A Doubter's Almanac (6 page)

“You recognized
me
?”

“As myself, you know.
You
—a young Hans Borland.”

“Well—thank you, Professor.”

“The way we both have of locating ourselves in the world. It’s a rare thing to witness.” Once he was seated, his movements seemed to liquefy. “And two lefties, on top of it all.” He lifted that hand, and with the other one reached across the desk to proffer a stoppered decanter. “Dry sherry?”

“No, no. No thank you.”

The professor poured a glass for himself, then pulled a sheaf of folders from a drawer and waved them over the blotter. “We turned down several hundred applicants with better records than yours, I should add. From far-better departments, too, obviously.” He tilted the sherry to his lips, closing his eyes. “Why? Because I had the feeling we should take a gamble on you, Mr. Andret.” The eyes opened and found their way over the lip of the crystal. “I’ll lay it on the line, young man: your exam was remarkable. Probably the best I’ve ever seen. No,
certainly
the best. I glimpse the possibility of greatness in you, Andret. My father was a chemistry teacher, too, you know, just like yours. Were you aware of that? East Scranton High School. What do you know about topology?”

Milo felt heat on his cheeks. “Professor?”

“What do you know about the field of topology?”

“I’ve read some Fréchet. And maybe some Euler. And a little Hausdorff.”

Borland stared at him. “That’s like an English Ph.D. saying he’s read some Shakespeare and a little Melville, and maybe a bit of Tolstoy.”

Milo flushed.

“And what did you learn from your little bit of Hausdorff?”

“I suppose I haven’t read it carefully enough, Professor.”

“That’s right, you haven’t.” The old man swept the folders back across his desk. “But let me tell you something—topology is your future, young man. I’ll have my secretary put a reading list in your box. Go home and study it.”

—

C
ALIFORNIA.
I
T SWEPT
him in. The drums in the parks. The racks of clothes for sale on the street corners. The ocean light billowing all the time in the sky like a sheet snapping on a line.

In North Oakland he found an apartment. A basement flat lit by two windows high apart on the wall, through which the constant milling commerce of Grove Street cast its shadows. A clicking stampede of pants and skirts and boots and heels that jettisoned cigarettes and coffee cups and sandwich wrappers at all hours of the day and night. Every morning he lifted the glass and cleared away what had gathered in the wells. The caretaker of a filthy, fishless aquarium. Yet at the same time there was an aspect to the outlook that was akin to the maple and beech forest of his childhood. The sense of a constrained world that nonetheless suggested a borderless one. His first weeks, he spent hours staring through the high frames of glass. Constant sameness. Constant newness. The swift legs of pedestrians scissoring a wobbling flame of sun.

It was a way to think.

He’d had a rocky start in the department. He was still drawn to numbers, of course. This part was unwavering. Number theory. The charismatic singularity of primes and semi-primes. The inevitability of numeric functions and their astounding analytic capture of the world—the V-shaped cadres of flying geese he’d watched as a boy, the fragmenting clouds yielding to disorder high above the two glass-paned graphs through which he now gazed hour upon hour. It was as though the numerals had been expressly fabricated, like more-perfect words, to elucidate the details of creation. He wanted to say this to someone. Instead, he sat in his thrift-store chair and watched the passing legs: steps, random crossings, the probability of flows. Mathematics not only described it all but could in large terms predict it. In his bed sometimes he wondered absently if it could be developed to alter it.

He was interested in other fields besides topology. Commutative algebra, for one: the work of Gauss and Cayley and Lasker. But whenever he spoke with Hans Borland about it, the old man seemed irritated.

“Topology’s the one for you, Andret,” the professor said brusquely, the next time they met. “I guarantee it. Forget algebra. Forget Gauss. Lasker was just a chess player with airs. Topology’s a perfect suit to your gifts, man. That sense we both have of the world. The rest’s a waste of your time.” He exhaled tiredly. “And talent.” A glass of sherry stood on the polished desk. He raised his left hand again, dramatically, as though it affirmed his conclusion. “Should have been a topologist myself.”

“You still can be, Professor.”

“A wasted thought.” He sipped from the narrow glass, then set it down carefully, as if balancing an egg. “Have you read Bott and Kuratowski and the rest on the list?”

“Yes.”

“And what did you find there to work on?”

“I guess I haven’t settled on anything. Nothing specific yet, anyway.”

“Well,” he sniffed, “focus yourself. Settle on something.”

—

N
OW AND THEN,
one or another of the graduate students in his department would stop by for an assignment or to discuss a problem. The conversation would begin optimistically; then it would falter, as conversations had all his life. Milo watched his colleagues walk back up the stairs of his apartment into the hectic brightness of the day. Satchels swinging at their sides. The spring-loaded door whooshing and slapping.

He needed something to fill the hours. Borland’s admonition breathed steadily in his ear. Startling. He’d somehow not fully appreciated that a graduate student was obligated to write a dissertation. How insanely idiotic to be unaware of a requirement like that. He needed something to unbind his thinking.

One day in Evans Library, he noticed that a girl was watching him. She was sitting at a desk by the window, and he was standing all the way across the room at the shelves along the far wall, looking at a book about Tycho Brahe, the great sixteenth-century astronomer. When he glanced over again, the girl was still looking in his direction. Dark hair and a man’s button-down shirt. That’s all he could see. He shifted his gaze.

Brahe had used a quatrant to define the orbits of the planets. Milo flipped a page and found a drawing of the instrument itself, which looked like a mariner’s sextant but was many times larger. Onto the palm of his hand, he sketched the spoked arms and the notched, chordal circumference.

When he walked back out of the stacks, she was gone.

That afternoon, at a lumberyard near the bay, he bought some cheap lengths of maple from a scrap pile. In a trash can behind the five-and-dime he found a sheaf of balsa. From there it was straightforward. A couple of weeks later, the arc and axis had emerged from the maple, and the calibrated rim from the balsa. It was scrupulous work—but so was a wooden chain and so was a mathematical proof. Puny assaults on the heavens.

Distractions, too. He was well aware that he needed to distract himself.

—

T
HAT SEMESTER HE
was taking algebraic geometry, Lie groups, and special topics in number theory. Six problem sets per week, each calculation illustrated, each solution rederived for accuracy. Again: distractions. He was also teaching two sections of undergraduate calculus. And all the while, the carefully angled quatrant stood alongside the door of his apartment, steadily emerging from its parts. Turning back to his desk with its piles of student quizzes to mark, he imagined himself in a world where the workings of the heavens remained a mystery, a world in which observation alone might propel forward the lot of man.

This astral machine was going to lead him to a discovery. That’s what he told himself. Not directly, but along an obliquity. The sun’s otherwise imperceptible climb across the equinoxes. His daily, progressive readings on the notched scales. The enterprise touching him, unexpectedly, with a remembered calm. This would free him.

Alone in a city that ran like an unclean river outside his window, he found, for the first time in his life, that he desired friendship. This, too, might free his thinking.

—

“W
ELL?”
B
ORLAND SAID,
offering the decanter.

“Okay, sure.”

A bright November afternoon. In the distance, a pale-blue Frisbee rose above the frame of the window, hovered like a flying saucer, and descended into shouts.

Borland filled a glass with sherry and slid it across the desk, then cast his glance where Milo had been looking. The Frisbee showed itself again. “An imperfect set of parabolic coordinates,” he said. “More aerodynamics than quadratics. Still, one of the benefits of the view.”

“Along with the Dopplerized shouts,” said Milo.

Borland chuckled, leaning back with his glass. He seemed to be enjoying the conversation. “Christian Doppler was a mathematician more than a physicist,” he said. “Son of a bricklayer, you know.”

“Is that right?”

His eyes found Milo’s. “Yes, it’s right.”

Something had soured the man.

“I just meant I didn’t know that about him.”

“Doppler’s work wasn’t particularly impressive,” Borland said flatly. “Not compared to his reputation.”

“I don’t know much of his history.”

“Evidently.”

“But I’d like to. I’d like to read more of it.”

“Nature never lies,” Borland said, leaning forward to refill his glass. “That’s what history tells us.”

“I see.”

“Men lie.”

“Okay.” Milo sipped the sherry. It puckered his cheeks.

“Do you know Lars Hongren, the number theoretician?”

Milo searched his memory. “I might.”

“You
might
?”

“Remind me.”

Borland leveled his eyes. “Never pretend to knowledge, Andret. Never do it. Learning is to be hungered for, not treated as currency.”

“I’m sorry.”

“Have you heard of Hongren then?”

“No.”

“Of course you haven’t. Lars Hongren was the most brilliant student we’d ever seen here. That’s who he was. That’s it, and that’s all.” He took a drink. “He was working on a new approach to the Catalan-Mersenne problem.”

“The double Mersennes are all primes,” Andret offered.

Borland waved him away with his hand. “He was closing in on a solution. Extremely talented young man. His dissertation was drumming up plenty of excitement. I’d phoned Stanford and Princeton about him. And then?”

Milo met his professor’s gaze. “Yes?”

“And then, I discovered what he’d been doing.”

Milo waited.

“Even recalling it disgusts me.” Borland closed his eyes. “I trusted the man at his word. Without checking on him. Suffice it to say that now Lars Hongren works in a bank somewhere, stapling together loans.”

The old man opened his eyes and let a silence settle.

“A sad story,” Milo finally offered.

“An
important
story. Lars Hongren stole his research, young man. He
stole
it. He lied to all of us. If you ask me, it should be a much-sadder story. He should be in prison. I tried to put him there, you know. But your country doesn’t deem it a crime.”

“Is that right?”

Borland leaned forward. “Look,” he said. “You have a talent, Andret. A significant one. Maybe like Hongren’s. Maybe like mine—that’s perfectly okay to say, by the way, if it’s
true
. Do you agree with me?”

“I can’t say.”

“This is what I’m trying to tell you, Andret. I
can
say
.
You’ve been chosen by God, young man. By humankind. By the cosmic order. By whatever you think runs this place, to translate a language. Topology is God’s rules, Andret. That’s what I’m telling you. And you’ve been called upon to translate them.” He tapped the desk. “Your talent is
that major
.”

“Thank you, Professor. I’m grateful.”

Borland looked across the desk again, this time blinking. He poured himself another sherry. “If you’re
grateful,
” he said, chuckling once more and turning back to his work, “then perhaps you misunderstand.”

Poets

O
NE MORNING, WHILE
thumbing a journal in the common room of Evans Hall, he came across a problem: the Malosz conjecture. In the early part of the century, Kamil Malosz had written to a friend wondering whether certain equations might have solutions in complex projective spaces; and over the years the question had evolved into a deeper and deeper problem. No mathematician had ever been able to find a solution. In
Zentralblatt für Mathematik,
Milo discovered a long history of attempts.

But that was what he was searching for, he realized: something that would earn Hans Borland’s respect.

Later in the afternoon, when he knelt to the quatrant, practicalities began to enter his mind. Even if he didn’t ever
solve
the problem, Borland could not fail to notice the attempt. He could just take on a small part of it, even, as he’d seen in
Zentralblatt
that other mathematicians had done. And the work itself might easily widen from there, into a dissertation, perhaps. Even a career.

At the time, most of his classmates hoped to find jobs at Xerox in Palo Alto or at IBM in White Plains or at one of the industry-funded think tanks that were popping up now along the coast; but to Milo, such ambitions were impure. He was not a practical person, but that day he decided he needed to be practical. The Malosz conjecture. His mind came to rest.

In his office, Professor Borland said, “Might as well start at the top.”

“How do you mean?”

“Submanifolds of complex projective spaces—it’s a famously difficult problem, young man.”

“Yes, I read about how difficult it is.”

“Ha! You read about it!” Borland seemed to be in a lighter mood today. He fiddled with the ascot at his neck. At that moment, somehow, they both turned to the window, where against the pewter-colored bay the sun abruptly turned the bridge’s suspension cables into a pair of shining silver parabolas.

“Ah,” said the old man. “Perhaps we’ve just witnessed a sign.”