2 January 2009 | Vol. 8, No. 4

The Poincaré Conjecture

by ALLEN LONG

"Please make love to me," I said, struggling not to plead.

My husband Dan jerked his chin to the right, meaning no. He picked up his Rubik's Cube from the nightstand and quickly solved the puzzle three times, his Holy Trinity. I gave him the puzzle on his last birthday, his thirtieth, shortly after they came out.

"I wish you'd play with me instead of that cube," I said. I'd been off birth control pills for a month and was ovulating.

"I'd love to, but I don't want a baby." His expression was regretful but firm. I considered seducing him, but assuming I was successful, I knew it would only worsen our situation. When we married a year earlier, we hadn't resolved the issue of having children. Now, at twenty-five, I felt a fierce, animal-like desire to have a baby.

Dan looked at me with eyes that resembled a tropical sea. I had his full attention. When my math genius husband was calculating, his blue eyes took on a frosty sheen.

The first time I looked into those eyes, I sat on the roof of my dorm at UVA late one night near the end of my freshman year, crying and feeling sorry for myself. I was exhausted from finals and lonely. Next thing I knew, there was a lanky guy on the roof with me with a map and compass, navigating by the stars. I sniffed too loudly, giving myself away, and he crossed to where I sat and brought his face close to mine. He took my hand, grasping it tightly as if I might escape, and smiled like a kind doctor until my tears stopped.

"You were navigating," I said. "Where were you headed?"

"Far Tortugas, but I'm happy I stayed here."

I loved the way he said "happy," like he couldn't quite believe it, his tone not quite disguising a hint of longing. That's how I thought of happy. We sat on the roof all night. Our almost ridiculously sad stories floated out and mingled in the darkness. When I was twelve, my mother died of ovarian cancer, and in the years that followed, my father drank himself to death. After that, my father's mean-spirited, unmarried sister, my Aunt Vera, raised me. She was a bitter woman and the pariah of our neighborhood—she used to puncture kids' rubber balls with a sharp cooking knife if they landed in our yard—so I didn't have many friends or much of a social life. By the time I reached college, I'd taught myself a tough self-sufficiency.

Dan's mastery of mathematics frightened his family. They never loved him; he was the family freak. When he was growing up, his plumber father was disgusted by his son's sensitivity and wouldn't even touch him, and Dan's sister, Barbara, hated him. She was plump and maliciously teased at school. The last thing she needed was a math geek brother. His mother was distantly kind; she treated him like an injured baby bird she expected to die.

"God, aren't we the pair," Dan said, laughing.

I laughed too, feeling an unexpected lightness. Here was a guy who fully understood my unhappy past and could laugh with me about it. I wasn't going to let him get away.

Now I straddled Dan and pulled off my nighty. "You see these breasts? You know what they signify?"

He stared at my breasts, then glanced at me, his face full of frustration.

"They signify that I know damn well how to be a good mother, a loving wife, and a hot lover all at once. I can multiplex. I can be particle and wave at the same time."

He was tempted. The rasp in his breathing told me he was thinking hard. Finally he said, "I need my swim." Dan swam every morning five days a week. After I went off the pill, he'd increased his routine from a mile to two miles every day. He was trying to exhaust himself to reduce his sex drive, but all the exercise was having the opposite effect.

—

On our way out of the apartment, we bumped into our upstairs neighbors—Sally, her husband Jim, and their two-year old son, Kevin, whose diaper bulged and stank. Sally smoked a cigarette and joked with Dan, who smiled and tried to hide his disapproval. Kevin, poor kid, almost always had a full diaper and red, watery eyes. Jim, a handsome and good-natured auto mechanic, stopped short of us, gazing across the shimmering parking lot as if searching for the perfect red convertible to drive to a new life.

I eased Kevin out of Sally's arms, picked up his diaper bag, and took him inside our place to change him. Sally followed me, jabbering about a ring she'd seen on a shopping channel. I changed Kevin's diaper, applying Desinex to his diaper rash. Sally never had any Desinex, so I kept a few tubes on hand. I slipped two into her diaper bag and then forced myself to hand Kevin back to her.

Outside, Dan and Jim faced the parking lot. Sally talked about her new hair stylist, but I eavesdropped on the men.

"I can't figure it," Jim said. "She was fine before we had Kevin; now she won't give me the time of day."

"You've been kid-ghosted," Dan said. "Scares the shit out of me. Any effect on your work?"

Jim looked surprised. "Yeah—I used to get these flashes about what was wrong with a car nobody in the shop could fix, but they're gone now. That's because I'm upset all the time."

"Shit. Any good news to having a kid?"

Jim smiled. "None, except you'll love him more than you can imagine."

The men turned toward us, sunlight glinting off their dark glasses, their expressions unreadable.

—

At the pool, I lay back in a lounge chair and watched Dan swim laps. He swam like a driven man, constantly overtaking other swimmers. As a youth, he'd won many trophies for swimming, but his father dismissed them. "Why can't you play a real sport like football?" he said.

I shifted in my lounge chair. We'd been silent on the drive over, except I'd asked Dan how many combinations were possible with his Rubik's Cube. Without glancing from the road, he'd said, "About 43 quintillion." I hoped talking about math might rekindle our conversation, but he was angry with me for teasing him with my breasts. Did I go too far? Where to draw the line? I certainly wasn't going to trick him into getting me pregnant.

I almost dozed off in the warm sun; then Rudy Banks came up from behind me and put his hand on my bare shoulder. Rudy was the pool manager and lived in a large white house on the hill overlooking the pool. Rumor had it he scanned the pool deck from a telescope mounted in his study and picked out babes to charm. If you didn't discourage him, he'd work on you all summer.

"Kate," he said. "So nice to see you. You're not here nearly as often as your husband."

I shrugged off his hand. "Nice to see you, Mr. Banks."

"Please—call me Rudy." He smiled. He was in his early forties and very handsome and athletic, and most women found him attractive.

Before I could give Rudy the brush-off, Dan stood next to him, water dripping off his tanned body. He held a three-foot length of metal tubing from one of the sun umbrellas; and though he held it at a casual angle, he looked angry and his knuckles were white.

Rudy nodded at Dan and moved off into the men's shower room.

"That was a little over the top," I said, "but quite impressive. With you swimming like Superman, how'd you even know Rudy was hitting on me?"

"Because I'm always watching to make sure you're safe."

I'd never seen Dan violent. "You wouldn't have hurt him, would you?"

"Only if he tried to lick your bikini off."

Dan dropped the tube and paced next to my chair.

"What's wrong?"

He snatched up the canvas bag that held his clothes. "Let's go."

"Why? Rudy's gone. You can finish your swim, and I'm enjoying the sun."

Dan jerked his chin to the right. "Because the pattern's broken. We have to find a new order. It's a basic rule."

"For math, maybe, but not for us at the swimming pool."

He frowned at me like he was talking to a child. His blue eyes looked slightly crazy. "Math radiates," he said gently. "It touches everything."

This was the scary side of Dan. Our life together was a complex web of invisible equations he'd woven around us. Every so often I unexpectedly bumped into one.

"You can be so rigid at times," I said.

"Maybe I wouldn't be so rigid if I could actually make love to you without worrying about having a baby."

"Maybe we shouldn't have a baby. Math seems more real to you than us or any children we might have."

Dan shook his bag impatiently. "I want to go."

I made a well-practiced pi sign with three fingers. "Here's the deal," I said. "We leave now, you take me someplace interesting I've never been. Mathematician's honor."

Wearily, Dan nodded and shuffled into the men's shower room.

A few minutes later, he started our white Ford F-150 pickup and pulled away from the curb. "I'm taking you to a place I used to go to think when I was in high school."

I nodded. Neither of us talked. I asked myself a question I'd been avoiding: exactly how happy had Dan and I been before I started pressuring him for a baby? The honest answer was we were disappointed in each other. I'd expected him to keep me company when I was home, but he sometimes calculated late into the night or on weekends. Worse, he'd get a call several times a month from his former thesis advisor at UVA, and he'd cram some clothes and a toothbrush into a small green duffle and squeal out of the driveway. His destination was a house in Warrenton containing virtually nothing except a giant blackboard, where he worked with two of his former professors and a former classmate on a math theory problem they believed might bring them great prestige—there's no Nobel Prize in math; the highest honor is the Fields Medal, and that's what they hoped for.

The year was 1981, and Dan and his team were trying to prove the Poincaré conjecture, an unsolved math theory problem posed in 1904. In a nutshell, the theory suggests that although we observe ourselves occupying a three-dimensional universe, there should be a way to mathematically test whether our observations are correct or whether a fourth dimension is present. To solve the seventy-seven-year-old puzzle, one had to mathematically devise the test, not determine whether a fourth dimension existed. When Dan talked about his work, his tone became reverent, as if he were describing his search for the Holy Grail or the lost city of Atlantis. He wasn't just trying to solve a math problem and win a prize—he was on a spiritual quest.

He came home from these work sessions ecstatic and exhausted. He refused to talk about them—he was afraid of jinxing his team's progress. I was Dan's anchor. Whenever he came out of Math World, I was there for him, and he was a wonderful husband. When Math World beckoned him, I used sex and any other trick I could to keep him from going back. That's how I disappointed Dan, by standing between him and what he loved as much or more than me.

Our other problems were miniscule. I loved the cinema, but Dan wasn't interested in movies. I went alone to films or with my friend, Cynthia, from yoga class. She was twenty-seven and pregnant.

"How'd you convince your husband to get you pregnant," I asked at Chariots of Fire.

She laughed. "Are you kidding? It was Mike's idea. I could've waited a few years, but he was eager to start his dynasty, as he calls it. He almost made it to the major leagues in baseball. He wants us to start producing the kids he's going to coach into the majors. He calls it his biological destiny."

This phrase disturbed me. Given Dan's reluctance to have children and how often he'd been branded a freak, I wondered if he considered himself a biological dead end.

—

We drove to Rock Creek Park in D.C. The greenery on the path down to the creek was soft and moist from the previous night's rain. The earth was damp. The creek was rushing. Dan's pace was almost urgent. Several bramble bushes crossed our trail, and Dan walked straight through them, and I followed. He took my hand and led me up to a semi-circle of damp grass jutting over the water.

"When I was a kid," Dan said. "I used to love watching the airplane arrival and departure boards flip like crazy. I could follow all the changes, picturing the routes of each airline in neon blue and determining each airline's system for numbering flights. I loved the way those boards changed so fast yet maintained order. My father kicked me once for lingering in front of one." Dan's voice contained a trace of bitterness.

"Except for the kick, it sounds wonderful." I was fascinated but tense. When he was nervous, Dan often revealed his inner world in a roundabout way, and I wanted to catch every word. I was afraid he'd stop talking and I wouldn't understand what he'd said.

"It was, but when I could drive, I found this place, which is an order of magnitude better. Close your eyes and tell me what you hear."

I closed my eyes and listened. "Rushing water—not only the creek, but streams spilling water into it at different pitches. A peeping frog. Birds—blue jays and sparrows. Wind in the leaves."

"Keep your eyes closed. Are we in chaos, or is there a rhythm to these sounds?"

I was used to silencing my inner voice for yoga. Here an unexpected lump of anger gradually melted; then I felt calm. I listened, breathed, and finally said, "Both."

I opened my eyes; relief lit Dan's face. "Oh you get it! I was hoping you would. This is where I used to come to prove to myself that order can exist in chaos."

I wasn't sure why we were here, but I had the feeling baby and chaos were synonymous in Dan's mind. I didn't say anything. I wanted to see where this was going. "Am I the only person you've ever brought here?"

He gave me a wary look and said, "When I was seventeen, I came here with a boy named Brandon from math class. He had a sharp, freckly face and buckteeth. If any geek was more reviled at school than me, it was him. I thought he might find peace here. Turns out he was gay, and he hadn't come to contemplate order and chaos. Even though I was straight, I nearly fooled around with him. Easing his pain and loneliness almost would have been worth betraying my nature. I ended up hugging him for a long time while we both cried. It felt like I was hugging myself. I never want to be that lonely again."

He gave me a nakedly hurt look—the pain he kept hidden most of the time.

I looked into his eyes. "I promise you'll never be that lonely again."

He kissed me in a way that normally would have led to sex in the wilderness.

We both pulled away.

"We need to talk," I said. "You first."

He nodded, looking at me with an earnest expression. "You want to know why I think you want a baby so badly?"

"I'm all ears."

"Because you want to mother a child the way you wish you'd been mothered after your mom died." Dan gave me a gently triumphant look. An inhabitant of Math World, he took particular pride in scoring points in the humanities.

The answer was true but so inadequate that pressure built in my chest like a volcano. "That's only a fraction of why I want this baby," I shouted, surprising myself. "Has it ever occurred to you how long I have to wait for you to come out of Math World and how quickly you go back? Do you have any idea how lonely I am? The real reason I want a baby is so I can love someone twenty-four hours a day and he or she will actually love me back. Is that too much to ask? I can't get that from you, Mr. Disappearing Math Wizard, so why can't I get it from a child?"

Dan held up both hands the way he does when I overload him with non-mathematical data. "Wait, wait, wait. What was that disappearing math wizard remark?" His expression bordered on panic.

"You know why you go crazy for those math work sessions? You're trying to go down in posterity, and you know why? You want to impress your father, who still won't give a shit, even if you win the Fields Medal. Have you thought about that? And by the time you've won that or some other prize, I'll be gone. I'm realizing this baby impasse is about our marriage and not about having a child. If we have a baby to make me happy or to hold us together, that baby's going to be born with a burden. There's no way I'm bringing a child into the world under those circumstances. You have to want one as much as I do."

Dan ran his hands through his thick brown hair until it was a tangled mess. "I can't handle this." He started walking in quick, stiff-legged circles. When he was really upset, Dan's joints became oddly stiff. When your brain's hot-wired for math, I guess you're not necessarily blessed with a normal nervous system. I wondered whether he was exaggerating his symptoms—Dan knew how hard it was for me to see him like this—but I couldn't tell. He set his mouth in a grim line and wouldn't look at me.

Finally, he said, "My math sessions don't have anything to do with my father. They're about me using my gift to help my colleagues solve a difficult problem, and if we succeed, the reputation we'd gain could set us up to have a family—if that's what we decide. Right now we're struggling to make it in South Arlington. With you teaching high school and me writing math textbooks, we're not exactly accumulating wealth. I'm trying to get us a foothold."

As soon as Dan said "family," a warm tide rose inside me, but it quickly receded when I caught a familiar manic glint in his eyes. I didn't know if he was telling me the truth or his warped perception of the truth. How many years would it take to solve the problem, and what chance did his team really have of doing so?

I sat down on a sun-warmed slab of granite and put my head in my hands, a gesture that made Dan fearful I'd say something irrevocable and leave him. He caught his breath, but I didn't budge. I was caught between impossibilities. I couldn't have a baby because it would be for the wrong reasons, and I couldn't turn to Dan for hope or comfort because I couldn't trust his motives and perceptions.

I thought about a teacher at school, a young widower named Bill. He was handsome with a kind face. He always sat next to me at faculty meetings, and his sad but hopeful expression was unmistakable. He was looking for a wife and a mother for his young daughter, Sarah. I'd squeezed his arm recently. I told myself it was out of kindness, but now I knew I'd taken out a small insurance policy in case things didn't work out with Dan.

The weight of these thoughts was almost unbearable, and I started crying. I pitched to the ground and tore up handfuls of damp grass. A real Academy Award performance, just in case Dan wasn't telling me his true motives.

Dan called my name in tones of increasing panic. When I didn't answer, he lay beside me and gently but firmly held my hands so I couldn't pull up any more grass. We panted and our hearts thumped wildly. A hawk called out, and a shade-cooled breeze blew across our hot faces. Dan pulled me into a hug. I felt good, even a little hopeful, but the balanced feeling I usually experienced didn't come. And when it didn't, I felt myself falling down a well of sorrow. I quickly stood and brushed myself off.

Dan stood, the expression on his face indicating he'd do almost anything to keep from losing me.

"Tell me about the math house," I said.

He didn't hesitate. "Professor Jenkins leads the team. He looks the same as he did at UVA, but his hair is more white and wild. We joke he's trying to look like Einstein. He's a good leader. Jeremy's the other professor. He's not very well liked. He's in his late thirties, and he's crazed to win the Fields Medal before he passes the age limit of forty. He's gone gray already. He constantly argues with Professor Jenkins—he thinks Jenkins is overly cautious and slow—and he's really impatient; he doesn't care about how airtight we need to make our proof. Also, he treats his wife Mary like dirt. She has the most bitter and unhappy expression I've ever seen. She's a devout Catholic and won't divorce him. Surprisingly, she comes to our sessions. So does Phyllis, Professor Jenkins' wife. I like her. She's a nurse, and she's warm and funny. My former classmate is Angie—you may remember her—she was my only real competition in the Ph.D. program. She's Chinese-American and a real workhorse. She lives somewhere near us. She goes down to the math house every day."

"Tell me about the blackboard."

"I've never seen one so big. It spans the length of the living room, which is longer than our apartment. We've completely covered it with equations. When we need to erase a section, we photograph it first."

I imagined the blackboard as a vast group mind. Would I ever get enough of Dan?

I thought about how much faith he put into his work. Besides working on a seventy-seven-year-old math problem, he was well aware of Gödel's theorems of incompleteness, which state that a math theory problem may turn out to be neither provable nor disprovable. What if Poincaré was a theory like that?

"Your math requires a tremendous amount of faith, doesn't it?" I asked.

Dan hesitated, sensing a trap. "Yes, I guess."

"You know what it takes to have a baby?"

"I'll tell you one thing—just when I think having a kid might be okay, your anger scares the hell out of me."

"I'm not angry." I couldn't look at Dan. My mind felt overheated and confused. Time to push for a resolution, like when I sometimes forced near-brilliant answers out of surprised, mediocre high school students. "I want to see your math house. I want to see who you are there and what generates so much faith."

Dan was uneasy. "That's a bad idea. You don't know what it's like there."

"That's why we're going."

—

Soon we were on Route 29 speeding toward Warrenton. During the hour drive, Dan suddenly spoke: "I saw how sad you looked after we hugged. I know I'm a big part of the cause—I'll try to give you more of my time, and I'll try to be more open-minded about having a baby. Also, I know you were missing your mother. I've been thinking—after we've proved or abandoned Poincaré, I'm switching to applied math. I want to improve the imaging capabilities of X-Ray and MRI machines. I want to do all I can to minimize illness and grief." He looked like he wanted to say more but didn't have the words.

My eyes filled, but I held steady. I unbuckled my seat belt, kissed Dan on the cheek, and snuggled against him. How could I be so frustrated with a man who had such a big heart?

We left the highway and wound our way through a woodsy, well-to-do neighborhood. Dan parked the truck in front of one of the smaller but still elegant homes and took my hand as we approached the front door.

After we knocked, Phyllis, Professor Jenkins' wife, opened the door, holding a fluted glass of champagne. She gave us a mischievous look. "Come in. We're celebrating."

"We solved the conjecture?" Dan said.

Phyllis' smile widened. "Afraid not, but we solved a problem almost as big—Jeremy."

"What happened?" Dan said.

"Last night, Jeremy thought he found a breakthrough in the proof, so he, Angie, and Jenkins worked through the night and up until a few minutes ago, when Angie discovered a flaw that unraveled Jeremy's hypothesis.

"Jenkins and Angie said they were wiped out and going home. Mary and I stood up to leave as well—we stayed the night because we thought the team would solve the proof. Jeremy blocked the door, calling us quitters in that crazy voice he uses when he's blown a fuse. That's when Mary approached him with this odd little smile. Then, WHAP! Jeremy was on the floor, and Mary was holding her right fist tightly in her left hand. She said, 'I've cast my bread upon the waters long enough, now the tides are bringing me what I need from this marriage.' Jeremy was pretty dazed, so we helped him into the kitchen for first aid. I got ice packs out of the freezer for Jeremy's swollen face and Mary's bruised hand. Next thing, Jenkins popped the corks on the Dom Perignon we've been saving, and everyone's celebrating."

"How's Jeremy?" I said.

"I examined him—he'll be fine." Phyllis stepped back, and we followed her into the house. Professor Jenkins rented it as a centrally located math workspace between Arlington and UVA in Charlottesville.

We passed the largest blackboard I'd ever seen. Just as Dan had said, it spanned the length of the living room and displayed a multitude of scribbled equations. I was pleased Dan didn't stop to examine the board. From my perspective now, I know those equations weren't nearly as important as we thought, given that Poincaré wouldn't be proved until decades later by a different team.

The nearby kitchen boomed with exuberant voices, laughter, and clinking glasses. When we entered, Angie, Dan's former classmate, toasted us with her nearly full glass. Phyllis and Jeremy's wife, Mary, also lifted their glasses. Professor Jenkins offered us champagne, but we declined, our attention drawn to Jeremy, who sat on the floor on the far side of the kitchen holding a blue ice pack to his face. Disheveled gray hair covered his features, except for a bloodshot brown eye full of anger and misery. He shivered as if he'd just been pulled from a meat locker.

"Don't worry," Mary said to us. "After I finish this glass to steady my nerves, I'll get him home and take good care of him. I shouldn't have hit him."

"Dan," Professor Jenkins said. "You've got to hear how Jeremy threw us off the scent. You won't believe it."

"Maybe later," Dan said.

Though I knew from Dan that Jeremy was a tyrant, I didn't like seeing him suffer. I was going to help him, but Dan shot me a surprising look: let me handle this. This was the first time he'd volunteered for anything this personal.

As if approaching an injured raccoon, Dan slowly lowered himself so he was near Jeremy but out of arm's reach. "You okay?"

"Fine, for someone who's now universally hated by his colleagues."

"Wow, a sentence with people in it. Encouraging."

"How about you shut your yap and help me up?"

"How do I know you won't bite?"

"I'm a changed man." Jeremy looked regretful, but his face still burned bright with ambition.

"Don't help him," I said. "He's crazy—we have no idea what he'll do."

"Mary, if I help Jeremy up, is he going to come after you?" Dan said.

"If he's honestly sorry," Mary said, "I'll take great care of him. If he comes after me, he's going to lose more face, American style." Mary and Angie laughed and clinked glasses.

"Ready to get back on your cloven hooves?" Dan said to Jeremy.

Jeremy lifted a hand, and Dan pulled him to his unsteady feet. Mary slipped in to support him. She stroked his back to relieve his shivers, and he gripped her waist. As they tottered before Dan and cautiously turned toward the front door, he smiled as if he'd just married a most promising couple.

Like us? I wondered.

—

As we pulled away from the math house, I looked at Dan, but his face was lost in evening shadow. We turned onto the highway access road, and I watched a long stretch of pines flash by in our headlights.