I wondered now why he had never published his results, these papers he’d apparently written and stored. I’d always assumed he hadn’t the talent, but it was clear now that he did—and that he’d worked on a wide range of subjects. I thought of what he’d said all those years ago, “Blood always tells.”
He’d been a real mathematician who’d never published a single one of his works. And neither had Henry—for all her glittering talent, she had abandoned her research as far as I knew, and she’d never finished a book. Would that have been different, I wondered, if I had exposed Karl all those years ago? Would it have freed them both? I thought of Peter, and the Mohanty problem, and how if I’d taken his gift, if I’d taken that proof and said nothing—it would have broken something inside me—it would have poisoned my relationship to my work.
Still, every time I saw Peter at a conference, as I did once or twice in each decade of my life, I wondered: If I had just been braver or wiser, if I’d had the imagination for it, could we have found some way to be together? He went on to marry someone else, a few years after Germany—a lovely, gracious woman, quite bright, by all accounts, a professor of art history who filled their house with beautiful things and interesting people, a born hostess. And together they had two children. My treacherous heart! It broke every time I saw him—clamored for him, turned in on itself in grief for him. That part never changed, but as I grew older, I also grew better able to bear it. An ache that I stopped fighting, a wound I carry to this day.
All the same, there were consolations. I applied my imagination and my energy in other directions, where I was able to make my mark. I am recognized now as one of the pioneers of dynamical systems and was one of the first to use it to solve problems in other fields like game theory and topology. At the celebrations I’m invited to talk at these days, the introductions of my life and work grow ever grander, and two years ago, an entire conference was named after me. These days I’m casually referred to quite often as a “genius,” and my name is invariably included in those “best of” lists that rank mathematicians by order of influence. And while I haven’t quite risen to the heights I once dreamed of, I can’t complain. My work has been used in fields as disparate as probability theory, thermodynamics, and most recently, artificial intelligence—my work remains at the frontier of new things.
But this return to the Riemann hypothesis by way of the Schieling-Meisenbach theorem is something else altogether. It is personal. This is the problem that brought Sophie and Xi Ling together, and in some ways split me and Peter apart—even as our contribution links us together in name, forever. The connection it claims between the distribution of prime numbers and the zeta function is so stunning as to be seen as miraculous, as improbable and astounding as life. This hypothesis, which has baffled mathematicians for over 150 years and lit up my youth with all its mystique, feels within striking distance again.
I wonder if this was what had kept Karl enthralled and kept him from publishing any more work. The hope that with these notes he would one day complete Sophie and Xi Ling’s work and be able to claim—potentially—the Riemann hypothesis himself. He must have known when he had Henry send me the box that I would not expose him, that my choice was made irrevocably, long ago. Not for him, but for Henry, who had come and cleaned my apartment when I was so low I thought I’d sink into the ground. Who’d said she’d adopt me, that her family would be my family, that she wanted to give me whatever she had.
I have spent the last three years putting the pieces of Xi Ling and Sophie’s final proof together and will spend the rest of my life chasing the answer. Whether I finish or not, when I die, I will send Karl’s notes to a colleague who will recognize them for what they are—I do not want to withhold them from the mathematical world. But if he’s scooped in the meantime, well, that’s how it goes.
AS FOR THIS PROOF I am working on now, I’ve come to think of it as my twin, conceived of by the same parents and carried across the same mountains. If I could, I’d travel in time to the night Sophie wrote her letter to Karl. I’d find her and Xi Ling in their stone cabin at the peak of the mountain they’d climbed and would later cross—the wind blowing around them and Xi Ling looking for wood. No candles, no light between them, just me, unborn, and the steel trap of hunger opening and closing their stomachs, hope clawing at their hearts, and love. I would keep them there forever, I would keep them there until the end of war.
But history has happened, and there is no such thing as safe forever. Perhaps there is no such thing as the end of war. Sophie and Xi Ling will leave their cottage in the morning. There are so many things they do not know. So much suffering and uncertainty ahead. And there’s nothing I can do, no burden I can take except for what’s left of their work. I will finish what I can: I will make it my life’s work, and gladly—but otherwise I must leave them here, grateful, at least, that they have found for this night some sort of shelter.
I cannot give back what was taken from Sophie or Xi Ling. I cannot restore them to life. But perhaps I can carry their final proof a little longer and try to give it a life. Perhaps I can complete this part of their journey. And if I succeed, their proof, like all mathematical truths, will live forever.
As I work through the proof, tracing the twists and turns of their logic, I often think of Peter. I see now what dedication it must have taken on his part to tackle the Mohanty problem for me—not mere commitment, but what painstaking love—to enter so wholly the architecture of my mind, to follow so carefully its rhythms. I can admit now how much I owe him for all that he taught me, and how he encouraged me, and how even that terrible betrayal he made out of love. I wish I could have told him I realized this before he died. I wish he could have known.
Sometimes I think of the problem he gave me in grad school, before we were lovers, when we were still just teacher and student. There’s a girl and a boy living on opposite sides of a lake. There’s a ferryman who can go back and forth, carrying a box on which you can put any lock. He’ll take anything over locked in the box, but anything else he will steal. The boy has a lock and the key to his lock, and the girl has a lock and the key to her lock. How can the boy get her a ring?
The solution is this: the boy puts the ring in the box, and then locks it shut with his lock. The ferryman takes it across. When it arrives, the girl puts her lock on the box, next to the boy’s, and sends the ferryman back. The box at this point is locked twice, and when it reaches the shore, the boy takes out his key and unlocks his lock, leaving only the girl’s. Then the ferryman makes his trip across the lake, back to the girl. She takes out her key and unlocks her own lock. And now the box is completely unlocked, and the girl opens it, and takes out the ring.
And so we’ve brought the boy and the girl together at last, but what they do next will be up to them. Because though we’ve found the solution, here is the lesson: in the end, we can only unlock our own locks, we have only the gift of ourselves.
November 1943
THEY ARE WALKING, JUST THE TWO OF THEM: SOPHIE and Xi Ling, Schieling and Meisenbach. It seems to Sophie they have always been walking and will never stop. After months of sickness, she has finally stopped vomiting, but she is always hungry and cold, her legs are tight and her stomach is heavy and sore. She can see in the bones of her lover’s face what a toll this journey has taken on him. When she runs her fingers along the bones of her own face, she can feel their sharpness under her skin. They have been reduced to the essentials, to so little, and now they are walking what she hopes will be the final leg of this journey, if only they can cross over to safety.
Xi Ling carries on his back the culmination of their work together. Each night while she lies on her side and tries to sleep, he stays awake and shuffles through their papers, muttering to himself, scratching out formulae that sometimes he whispers into her ear as she lies in the dark, wishing he would lay himself against her and wrap her in his warmth. But she does not interrupt him. There is an urgency to his whispering: she knows he is racing ag
ainst time. They feel it always, running against them. Sophie sighs. She has not contributed to their work in months. Her mind is foggy, her body taken over by the needs of the life she carries in her belly: heavy, rounded, as full as the earth. She runs her hands over the swell of it: the one part of her not left emaciated—the one part of her that is not, really, herself.
In her pocket is a notebook, just one small notebook that fits in the palm of her hand. All around them are mountains, beautiful to look at and treacherous to cross. Beside her is the man she loves, who has pledged never to leave her, his child stirring in her belly. In her mouth is a sour taste, tinged with blood. The world, she knows, is coming apart around her. Everything in Sophie’s mind comes down to this: these mountains, this man, this promise of a life that will come, regardless. Let her be all right, she thinks. This child of mine. It is her only thought. I would gladly trade in everything. Anything that is mine to give I would give, willingly, with gratitude. She is not creating life, this much she understands. She is the vessel: life is creating itself inside her, using her, making itself out of her.
What wouldn’t she steal, who wouldn’t she murder, what wouldn’t she do for this child? She would die right now, this moment, give up meeting this bit of life that has so hungrily devoured and distorted her body. What wouldn’t she give? She clenches her fists. Let her live, she thinks. Let me safely deliver this child into her life.
Author’s Note
I am indebted to a number of texts, including, in particular: The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus du Sautoy, Birth of a Theorem by Cédric Villani, Prime Obsession by John Derbyshire, Emmy Noether, 1882–1935 by Auguste Dick, Hilbert by Constance Reid, Men of Mathematics by E. T. Bell, A Mathematician’s Apology by G. H. Hardy, Geons, Black Holes, and Quantum Foam: A Life in Physics by John Archibald Wheeler with Kenneth Ford, The Apprenticeship of a Mathematician by André Weil, The Man Who Knew Infinity by Robert Kanigel, The Honors Class: Hilbert’s Problems and Their Solvers by Ben Yandell, “The Sexual Politics of Genius” by Moon Duchin, “The Pursuit of Beauty” by Alec Wilkinson, Asian Americans in Michigan, edited by Sook Wilkinson and Victor Jew, Higher Education for Women in Postwar America, 1945–1965 by Linda Eisenmann, and Keep the Damned Women Out by Nancy Weiss Malkiel.
I am also indebted to the Institute for Advanced Study in Princeton for the use of its archives, and the library at the University of Göttingen, where I was able to read letters and notebooks relating to (and written by) Emmy Noether, Albert Einstein, David Hilbert, Abraham Flexner, Oswald Veblen, and Hermann Weyl.
The historical figures mentioned in The Tenth Muse are almost entirely made up of mathematicians and scientists. The stories of Emmy Noether, Sophie Germain, Sofia Kovalevskaya, Maria Mayer, Hypatia, Srinivasa Ramanujan, Alan Turing, and David Hilbert and his program in Göttingen make brief appearances in this book. Also mentioned are Jocelyn Burnell, Chien-Shiung Wu, Rosalind Franklin, Lise Meitner, Albert Einstein, Felix Klein, Alexander Grothendieck, Charles Fefferman, Kurt Gödel, G. H. Hardy, Carl Friedrich Gauss, Richard Courant, Henri Poincaré, Niels Bohr, Ludwig Boltzmann, Janos Bolyai, Leonard Euler, Otto Hahn, Werner Heisenberg, Andrey Kolmogorov, Edmund Landau, Adrien-Marie Legendre, Nikolai Lobackevsky, Edward Lorenz, Hermann Minkowski, Friedrich Nietzsche, Atle Selberg, Stephen Smale, Lawrence Summers, Hermann Weyl, and Ludwig Wittgenstein.
I make use of a number of real equations and formulas:
In the 1700s Carl Friedrich Gauss shocked his elementary school teacher by solving the sum of numbers between 1 and 100 in a matter of minutes (some say seconds). Like Katherine, he recognized a pattern—that if he split the group in half—the numbers from 1 to 50 and 51 to 100, and added them together from opposite sides, he’d get 1 + 100 = 101, 2 + 99 = 101, and so on and so forth until 50 + 51 = 101. Since there’d be fifty such pairings, 50 × 101 = 5050. Thanks to Gauss we even have a shortcut to this shortcut in the guise of a formula for calculating such sums: S = . So the sum S of Gauss’s elementary school problem up to 100 is: S = = 5050, and if you remember Katherine’s much simplified problem, the sum of the numbers 1 through 9 is = 45.
The Boltzmann equation and the zeta function, also mentioned in this novel, are real, as is the Riemann hypothesis, which was first proposed by Bernhard Riemann in 1859. In 1900 David Hilbert included it in his list of twenty-three problems to define the future of mathematics at the International Congress of Mathematicians, and it remains unsolved to this day. In fact, the Clay Institute is offering $1 million to the person who solves it first. The Schieling-Meisenbach theorem is pure fiction, as are the Mohanty problem and the Kobalesky formula, and so—alas—will be of no help at all in the search for a final proof of the Riemann hypothesis.
Acknowledgments
I am grateful to so many people without whom this book would not exist. To Karen Uhlenbeck, meeting you was a great stroke of good luck. Thank you for your time and generosity, and above all your warmth and kindness, and for whispering the name of the Riemann hypothesis into my ear.
To Amanda Folsom, thank you for your perspective and insight, and for explaining some math to me in a moment of critical need. To Ben Recht, for gamely answering my preposterous questions with the same patience and generosity as when you were my favorite math TA in college, thank you.
Thank you to Pablo Londero for explaining Hawking radiation to me, and to Freeman Dyson for telling me about chaos theory and Mary Cartwright, and to all the people who talked to me about math and science over the years: this book is infused with the spirit of those conversations. To Debbie Endo and Keith Endo for your heroic last-minute reads and consultations about some of the times depicted in this book, I thank you. Thanks also to Michael Lynch, James Walsh, and Michelle Cahr for so kindly and quickly responding to my questions. Any errors or crimes against mathematics or history that may remain are unintentional and mine alone. Thanks to Jen Gann at The Cut, for publishing a tiny piece of mine containing a nod to a line from this book.
To Jin Auh, thank you for reading this book five million times! You are equal parts ferocity and grace, and I am so grateful for you. To Jessica Friedman, thank you. To Andrew Wylie, Alexandra Christie, Alba Ziegler-Bailey, Sarah Chalfant, Jessica Bullock, and everyone at the Wylie Agency, you guys are the best of the best.
To Megan Lynch, I am always amazed by your keen eye and killer instinct. Thank you for making this story so much better, and for ushering another one of my books into the world. To Daniel Halpern, Sara Birmingham, Laurie McGee, Miriam Parker, Meghan Deans, Ashlyn Edwards, Allison Saltzman, Suet Chong, Nyamekye Waliyaya, Kristin Bowers, Caitlin Mulrooney-Lyski, and David Palmer, I am so deeply grateful. Long live Team Ecco! Thank you to Joan Wong for the beautiful cover. Heartfelt thanks to my UK editor, Clare Smith, at Little, Brown.
Everlasting gratitude to my brilliant readers and beloved friends—thank you for reading the earliest scratchings of what became this book, and for selflessly reading drafts in the midst of your own busy schedules to drag me across the finish line: Autumn Watts, Ben Warner, Dwayne Betts, Helen Oyeyemi, Kimberly (Panshee!) Capinpin, Lauren Alleyne, Matthew Salesses, Pilar Gómez-Ibáñez, Rita Zoey Chin, and Seth Endo, thank you. Where would I be without you? (To shamelessly misquote Whitman: We were together. I forget the rest.)
To the Institute of Advanced Study: you are a dream made real. Thank you for the months I spent happily buried in your archives. Thanks especially to Robbert Dijkgraaf and Stephen Adler for the inspiration and wisdom, and for bringing me there, and to Sebastian Currier for his friendship. Erica Mosner, your help in the archives was absolutely essential, and your kindness and radiance of spirit still warm my soul. To Anthony Adler—thank you! To Piet Hut: I thank you in particular for the beautiful memory of the afternoon when we sat in the shade and you showed me how to look at the trees and the buildings, and how they were regarding me back in return.
Thank you to Thomas Richter, for your tour of Göttingen and your spectacu
lar generosity. Thanks also to Bernhard Hackstette, Samuel Patterson, Susanne Ude-Koeller, Doris Hayne, Roswitha Brinkmann, and Esther von Richthofen. To Helmut Rohlfing, thank you for letting me hold the letters of Emmy Noether, Carl Gauss, and Albert Einstein in my hands.
I am forever grateful for the support of the National Endowment for the Arts, the Frederick Lewis Allen Room at the New York Public Library (and especially Melanie Locay), the Jerome Foundation, the MacDowell Colony, Yaddo, UCross, VCCA, and Civitella Rainieri (O you beautiful castle on a hill!). Dana Prescott, your warm encouragement and our poetry exchange have meant so much to me. To MEC, thank you for letting me write in your office, where I got so much done. I am grateful to the memory of Lee Hall, whose encouragement and faith warmed these pages.
Thanks also to Sam Grogg and Adelphi University for the research support to write this book, and to Peter West and Judith Baumel for encouraging me to take the time. Thanks to my students there and elsewhere for your hunger and ambition, and for reminding me that stories are always being born anew.
To Clare Wu, Frances Cowhig, and Faye Chiao, thank you for running away with me on our writing retreats. To Steph Opitz, Michael Lowenthal, Christine Hyung-Oak Lee, Sharon Guskin, Alex Chee, Meakin Armstrong, and Téa Obreht, your friendship and faith mean the world to me. Thank you to Mingmar Lama and Judy Brown-Steele.
To Lena Mushkina-Livshiz, Michael Livshiz, and Toma Livshiz, thank you for your enthusiasm and warmth, and your spirited conversations. I love being part of your family.
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