The Mathematician’s Shiva (18 page)

Two visual images that came to me back to back were the foundation of all my significant work on turbulence. At face value, they are trivial images. But that’s how it always works. Just like life has its origins in ordinary elements, the foundation for human creativity is ultimately quotidian.

It all happened on one summer day. We were studying geography at the time and I suppose that our teacher wanted to make the maps of our lessons come to life. She asked all of us, “Who has seen the ocean?” No one raised a hand. Later I found out that my teacher had never seen the ocean herself. I had seen the Black Sea, of course, when we briefly lived in Odessa. But the ocean seemed to be of an unfathomable scale to my mind. It seemed improbable that I would ever be able to touch its waters anywhere, much less in the remote location to which we had been banished.

Somehow Mrs. Sharekhova, the woman whose observant eyes were responsible for the strong foundation in mathematics given to me, managed to gather three military vehicles to take us to the coast. It was a distance of approximately twenty kilometers along a crude road.

Many of us were excited, even giddy, on this trip. It was perhaps the first time any of us felt like we were real children, not adults of small size, since the war began. When the ocean came into view, I could not believe what my eyes took in. I did at first think I was seeing a mirage. You can talk about the vastness of the ocean, but until you actually see it, any discussion is inherently meaningless.

We climbed out of the vehicles onto the sand. I reached down and felt its coarseness with my fingers. I had never touched anything like it, damp and full of organic debris from the sea. The air also was new, like nothing I had smelled before, a mix of sulfur, moisture, and salt. Mrs. Sharekhova led us on a walk to the ocean while the drivers stayed near the vehicles, smoking and chatting. For us, it was magical, truly, to spend this little time on a beach, something that seemed like a vacation. It was a wonderful respite from our squalor. The wind blew against my face as I looked out at the waves crashing in the distance. The sand was mixed with seaweed. Wind-weathered branches were scattered along the shore.

We got closer and closer to the waves and I could feel the adrenaline surge in me as I looked at the water. It was so noisy and dramatic. I watched the waves swell to what seemed to be an impossible height and then crash perhaps ten meters offshore, their progress apparently impeded by an invisible sandbar.

Our teacher walked with us to the edge of the water. I could sense that she felt brave doing this. I could also sense, or perhaps I’m projecting, the pride that she felt for finding the wherewithal and tenacity to take us all there.

Mrs. Sharekhova reached down and felt the water against her fingertips. Most of us followed her lead. She took the salt water from her fingers to her lips and tasted it. I’ll never forget my own taste, so remarkable it was to sense the ocean this way. I watched the waves crash and crash again. This was turbulence, the movement of water ascending and curling, the cascades coming one after another. How can one possibly solve the problem of turbulence without actually seeing it first, not in a picture, but in real life in all its dimensions and scope? I would have made no progress at all that summer and fall had Mrs. Sharekhova not made the extraordinary effort to let us witness this wonder of nature.

I don’t know how long we actually stayed there. Perhaps we walked, ever so carefully, along the shore for an hour. The weather was not pleasant, but we didn’t care. This visit was a true gift for us, a cherished freedom. We might have stayed longer, but the soldiers were getting so impatient that they shot their rifles into the air, and we knew it was time to go back. I cannot describe how wonderful I felt as we made that final walk.

As we approached the vehicles, insects, which apparently didn’t like the salt water at all, returned to hover around our faces. They were as voracious as ever. I watched their motion intently, like I had never watched them before, because it was something fresh again and strangely unexpected. No, their paths were not, in any pure sense, turbulent. But these little creatures did not move independently. They were constrained, somehow, by the motion of their neighbors, who were in turn constrained by others more distant. Their dance was somewhat akin to the dance of fluid particles. Like the particles that made up the waves in the ocean, their ascent and descent required some information to transfer from one insect to the next. I, of course, did not like the presence of these biting bugs one bit, but I was learning something from their patterns of flight. The equations that could describe insect flight behavior were not the same as those that described fluid flow, but they were related. The horrible bites had a silver lining. If there had been no clouds of insects as we left the beach (and in our camp always), I likely wouldn’t have been able to start to truly understand turbulence. Insects and an ocean. I needed both.

I drew pictures of what I saw at the ocean that day. I can still recall those images on paper from when I was young. They became the foundation of decades of work. At the age of eleven years, far from any semblance of civilization, while my father did all he could to keep us alive another day, I began an intellectual quest to understand turbulence. More formally, I wished to prove the following: in three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes equations. It would be both the most difficult and most private achievement of my life.

DAY 2

“H
ow does she do it?” Yakov asked, slurping his bowl of
ukha
at the kitchen table. I sat across from him. He looked like the happiest man on earth.

“Recipes, I guess,” I said.

“There are recipes and then there is talent. This woman has a feel for Russian food. Beautiful, too. I talked to her when she came this morning. She has a head on her shoulders. You sure she really is an American? Maybe she came here when she was very young.” Yakov again sounded hopeful, despite the long odds.

“Definitely, she was born here. I think she was going for a Ph.D. in chemistry before she had kids. You in love?”

“Maybe. There is not one woman in all of Nebraska like her. I know. I’ve already looked.” Yakov looked down at the soup with a baleful expression, as if the reflection from the bowl was playing a movie of his failed attempts at romance.

“I don’t think you’ve looked that hard, Yakov. Not to diminish the allure of Jenny Rivkin.”

“You have no idea. I’ve looked plenty. Here, taste.” Yakov held out his spoon.

“You’re going to do this every day? Feed me?”

“Were you disappointed last time?”

“Can’t say that I was.” This was undeniably true.

“Then taste again.” I reached down and Yakov fed me like I was a baby.

“Really is good. You’re right.”

“Of course I’m right. And it’s healthy. Perfect. I don’t understand how you can live in bumblefuck Alabama when you could live here, be with a woman like Jenny, and be with your family. You’re crazy.” Yakov was trying to claim an edge over me in hard-won wisdom.

“I wouldn’t have a job.”

“Now that’s a lie. The people in your department here. What do they call it, meteorology?” There it was: the dismissiveness of all mathematicians toward any field that wasn’t mathematics.

“The department of atmospheric and oceanic sciences is what it’s called.”

“Whatever. People add ‘sciences’ to a department name only when they are worried they aren’t really scientists. They would hire you in a heartbeat. Your mother told me. You broke her heart staying away, you little
moshennik
.” He took his free index finger and scolded me. If I had reached over just a bit, I could have chomped on his reprimanding finger and ended our little discourse instantly.

“My mother understood my reasons.”

“You and your mother. I never understood either of you. Your mother was too smart. And you, you’re just perverse on purpose.”

“I hope you’re wrong. But it could be so.”

“You have Jenny’s number or am I going to have to find it in the phone book?”

“The phone book would be best. You had better be careful. She is a smart one, you’re right. Used to come to our house and talk with my mother about men, right where you are sitting. One of my mother’s little sisters, coming here for advice. She’ll see right through you.”

“My intentions are entirely honorable, Sasha.” Yakov looked genuinely affronted.

“I see. What happens if she burns the food one day, Yakov?”

“Everybody makes mistakes.”

“Maybe you are in love after all.”

It was day two of the shiva, and people already looked tired, except for Yakov, who seemed to be thriving in the presence of all of his friends, heroes, and the good Russian food of Jenny Rivkin.

“What did you mathematicians do last night, anyway?” I asked Yakov. I was truly curious.

“We went to the conference room at the department. All of us. We drank. We worked. What the hell else can we do? It’s who we are.”

“And what exactly were you working on?”

“Do I have to tell you?”

“Navier-Stokes is hard, isn’t it?”

“Shut up. Of course it’s hard.”

“But that’s ridiculous. It would take years to come up with a solution. Plus, there are too many cooks working on the recipe. Over a dozen of you together in one room. It’s madness.” I had hit a nerve. Mathematicians are solitary creatures by nature. They sit in their offices day after day looking into space, and wait for the moment when they can actually use the paper and pencil they have on their desks. It can take days, months, and maybe years before they actually write anything meaningful. A mathematician will even dream about his or her unsolved problem, often to no avail. Then one day it happens for no apparent reason. Just like that a solution appears, or a major step to a solution is envisioned. To the unfortunate mathematicians tackling major problems like Hilbert’s sixth, though, it’s likely that no solution will ever come. Success will always be beyond their grasp.

“I know, I know. It sounds like something doomed to failure. But in other fields people work in teams nowadays. It’s the new way of business. Look at physics. They are like armies now, with soldiers and officers clawing their way to solutions. Maybe it’s time for us to work that way. That’s what Zhelezniak says, at any rate.”

“So by day you mourn and by night you try to solve something my mother supposedly cheated death to finish.”

“Did she finish it?”

“Maybe.”

“Now you’re torturing me.”

“That memoir I mentioned at the funeral? Last night, I translated some more. It seems she was working on it.”

“I knew it!”

“Yeah, she was working on Navier-Stokes sixty years ago when she was eleven years old. That’s what her memoir says.”

“We already know that. Everybody knows that.”

“I didn’t. It was news to me.”

“She never told you?”

“It didn’t come up in conversation, no. My father there last night?”

“No. He was busy, he said. He is a doting great-grandpa now, did you hear?”

“I’ve noticed, yes. He’s hogging my
nakhes
. I’ve barely seen my daughter and granddaughter because of him. This morning he called to say he would be late. He’s making blini for Amy.”

“Really? I haven’t had good blini in such a long time.”

“Maybe you should marry my father. He’s actually a pretty good cook.”

“Your father would be hell to live with.”

“Tell me something I don’t know.”

The geographic distribution of people in the house had changed overnight. The mathematicians, both the men and women, were all in the living room. My small family, sans my blini-making father and his granddaughter and great-granddaughter, were in the dining room.

“The parrot likes Jenny, I noticed,” Yakov said. “She moves to the cage door when Jenny is near, like she wants to get a better look. I can sympathize.”

“Pascha likes all women.”

“She doesn’t like you?”

“Tolerates me, but no, she doesn’t like me. And she’s known me for over thirty years.”

“You were competition for your mother. Why did your mother have a bird, anyway? A little thing in a cage like that. I would think it would be depressing.”

“A Polish friend was moving to Israel and they couldn’t take the parrot. Pascha knew a few Polish words. My mother taught her many more. She’s pretty amazing, actually.”

“Not like you.” Yakov gave a hearty chuckle. “Your Polish is probably as bad as your Russian.”

“No, it’s worse.”

“It’s hard to believe you were born in Moscow. Impossible, actually.”

“It’s been a long time. I can hardly believe it myself.”

He got serious and looked toward Pascha. “We’re very interested in that bird, Sasha.”

“Who is we?”

“The people in the living room.”

“You think Pascha holds secrets?”

“We are desperate enough to think anything.”

“A bird brain is not going to solve Navier-Stokes.”

“No, it won’t. But that bird might have memorized a few important lines from a human being who probably did solve that problem but kept it to herself.”

“Now you’re starting to talk like Otrnlov.”

“That man is crazy. We’re just crazy about mathematics. There is a difference.”

“I can’t tell any difference at all, to tell you the truth.”

“Enough. The heavenly soup is gone. I’m going to call up Miss Jenny Rivkin, tell her how much I admire her cooking. If I’m lucky, I’ll eventually convince her to move with me to my wholesome state in this gentle, warmhearted, but somewhat ignorant country. Go. You’ll kill the mood. Now it’s time for you to go to the living room. They want to talk to you there.”

“For what it’s worth, there is a list of numbers on the corkboard next to the phone. It’s a good guess that Ms. Rivkin’s phone number is on that list.”

Someone from the synagogue Sisterhood had been thoughtful enough to bring folding chairs from the synagogue. Otherwise, there was no way this collection of geniuses and the merely supersmart could have all fit into a single living room. The only upside of their presence was that I did not have any trouble getting together a minyan for evening prayers before dinner. I stood before the mathematicians, leaned against a doorjamb—where pencil marks and notations indicated my growth as a youth—and listened.

“She should have won the Fields. It’s a travesty that she didn’t,” Eva said. The Fields was the aforementioned Fields Medal, the international award given to two to four mathematicians under the age of forty-one every four years.

“It’s true. But the time was not right,” Zhelezniak said.

“What do you mean? They had until 1970 to do it. That’s twenty years of her waiting in vain for an award she should have won in 1950,” Eva pressed on. This was her style, and it’s probably why she was successful working with my mother. She wasn’t intimidated by anyone.

“It’s because she was a woman. Those bastards wouldn’t give it to her,” Virginia Potter said.

“That’s not true,” one of the Karansky triplets said. “Vladimir should have received the Fields, too. It was because of the Cold War. No Russian won the Fields until 1970. All those years and just one medal.” The other two triplets nodded their head in approval. It’s no wonder I couldn’t tell them apart.

“Enough. It’s all in the past now,” Zhelezniak said. “I’ve had a good career. I have no complaints.”

“I was there in 1970 when the announcement came about Novikov,” Eva said. “Rachela was furious. That was her time and the committee wouldn’t do it. Novikov. It’s like they knew just which name would make Rachela angriest.”

“The committee blackballed her. Boyle from Oxford. An anti-Semite bastard,” Ben-Zvi said. “It wasn’t because Rachela was a woman or a Pole.”

“Axton Boyle, oh my god,” Eva said. “I forgot. Rachela hated him.”

“I don’t know if he was an anti-Semite, but Abraham is right, he was a bastard,” Zhelezniak said. “There wasn’t much to like about Boyle.”

“Shackleworth was a student of Boyle’s, wasn’t he? So ironic. Him tutoring Rachela’s great-granddaughter,” Eva said.

“Shackleworth is a different man entirely,” Zhelezniak said. “A true gentleman in every way.”

“True, but still,” Ben-Zvi said. “Can you imagine if Rachela knew about a student of Boyle’s teaching her own flesh and blood? She would be on the first plane to Berkeley to strangle him.”

It brought back memories, hearing the fractious gossip about this and that mathematician being discussed in the room. That anyone, much less these thirteen people, could devote such passion to opinions about others, which in effect absurdly elevated those others to celebrity status, was kind of amusing. Famous mathematicians are about as common and likely as famous auto mechanics.

“Yakov said you wanted to talk to me,” I said.

“Yes, we do,” Zhelezniak said. “We need your help, Sasha.”

“I think I’m the one who should be getting help, quite frankly,” I said. “From all of you. Or at least the small favor of not getting in the way while I try to be a good son and grieve over the loss of someone I loved.”

“Everyone grieves in their own way,” Ben-Zvi said.

“You’re telling me that you are grieving by solving Navier-Stokes?”

“It’s a way to celebrate your mother,” Ito said. “It’s our way.”

“It’s the only way we know how, Sasha,” Eva said.

“Even so,” I said. “If she solved the Navier-Stokes problem—which I doubt she did—and, as Yakov says, chose not to share it, then you are trespassing on her wishes.”

“That would be my fault, if that was her wish,” Zhelezniak said.

“My mother didn’t like you, but don’t think you’re so important that you influenced her worldview.”

“It was beyond not like. It goes back a long time. When you were young.”

“A problem from Hilbert. I know about that one.”

“You don’t know about it. No one here does.”

“Actually, I do know more than a bit. I heard my mother on the phone a long time ago. I was little, it’s true, but I remember the shouting.”

“With Kolmogorov?”

“You know that conversation?”

“Yes. I was listening, just like you. That’s funny. Except I only heard what Andrei said, not your mother.”

“It wasn’t so funny, Vladimir.”

“What’s this about?” Virginia Potter said.

“This is old history,” Ben-Zvi said. “Ancient rumors about stolen papers.”

“It’s not really a rumor. It’s the truth, isn’t it, Vladimir?” I asked.

“You and I both know what is true and what isn’t,” Zhelezniak said.

“What?” Peter Orlansky shouted out of nowhere, his temper flaring. “Vladimir, you really stole Rachela’s papers on Hilbert’s thirteenth problem?”

“It wasn’t like that. I didn’t steal them. Andrei had the papers. He gave them to me. I did use them, yes, but there was much more work to be done. Even Rachela, bless her soul, would admit this. Ask Viktor. He undoubtedly knows.”

“Your career is a fraud, Vladimir!” Peter Orlansky jumped out of his chair. I had never seen him so angry. The others present were as surprised as I was. The ugliness of what was being revealed turned Peter into an instant champion in their eyes. All the slights and deceits these competitive people had endured over the years in their quest for recognition in mathematics were being relived.

“Calm down, Peter. Andrei had some drawings and initial work from Rachela. You know how she loved to draw. We looked at her work for months. Without her work, I would never have been able to solve that problem, I admit.”

“All these years, Vladimir, taking sole credit. It’s criminal,” Peter said.