and the universe, Simone argues, present-day mathematics has become inaccessible.

  In 1886 Lie moved to Leipzig to take over a position from Klein, who’d been hired by the University of Göttingen. There he became known for his informal way of teaching (Felix Hausdorff was one of his students), yet he began to chafe at the workload and the fact that his colleagues treated him as Klein’s disciple. He found it harder and harder to sleep at night, and finally he suffered a nervous breakdown. At a psychiatric hospital near Hanover, he resisted the prescribed opium treatment. Although he returned to teaching the following year, it took that year and the next before his insomnia passed. “With sleep,” he wrote, “the pleasure of life and work has returned.”

  But he was a changed person, touchy and irritable. He wrote to friends in Norway that he longed to return there. He clashed with Klein, his friend of more than twenty years, then publicly attacked him. (“I am no pupil of Klein, nor is the opposite the case, although that might be nearer to the truth,” he wrote.) Klein found this “both painful and incomprehensible,” Anna would recall, but “soon my husband understood that his best friend was ill and could not be held responsible for his acts.”