A Doubter's Almanac (15 page)

That year, having accepted every lecture invitation in his mailbox, he’d discovered himself to be an able speaker; but this audience of submissive-looking teenagers didn’t seem to care. Looking out at their vaguely distracted faces, he suspected that whatever would induce them to their historical destiny hadn’t yet arrived on the scene. Sometimes he would lower his voice to a whisper—a trick he’d learned while on the lecture circuit—or pause for several long moments, waiting for their attention to return.

He always had a drink or two before he taught, and now and then his mind would bring him back to his own undergraduate days in East Lansing. By the time he was a sophomore, he was already taking upper-level courses, sitting at the back of the room considering the differences between himself and the boys he’d gone to school with in Cheboygan, most of whom were by then a good way through their second or third tours in Vietnam. He himself had been kept out of the draft by collapsed arches in his feet—he still sometimes wondered if the doctor had exaggerated his condition—and then in Berkeley had been given another five years of reprieve. Now the war was over, and the draft, too, and when he looked out over his classes, the seriousness of those times seemed no more relevant to the students in front of him than some ancient Assyrian epic. In East Lansing in the sixties, the men—future or past soldiers, nearly all of them—had worn ties to class, and there was a gravity to the tone that he hadn’t found since. Not even in graduate school at Berkeley, the thought of which, as he stood before his class one day recalling the incense that burned in the back corner of the Lime Rose, deflated him. On the other side of the podium, in the rows of sloping, newly upholstered seats, sat the sons of attorneys and financiers. They rested their sneakers on the chairbacks, passed notes across the aisles, and opened sodas loudly as he spoke. A few of them fiddled with skateboards.

—

A
T THE BACK
of his mailbox one morning, a sealed envelope with his name on it.

I was buying a humidifier for the office. (I never shop there myself, the prices are ridiculous.) If you need anything related to the department I’m available and hope you will ask. I’m not ashamed about what happened but I’m not proud on the other hand, and it’s better if we just ignore whatever it was and pretend it didn’t occur. (Of course I mean better for
both
of us.)

Dogs and Horses

T
HERE WAS NO
denying it: time had passed, yet he’d made almost no progress on the Abendroth. He wouldn’t be able to overpower it the way he’d overpowered the Malosz.

Tactic rather than force: that’s what he would need.

At first consideration, the problem appeared almost facile; but the essentials quickly hid themselves. He’d begun to conceive of the proof as a fortified castle pierced by ten thousand brightly painted doors, each of which was designed to deceive him. All ten thousand would open—this wasn’t the issue—but so far, none of them had allowed him entrance.

Perhaps none ever would.

After a year and a half of effort, he realized that it made sense to limit his aspirations. Perhaps it would help to give up on a solution altogether and focus instead on merely locating the proper vulnerability.

He also understood with a dull sense of foreboding why so many gifted men had been circling the problem for most of a century. The work to all of them must have seemed a seductive lover. By now he’d grown accustomed to waking in the middle of the night with some electrifying premonition, to hurrying through the dark to the mathematics building, to laboring alone in the predawn hours while the radiators clanged around him, as though being hammered by Ulrich Abendroth’s own imperious ghost. But over the course of the pale, tree-obscured sunrises, which turned his office from blue-gray to dusky orange to a brightly nauseating yellow, his thrilling premonitions uniformly faded. He could find no passage in.

—

O
NE AFTERNOON NEAR
the end of his second year at Princeton, there was a knock at his office. He ignored the interruption; but a moment later it came again. When he opened the door, a woman said, “Oh! You’re here then.”

“Forgive me,” said Andret.

She was well dressed, almost prim—his own age or slightly younger. Dark red heels and a somewhat-dowdy outfit of a related color. Although he never trusted his memory of people, he was fairly sure he’d never seen her before. “I must have been lost in my work,” he said.

“I’d give anything for that.”

He looked again. A rather pretty face. Maybe a little rebellion in the eyes. No: they’d certainly never met. “Would you?” he said.

“You must be busy. I just wanted to see if I could make an appointment. But I can come back another time.”

For a woman who’d arrived uninvited she seemed insistently timid. Yet on the other hand she made no move to leave. In fact, she appeared to keep herself before him by some calm demonstration of will, like a mystic holding a finger in a flame.

He lifted a box of work off the visitor’s chair and gestured her in. “The problem,” he said, taking the seat at his desk, “is that I might be even busier the next time. You’d have to estimate
those
odds, too. What may I do for you?”

“You’re absolutely sure?”

“No.” He shook his head. “Nothing is absolutely sure.” He smiled. “But now’s as good as ever.”

She sat down across from him and returned his gaze. Her name was Annabelle Detmeyer, and she was an associate professor in history. “My husband, Yevgeny Detmeyer”—she paused—“my husband is the chair of economics and the co-chair in political science.”

“Ah. A double threat.”

She laughed, pleasingly. Her laugh wasn’t so dowdy.

“How can I help you?”

“I just wanted to learn what a mathematician does,” she said. “It’s so different from what I do, and I’d heard so much about you. I read an article about the Malosz theorem, and I was intrigued. I actually tried to read it. The proof itself, I mean.” She laughed. “I got about a half a sentence in.”

“Well, history might be beyond me, too.”

“Your work is quite mysterious to someone in a field like mine. That’s all. I was just wondering what a person like you does all day. As a historian, I know what I do.”

“Which is what?”

“Travel and search out sources and make notes. I teach. I write. At the moment I’m stuck in a little skirmish over Sigismund the Third, of Poland. But I don’t imagine you spend your days fretting over anything quite like that.” She rose. “Well, you’re obviously busy. I can come back another time.”

Beneath the dowdy clothes, he glimpsed a body that, like the laugh, wasn’t dowdy in the least. “No, no,” he said. “In fact, I welcome the interruption, and I’ll almost certainly be busier next time. As it so happens, I was just trying to figure out the same thing myself.”

“The same thing?”

He leaned back in his chair. “What someone like me does all day,” he said.

—

S
HE WAS A
farm girl. Had spent her childhood on a cattle ranch half a day’s drive from a library and now found it ceaselessly amusing to be teaching seventeenth-century history in the Ivy League.

In a way, then, she was like him.

She told him all of this the next afternoon, when they met for a drink on the patio of a place on Chambers Street. She ordered a glass of wine. Spring had appeared. They sat along the sidewalk, where bikes and baby strollers and students on roller skates ambled past. The peak of the afternoon in the peak of the season. As they talked, she greeted a number of professors who stopped at the table to talk. Most of them looked quite senior—cuff links and bow ties—and all of them forwarded their regards to her husband.

When another of them had walked away, Andret lifted his bourbon and examined her over the top of it. “Your husband is evidently quite an important man,” he said.

“Yes, indeed he is.” She glanced to the side, then brushed at her hair. Then she took a swallow of wine.

“Ah, yes,” said Andret. They were silent for a few moments, during which the air seemed to grow warmer. “Why don’t we go someplace a little less public,” he finally suggested.

—

O
NE DAY, ALMOST
three years along in the task, he happened upon a paper by Paul Erd
ő
s. There was an older theorem of combinatorial topology that Erd
ő
s and a colleague named George Breville had proven rather magnificently, but also rather whimsically, by modeling their analysis on a children’s game. The game was called Kutyák és Lovak, or, loosely translated, Apples and Oranges. Erd
ő
s had played it as a young boy in Budapest.

In the game, one child named an object—a lemon, say, which was sour—and the other child countered with an object that possessed some antithetical quality: a cube of sugar, say, which was sweet. The first child then had to do the same for the cube of sugar, but in relation to another quality—it was orthogonal, for example, while a sheet of paper was flat. And so on, until one of them accidentally named an item that shared one of the previously named qualities with the lemon. The game, as Erd
ő
s and Breville pointed out, was simple enough for children; but if one began with multiple objects and multiple players, it became more difficult. There were other permutations, too. If after a certain number of moves, for example, the players were allowed to secretly change direction, so that one group might be racing away from the lemon at the same time that the other group raced toward it, it became devilishly complex. This last incarnation was the drinking game that Erd
ő
s and his friends had played for bar money at university.

The paper was printed toward the rear of the
Journal of Combinatorics,
spring 1978. Andret read it late on a Monday afternoon. He stared at the poorly printed page in the small lounge where the department shelved its literature. A steeple bell chimed the hour; but he hardly heard. Someone entered the room, poured a cup of coffee, and left. Andret turned back the page and reread the paragraph.

Erd
ő
s had cited a method he’d developed to discern the probability that one of his opponents had changed direction in the game. Andret closed his eyes for a moment, then went back and read the paragraph again.

This was it. This was the way to breach the Abendroth.

He shut the journal, set it back in its place on the shelf, and looked around, as though he’d been caught at something.

Another Roof, Another Proof

N
OW HE BEGAN
a search. The instructive theoretical instance was the prize he sought. When he arrived at the office, he would clear the desk of yesterday’s notes, then spend the morning in thought—it might take three or four hours to assemble a single figure in his mind. In the afternoons, he would sketch, committing what he’d composed to memory. At first he tried to conceive of a figure that might invalidate the approach he’d picked up from Erd
ő
s; but then, as such figures one by one proved assailable, he began to work on examples that might support it. At home on his bed table, he kept a pen and paper, in case something came to him at night.

For months, he pushed on. The proof might require another three years of work, maybe four—but so what? To solve two great problems in a lifetime would bring him to the pinnacle of his profession. He asked one of the secretaries to buy a half-dozen cartons of laboratory notebooks. When they arrived, he numbered their bindings, then filled the pages with variations on certain theoretical shapes—3-manifolds parsed into every sort of Heegaard Splitting and torus decomposition that he could imagine.

Then he began, slowly and at first by negation, to bridge the moat around the problem. It was tedious work, but the moat needed to be crossed before he could scale the walls. In a day he could fill an entire pad with drawings. As he worked, he felt warmed through his body, down to his hands and feet, as though it were vigorous physical exercise he was performing and not a motionless feat of endurance, sitting still at his desk for hours. He kept a window open for air. Sometimes he grew breathless nonetheless, as first he sensed himself moving to the edge of the moat, then starting across it. On he went. Into numbered file boxes he laid numbered notebooks, which in turn he stacked in numbered sequence along the bare walls of the office. He didn’t need his own drawings in order to think, as did some of his lesser colleagues; but he knew that he would need them later for reference—a year from now, five years from now—when he went over the wall into the castle.

—

I
N
O
LGA
P
ETRINOVA’S
basement apartment, the radiator was set so low that she kept a sweater on indoors. But underneath the sweater was always a dress, pleasingly stretched, and underneath the dress a queerly sewn, Bolshevik-style undergarment that he grew to crave. Between the knees and breasts it was a thick gray wool, but around the hems ran lacy sinuations of black silk that might have come from a shop in Paris.

The revived state of his work made him ravenous.

Afternoons, he visited. At the door, which was down a flight of stairs, she would greet him with her hands on her hips, her breasts thrust forward through the sweater in a way that was both flirtatious and accusatory. Was this only in his mind? At the small thrift-store table by the refrigerator they would drink a pair of bourbons, then slide the chairs across the room to wait for the sun. Her hands smelled of fennel, and there was something not quite real about the color of her hair, but her bony beauty never failed to catch him. By 2:30 or so, when the sun and the bourbon had warmed her enough, she would narrow her eyes.