The Weil Conjectures

by Karen Olsson

  Yes. But I think I understand you.

  They go on talking for hours, until three or four in the morning. Refusing a bed, she retires to the next room to sleep on a mat.

  Later she’ll write a terrifying prayer in a notebook:

  Father, in the name of Christ grant me this. That I may be unable to will any bodily movement, or any attempt at movement, like a total paralytic. That I may be incapable of receiving any sensation, like someone who is completely blind, deaf, and deprived of the other three senses. That I may be unable to make the slightest connection between two thoughts, even the simplest . . . That I might be insensitive to every kind of grief or joy, and incapable of any love for another being, another thing, even for myself, like old people in the last stage of decrepitude.

  At long last Simone and her parents receive permission to travel to Casablanca, where they spend seventeen days in a camp for transients. The sun beats down; there aren’t enough chairs; refugees fidget and quarrel and piss into the sand; they snatch up the newspapers, any morsel of information like scarce food. Simone, suddenly territorial, occupies a chair all day long, and when she must get up, one of her parents will sit in it, but only to save it for her. She writes all day, hard at work on an essay about the Pythagoreans.

  From there they board a ship bound for Bermuda, then New York. The crossing lasts a month, and the other cabin-class passengers, released from the deprivations and anxieties of war, indulge themselves in every way. Simone, irritated by all the carousing, tries to sleep in steerage, then on the deck. She calls the boat “the floating brothel.”

  André meets the ship when it docks in New York. The wharf is raucous with the blasting of ships’ horns, and bleating gulls, and paperboys, and he fears he won’t find them, he searches among the descending passengers for his family until he sees his parents, two little gray birds stepping gingerly down the gangplank, threading their way through masses of larger people. He doesn’t see Simone, until abruptly she comes tearing toward him.

  Her skirt is rumpled and her eyes shine out from hollows of skin and vein. They’ve barely said hello before she starts in on the possibility that she might be able to meet with President Roosevelt to discuss her plan for front-line nurses, and she talks so incessantly he finds himself losing track of what she’s saying. In another family, less accustomed to obsession, Simone might be considered insane. Another family would say that she’d lost her mind.

  PART FOUR

  9.

  Over the course of the 1940s, beginning with his time in the Bonne Nouvelle prison and then in the years that follow, during which he emigrates to the United States and becomes a father to two daughters and loses his sister, André develops a set of ideas—proposals supported by solid theorizing but not yet proved—that become known as the Weil conjectures. They develop the analogy that he described to Simone in his letter from prison, relating the fields of topology and number theory. Since topology is a mathematics of space and number theory concerns the properties of whole numbers, it’s far from clear that one should inform the other. But the Weil conjectures establish a surprising link between them, and so span the gap between the continuous (space) and the discrete (whole numbers), seemingly quite distinct and yet somehow part and parcel of the same truth.

  These conjectures are outside the limited terrain I covered in college. How little, really, I learned in college, less and less every semester, since whenever I studied one thing, all sorts of further potential studies would reveal themselves, so that instead of making progress I only walked a short way up a sand dune that continually grew larger as I went along.

  No sooner has she made it across the ocean than Simone wants to go back home. Her hope in coming to America was that she might somehow find support for her airborne nurses, but that dream doesn’t last long. I can’t go on living like this, she tells her parents.

  She spends hours in the waiting room at the French consulate, intent on convincing an official to send her back to Europe, to work for the Free French. A fixture in yet another waiting room, silently chain-smoking, scribbling in a notebook.

  She takes a nursing course in Harlem.

  In grade school we are introduced to the number line, presented as a black horizontal strip. Arrows at both ends tell us that the line extends infinitely to the left and infinitely to the right, the positive and negative numbers running off in opposite directions, the zero presiding coolly in the middle. Over the course of however many years of school we populate this line, first with whole numbers, then with fractions, then with irrationals, like √2 and π, and arrive at what are called the real numbers.

  A strange, bold term, real numbers, as though we could hold π in our hands. Sometimes they’re called simply “the reals.”

  Now (bear with me) consider just a section of that line, for example, the interval [0, 1], stretching from 0 to 1, containing 0, 1, and all the infinitely many numbers in between. And let’s say you have a map f(x), a function that takes any value of x in [0, 1] to another point f(x) within the interval [0, 1]. Starting from that section, the function will land us back in the same section.

  There’s a theorem, one that’s not hard to demonstrate in this one-dimensional case but that I still find pleasing, about any such function that is continuous, meaning roughly that if two points x1 and x2 are sufficiently close together, then f(x1) and f(x2) will also be relatively close together. In that case, there will always be at least one point fixed by this map. There’s always at least one x for which f(x) = x. No matter how you act upon the interval, no matter how you twist and torture it, there will be one or more points left in place. And there are analogues to this in two or three or any number of dimensions.

  An extension of this result, in all finite dimensions, was the Brouwer fixed-point theorem, and after that came the Lefschetz fixed-point theorem and a theory of how (roughly speaking) to count the fixed points of continuous maps. Unlike the great majority of math theorems, which I take to be strictly about math—that is to say, I don’t think of them as suggestive of anything beyond math—in my mind the fixed-point theorem ripples outward. Even the most abstract of transformations, the mathematician’s perfect, disembodied maps, cannot change everything at once. Although I’m now resorting to loose, messy analogy, I’m abandoning rigor in a way that might make a proper mathematician cringe, this inclines me to think that any transformation in any realm has a fixed point; I think of Kafka’s Gregor Samsa—of the part of Samsa that remained Samsa after he became a cockroach—or of the pieces of myself that remain constant over time, in spite of so many physical and personal alterations, in spite of the fact that over the course of however many years every cell in the body renews itself, and while none of this is in fact even remotely implied by the Brouwer fixed-point theorem or the Lefschetz fixed-point theorem or any other theorem, I nevertheless associate the idea that not everything changes at once, that in the middle of no matter what catastrophe there’s something that stays still, with the theorem’s f(x) = x.

  And then again I am tempted to say, even more loosely and unforgivably, that math itself seems to me a fixed x, a still point surrounded by human frenzy, the eye of the storm.

  Although this hardly serves to justify my gauzy appropriation of the Brouwer fixed-point theorem, I’d like to note that Luitzen Egbertus Jan Brouwer, a Dutch mathematician born in 1881, was himself drawn to mystical thinking. He had to be persuaded by his adviser to cut a section of philosophical musings from his dissertation, and on top of the math lectures he delivered to students in Amsterdam, he gave a series of more-wide-ranging talks later published in a book called Life, Art, and Mysticism.

  He married a divorced older woman, and they built a weekend cottage in Blaricum, a proto-hippie town by the ocean, where they could pursue their interest in herbal medicine and practice nudism. As he grew older Brouwer invited many fellow mathematicians to Blaricum, and though maybe he wore clothes during those visits I still imagine otherwise, the renowned topologist serving
his guests a bitter tea made from roots or sticks and offering his thoughts about homotopy theory, naked all the while.

  When Simone comes to visit, Eveline expects that she might be tentative with the baby, but no, not at all. “How beautiful you are in your angel’s robe!” she crows as she shifts Sylvie into her hands. For a moment Eveline and Simone both hold her, Eveline not quite sure, not quite ready to let go, and when she does release the baby she has to check the impulse to instantly grab her back because of the way Simone has taken possession. As though staking a claim. Simone draws her niece’s soft lump of a body to her bony one and nuzzles the baby’s head, her dark curls spilling down over tiny Sylvie.

  That a person could ever be so light and vulnerable, it’s impossible to reconcile with the weight we suppose ourselves to have as adults. It exposes the illusion of that weight. Now Simone’s lips are moving, muttering something to the baby, more intentional than simple coos and hellos and yeses. Verses, maybe, or prayers?

  Sylvie starts to cry, and Simone lifts her head, not all the way but just enough to reveal a curious, strained expression that Eveline has never seen before. She’s nearly cross-eyed. It’s as though rather than comfort Sylvie, she’s trying to absorb the crying into her own body. The baby pauses, they all pause, four hearts beating. Then Sylvie starts to cry again.

  Eveline thrusts out her arms. How reluctantly Simone surrenders. She has never looked worse, Eveline thinks, and she has never looked more alive, the way her eyes radiate out of her caved-in face.

  Simone and her brother trade short phrases in what Eveline assumes is ancient Greek. Half the time she can’t even follow what the two of them are saying. It comes as a relief that there is another person in the room, alien creature though this baby may be, an ally nonetheless. Even as her tiredness and the bottles and the diaper changes make it all the more impossible to keep up with the talk between brother and sister, now it doesn’t matter—so be it—for here’s this girl who is her own, hers and André’s.

  Isn’t she? Only look at that mat of ebony hair on Sylvie’s head, soon to become curls, exactly like her aunt’s.

  The seeming fixedness of mathematics is surely one of the reasons I’ve felt drawn back to it, given our present-day world’s particular instabilities and alternative facts, but another reason, a stronger reason, is that my son likes math, which is not to say that I need to relearn abstract algebra for his sake but rather that his excitement has reminded me of my own old excitement, has made me want to blow on the embers—has made me realize there are embers. And as I do, what strikes me are the dialogues, the exchanges, whether it’s me talking with my son about numbers or Benedict Gross’s performance of algebra. Even as mathematics presents itself from afar as an austere architecture dreamed up by singular geniuses, up close it’s a torrent of transmissions, teachers lecturing, college kids trying to solve problems together, colleagues at conferences, André writing to his sister. For every solitary discovery there are massive systems of relationships, which I begin to think of as a kind of giant math ant colony, or math hive, and I even begin to wonder whether (or conjecture that) the desire for mathematical revelation, the wish to dwell in a perfect, abstract world, is secretly, unconsciously twinned by another desire for communion. One the negative imprint of the other. Abstraction the flip side of love.

  “Nothing which exists is absolutely worthy of love,” wrote Simone.

  “We must therefore love that which does not exist.”

  For three days she stays with him, and for three days they argue. Few people listen to her like he does. He has always felt it would be cruel to tune her out, and for that matter he likes the challenge, stress-testing her intricate arguments, objecting to them, developing them further. Even when she crawls far out onto a rhetorical ledge that’s beginning to crumble underneath her, still he goes along, keeps her company until she’s ready to come back.

  His armchair engulfs her as she glares at the ceiling, smoking, still wearing her rumpled black coat; its wide collar, flecked with ash, droops down from her shoulders. I’m through with this, she says, I should’ve never left France.

  You’re safe here.

  If I’m forced to stay I will go south and work in the fields with the blacks. I don’t care if I die there.

  I think the fieldwork here is more difficult than in your Catholic vineyard.

  Good, I hope so.

  You might harvest tobacco, he muses, as more ash drifts onto her coat.

  Yes, you’re right!

  It’s settled then, he says, and sighs.

  She ignores his irony and nods once. May I give Sylvie her bottle? she asks.

  Simone can’t get enough of feeding the baby; she loves to watch the spot beneath Sylvie’s chin that pulses in and out as she swallows, loves to hear the little clicking gulps. Meanwhile Simone eats almost nothing herself, nudges her food with the very tip of her fork.

  Isn’t there some way to serve the war from here? asks André.

  Doing what? Knitting socks for soldiers? You know I’m hopeless for that sort of thing.

  A train lumbers by, laden with the steel beams that are forged in Bethlehem and sent off into the depths of this country that is continually building itself.

  There seems to be plenty of factory work, he says, only half kidding.

  I’m happy for you and your family to know this kind of peace, but I have a different purpose.

  What is it?

  If my plan for the nurses is not approved, I am going to request that I be sent into France on a mission of sabotage. Even if it’s one I won’t survive.

  Half in love with easeful death, he says in English.

  It’s not only you who know your own dharma.

  He’s quiet.

  For almost the entire hour before she leaves for the train station, Simone stands over the bassinet, gazing at Sylvie as she sleeps, whispering to her, bringing a hand to her chest to feel her breathe.

  It will be another six weeks before Simone’s British visa comes through. On the phone with André, or in the apartment with her parents, she’ll talk about the baby incessantly, her normally slow, severe voice becoming faster and higher pitched. The lovely Sylvie, the delectable Sylvie! She’s nearly as besotted as Eveline and André, or maybe more besotted, since she can love the baby purely, a distant crush, without being exhausted by her. She’ll warn her brother—though Sylvie is still an infant!—not to let her become a flirt.

  Just as she is leaving her brother’s house, stepping out the door with her bag full of books, Eveline says to her, We’ll be seeing you, Simone.

  Behind her glasses Simone’s eyes narrow. A sad pitiful smile spreads across her face. No, you won’t, she says. This is goodbye.

  10.

  Between not knowing and knowing, what is there? A doorway, a slim threshold—or maybe a dark, rocky path joining distant points: the journey from one end to the other might take years, it might take centuries.

  A conjecture forges a trail, shines a torch and clears the initial stretch. It’s a setting forth, full of plucky confidence, and at the same time it’s a reminder that the destination is not yet in sight and might well be unreachable—an aspiration that may never be realized, an arrow spinning in the wind.

  “I have had my results for a long time, but I do not yet know how to arrive at them,” wrote Gauss.

  I take it back: the region between the unknown and the known isn’t really a path, better to call it a wide expanse with very few directional markers, a field through which you beat your own track, uncertain of where it will lead and barely noticing how that choice of one track causes other tracks to disappear. All but oblivious to the quiet but tremendous collapse of countless other possible routes, avenues that will go untraveled as other theories are not developed, as all the possible books I might have written, perfect in their nonexis tence, are replaced by this flawed one. (Or take this sentence itself, knocking off whatever else I might have come up with, as I keep feeling my way toward an
unknown that will only ever recede.) Along with the optimism of conjecture, that faith in what seems true, perhaps there is or there should be a distinct unease, a regret for all that’s been lost in the process—or maybe one of those compunctions that come upon me in dreams, that sense of having forgotten to do something. In this case having forgotten to regret what was lost in the process.

  I’m leaving next week, Simone in New York writes to André in Pennsylvania. Impossible to go to Bethlehem. I’m terribly busy. I so regret not being able to say goodbye to Sylvie. I hope to see her again before she’s married. In any case I’ve cast a spell on her, you’ll see in a few years.

  André reads the letter in his living room, standing by the window with baby Sylvie in the crook of his arm. When he’s done reading he shifts her into his two hands and holds her to the window so that she can see the rain.

  Though I didn’t go far enough in math to understand the Weil conjectures, nevertheless I wonder, to what extent could I appreciate more about them? A bee in my bonnet, a dubious goal: maybe I could try to apprehend something of their flavor, I speculate, but at the same time I don’t know what that would mean. What sort of apprehending would it be? I can only recall André admonishing his sister that explaining his work to a nonmathematician would resemble explaining a symphony to a deaf person. One has to resort to mere metaphor.

  Still, I think, I could try to suss out the metaphor. I decide to look for a mathematician who might be willing to talk informally about the conjectures, and so I scan the website of the University of Texas math department. With a nervous prickle I realize that one of the professors at UT is someone I met long ago, at that math summer program—I wasn’t sure what I was doing there, while his destiny was clear: he would be a mathematician. Back then he had the pale, soft look of a person who spent his days indoors; now, to judge by the photo on the website, he’s a bit tanner and slimmer, but of course it’s him, absolutely the same guy.