“Yes, I see,” said Borland. “Well, listen now, Andret. You have a limited amount of time. That’s what I called to tell you. To warn you about. I’d hazard ten years, on the odds. Then things will begin to cloud over. I was at the doctor’s yesterday. I’m sixty-two years old now, did you know that? I guess one can’t expect a clean slate forever.”
Milo heard the clink of a glass.
“Is everything all right, Professor?”
“Well — thankfully — yes, it is. But I did have a scare. I jog five kilometers every day, you know. Have for more than twenty years.”
“You’ve nearly run around the world, then.”
The old man laughed. “At the equator, that’s correct. At Berkeley’s latitude, I’m actually on my second lap. But I mention it because it’s set me to thinking. How old are you now, Andret? May I ask?”
“Thirty-two.”
“Well, you have five or ten years then, in all probability, and then perhaps a standard deviation. Two at the outside.”
“Yes, sir.”
“To finish your work.”
“I understand.”
“Your life’s work, Andret. You need to start something new. Something as great as what you’ve already done. Preferably, greater.”
Andret paused. “I’m already working on something.”
He could hear the old man breathing.
“What I’m trying to tell you,” Borland said, “is that the Malosz theorem was merely the beginning of what Milo Andret can do. Of what he will do.”
Andret couldn’t speak.
“Just the beginning,” Borland repeated.
“Yes, I heard you.”
He hadn’t intended to sound so sharp.
“I see,” said the professor. After a time, he added, rather clumsily, “Well, that’s all I wanted to say. Goodbye then, Milo.”
—
AT THE OFFICE now, he had difficulty facing Helena Pierce. He quickly discovered that she had the same difficulty facing him.
By the end of his first month, they hadn’t even spoken again. Was she angry? He didn’t know. Wounded? Was it a triviality to her? He had no idea. Could it have been the crucifix? No — if that was what was bothering her, she would have mentioned something. It could have been that she was merely shy. Certainly she was inexperienced, and more than likely she’d been drunk. They’d both been.
Whenever he appeared in the mathematics offices now, she was there at her desk, but always at the rear, her head lowered over her typewriter. It was as though she could sense him through the two concrete walls, the carpeted anteroom, and the pair of frosted-glass doors that led from the hallway. When he entered, the blonde secretaries in front continued their noisy laughter and their cheeky asides, but Helena Pierce no longer rose to defend him.
—
HE COULD WORK on something related to the Malosz, as plenty of other mathematicians would have done in his situation. But Hans Borland was right: there would need to be something greater.
That winter, as he perused the journals, his attention landed on the work of a man named Ulrich Abendroth, a midcentury Austrian who at nineteen had proposed an eminent problem. Abendroth’s precociousness itself had been the stuff of legend: at sixteen he’d been appointed to the faculty of both Cambridge and the École Polytechnique; at eighteen he’d fathered two sets of twins with two different women on two sides of the English Channel; and at twenty, a month after he’d proposed his conjecture, he’d been found dead in a coffeehouse. His problem had entered the canon with a flourish of intrigue. In fact, if mathematicians had believed in any sort of superstition, they might even have considered it cursed. It was famously difficult — very likely as difficult as the Malosz — and in the years since its appearance it had resisted every advance.
All of this sat fine with Andret. He was, in fact, superstitious — but in reverse: what was supposed to be cursed attracted him.
The central puzzle of the Abendroth conjecture concerned a subset of Whitehead’s CW-complexes that were infinite yet finite-dimensional. Clear enough. Though it was considered part of algebraic topology, Andret had a feeling that its solution — if it was going to be solved at all — would come not through equation but through the ability to visualize strange and unearthly shapes.
At this he was quite adept.
In those days, as it happened, the broader discipline of topology was at the apex of its ascendance. The field had become prominent at the turn of the century with the publication of “Analysis Situs,” and in the following decades it had only grown in eminence, not just among mathematicians but among scholars of every branch of the natural sciences. The years leading up to his arrival at Princeton had charted themselves perfectly for the wave of new thinkers who were beginning to populate the upper levels of the universities. These men were no longer bound by symbology but instead spent their days constructing complicated hypothetical shapes that had never before been seen — nor likely imagined — by the human mind. Topologists built undrawable figures in their imaginations, then twisted and folded them. They devoted their time to inventing a cosmology in which the world as it was known — the world of earth and sea and sky — was no more than the three-dimensional rendering of an infinitely higher-dimensional space, much as a two-dimensional movie screen might appear to hold a three-dimensional tableau. In the new paradigm, sensory experience counted for nothing. Pure mathematical ingenuity — the ability to ignore common understanding, to construct a world solely from derived principles — had begun to supersede empiricism.
The field itself required a particular mode of thinking. Not merely the standard mathematical skills but a visual dexterity that could retain complex constructions in the mind for long periods of time, transforming certain parameters while leaving others intact. It was a strenuous and disobliging intellectual endeavor, a sea change of thought in which the brain performed multidimensional mapping. There were topologists who could build architectural structures in their imaginations, then turn them over, then flip them inside out, then spin them around, then open them up and go inside them.
Andret’s own gift for such internal rendering seemed to him to be a derivative of the old positional sense that had once located him in the woods. And not only did it guide him now when he pictured objects in his mind, but also when he drew them with his pencil. He found that he could begin any topological rendering at the upper corner of a sheet of paper and proceed diagonally down to its opposite. No matter how complex the figure, no matter how many layerings of foreground and middleground and background interceded, he could steadily bring to life an entire theoretical construction, with all its dapplings and stipplings, depicting shadow and volume and transformation, in a single, angled pass. It was an astonishing aptitude, really. As far as he knew, nobody else in the department possessed it.
He’d been equally capable of it during his years at Berkeley, of course, but he’d rarely had the opportunity to use it; none of his peers and none of the faculty had even been aware he could do it — not even Borland — and Andret himself had been no more impressed by it than he would have been on any given morning to see his unexceptional face in the mirror.
At Princeton, on the other hand, midway through the semester, he’d been approached at an outdoor café by one of the endowed professors and asked to produce a fully rotated rendering of a Steiner surface, which was formed from the smoothed union of three hyperbolic paraboloids. Andret had complied immediately, sliding a cocktail napkin to the center of the table and using the pen from his jacket pocket to move without hesitation from the top left of the paper to the bottom right. When he’d finished, the professor said, simply, “Remarkable.” For a few moments, the two of them exchanged pleasantries. Then the professor left, taking the napkin with him.