The Poincaré Conjecture
Cheong Wa Dae. Bang! “You call that a proper answer?” The solid wooden table where the teacup sat rang loudly under the palm strike. At the sound, the ornate door swung open and several burly men in black suits rushed inside, only to freeze in embarrassment. The noise had suggested a violent quarrel, yet nothing of the sort was happening.
“Stand down.” At the gesture from the man of highest rank, they offered a quick bow and withdrew. This was a private summit between President Roh and Kim Young-sam, chairman of the Democratic Liberal Party. The meeting had been arranged in a flash on September 1st so that Kim could keep his word to confront the president directly in front of the press.
The atmosphere had been sour from the start. Kim Young-sam was visibly furious, while President Roh maintained his usual unruffled expression. The president had nothing to lose; Kim was the one whose feet were being held to the fire. That imbalance was only natural. Worse still, the relaxed president had no intention of altering his retirement plan—namely, pushing Sunkyung’s entry into the mobile telecommunications business to the very end.
Since open favoritism had failed, Plan B—the privatization of Korea Telecom—had been drawn up and was already moving through the working level. “Do you truly not know what the public mood is right now?” Kim Young-sam finally slammed the table again, his voice rising.
President Roh’s Plan B was not unrelated to his own post-presidency security; it required the next president to lend a hand. “I know. But what of it? It won’t affect the overall trend.”
Kim’s voice grew louder still, yet the president did not so much as blink. Though the public still called him an “ordinary man” and “Moo Tae-woo,” and though the approaching election had already brought on a lame-duck effect that diminished his presence even among politicians, he still wielded formidable power.
“And if it does affect things? Can you not see Assemblyman Jeon Myeong-heon’s approval ratings soaring in the polls?” Even Kim Young-sam, all but confirmed as the party’s presidential candidate, was forced to plead.
“I understand your concern, Chairman Kim. So prepare thoroughly and make sure such a thing never happens.”
Kim’s complexion changed at once. As a veteran of nine political lives, he knew exactly what the president’s words implied. “We have a solid base of support. No matter how high Assemblyman Jeon’s numbers climb, they won’t translate into actual votes. So rest easy, Chairman Kim, and simply keep your promise.”
Plan B, hastily assembled after Sunkyung abandoned the second mobile license, required the next president to exert a little influence. Final approval for the privatization of Korea Telecom rested with the president’s signature. Kim Young-sam’s face shifted several times in a single moment. He wanted to storm out, yet could not; shame burned alongside anger at those who had put him in this position. As time passed, another feeling grew: the desire for compensation. His political goal was the presidency. Once there, he was confident he could govern far better—on an entirely different level—than any military man. For that goal he had already risked everything, even the three-party merger. It was too late to turn back now.
“I expect you to keep that promise.” In the end, the word of compromise left Kim’s mouth.
“Of course.” President Roh, who had been waiting for exactly those words, answered with a bright smile.
“Jeon Myeong-heon!” Inside the car returning from Cheong Wa Dae, Kim Young-sam ground his teeth and spat the name. At this moment, that name was more hateful to him than even President Roh’s. It was one thing to target Sunkyung, but why drag him into it?
“Yoo Jae-won!” Wherever Jeon Myeong-heon went, Yoo Jae-won was never far behind. Everyone knew the Tongil National Party’s upset had been backed by Jae-won’s support. Kim Young-sam, positioned to access far more—and far more accurate—information than ordinary citizens, understood better than anyone just how much Jae-won had accomplished. He was now almost certain that Jae-won’s design lay behind this latest development as well. Jeon Myeong-heon was not the sort of man to speak of “memories of righteous success” or “positive sharing.”
Kim Young-sam’s first impression of Yoo Jae-won had already been poor. Now it had grown even worse.
September 1st. Yoo Jae-won hummed as he prepared for school. Jordan sneakers, loose jeans, a checkered shirt layered with a Stanford “S” knit—altogether the most ordinary outfit imaginable. He merely blow-dried his hair back so his forehead showed, no styling products. A backpack containing only a single shell notebook completed the look. On closer inspection the brands were anything but ordinary, yet at a glance he appeared perfectly unremarkable.
Standing before the mirror for one last check, Jae-won sighed. He looked nothing like a university student. If anything, he still passed for a middle-schooler. Americans already tended to see East Asians as younger than they were; the worry only grew. If only he could grow a beard—yet his upper lip still held nothing but peach fuzz. Being seen as young was no advantage in America, though it would have been a plus back in Korea. Since time would solve the problem anyway, he decided not to fret.
Leaving the mirror, Jae-won headed straight for the entrance. He took the electric bicycle parked beside the shoe rack and pressed the door-open button.
“Good morning.” Two bodyguards waited outside: Ken and Greg, the pair who provided close protection. Jae-won had said campus escort was unnecessary, but Remington and other executives had flatly refused. Accidents, they insisted, struck when one grew careless. Jae-won agreed. Recently he had even called Jeon Myeong-heon to advise extra caution and tighter security. Ultimately, refusing protection would cost Ken and Greg their jobs, so he accepted the inconvenience. In compromise, the two men left their sidearms behind and carried only collapsible batons. Carrying firearms onto a Stanford campus was unacceptable even in America—the very reason the nation had been shocked by the recent shooting. Fortunately, in Jae-won’s previous life, no such incident had ever occurred at Stanford.
“Shall we go?” All instructions and contingencies had been covered the day before, so no further discussion was needed. After exchanging greetings, Jae-won mounted the bicycle and set off, the bodyguards following in a large SUV stocked with spare wheels, batteries, and room for the bike itself—precautions against mechanical trouble or sudden weather changes.
Because it was the first day of classes, the schedule was relaxed; few professors piled on heavy work immediately. Still, the word “few” mattered. Stanford had its share of eccentric professors as well as eccentric students. One required mathematics course in the Electrical Engineering department began racing through material on day one. Having already mastered the content in his previous life, Jae-won had no difficulty. Gilbert Owen, however, who had let his guard down on the first day, was thoroughly flustered—especially when the professor called on him unexpectedly.
“Starting the syllabus on day one? That’s just cruel!” Gilbert complained as they moved to the next lecture hall. Jae-won would have confronted the professor directly; the timid Owen could only vent to his new friend.
“It’s fine. You didn’t get the answer wrong.” Jae-won answered like an adult. The question had been high-school level; anyone reasonably prepared could solve it. Owen was an excellent student, so his answer had been correct—he had merely stammered from nerves.
After chatting a while, Jae-won turned. “I have somewhere to be.” It was nearly lunchtime; Owen had been heading to the dining hall, but Jae-won still had business.
“Yeah? See you later, then.” In Korea one might hesitate to eat alone, but American individualism made it unremarkable. Owen accepted it easily and went his own way. Jae-won watched him for a moment, wondering if he had taken offense, then moved on.
“Come in.” The people waiting for Jae-won were the Electrical Engineering professors. Every instructor whose course he had registered for was present, along with the department chair and several others drawn by the unusual spectacle. Applied Mathematics and Bioengineering—departments friendly with Electrical Engineering—were also represented. More than twenty professors filled the lab, yet the number of absent faculty was even larger.
“Hello, everyone.” Unable to single out individuals in such a crowd, Jae-won offered a general greeting. It was highly unusual for so many professors to gather near lunchtime, and what Jae-won intended to do was equally unusual.
“Oh, so you’re that famous young chairman? Seeing you in person after only television is quite a shock. You’re far too young!”
“Indeed. At that age I was still taking radios apart. Already in university—times have certainly changed.”
“Where did you say you were from? North Korea, was it?”
“South Korea!”
At Jae-won’s greeting the professors erupted in overlapping chatter.
“Hey! Are we doing the test or not? Lunch waits for no one!”
Fortunately, someone stepped in to calm the room—Daniel Kamen, chair of Electrical Engineering.
“Jae-won, you’ll have to forgive them.” Jae-won had asked Professor Shucheng Chang about earning attendance credit through special projects. He was not the only student with such ideas. Stanford attracted eccentrics and prided itself on accommodating them. Among those eccentrics were inevitably students who wished to breeze through classes yet still receive a diploma. For them the university had created an unofficial—but entirely legal—Fast Track examination.
“It’s been so notorious that almost no one has attempted it in four years.” American universities made admission easy and graduation hard; Stanford was no exception. Its reputation only made entry more difficult and graduation stricter still. The workload, reports, and exams were brutal. To skip the normal curriculum, one had to prove equivalent ability.
“The test is simple. Choose any problem on the blackboard that appeals to you and solve it. Pass, and you’re done.”
Professor Kamen pointed to one wall of the vast lab. What had looked like random graffiti from afar resolved, up close, into densely written, obviously difficult problems.
“How much time do I have?” Jae-won asked, scanning from the left end.
Professor Kamen burst into hearty laughter; the other professors joined in with quiet chuckles.
“Time limit? That would make it seem too short. We should say ‘period.’”
“Period?”
“At most, until the end of the semester. You must pass before final grades are posted. If you cannot solve it, you would do better to abandon the Fast Track and focus on regular coursework. Concentrating on the Fast Track only to fail both it and your regular exams would be your own loss. And even if you solve a problem, it won’t earn you a high grade—passing and studying diligently are separate matters.”
Ah. So the professors were confident the problems were too difficult to solve easily. Well then, let’s see just how hard they really were.
“My word!” The professors had not exaggerated. The problems on the board were exactly the sort that warranted a warning. Yet many seemed written more for amusement than for any realistic expectation that a student would attempt them: the Riemann hypothesis, the P versus NP problem, the Hodge conjecture, the Poincaré conjecture… the seven Millennium Prize Problems, the ultimate bosses of mathematics, all listed here. It felt like a Stanford-style joke.
Moving farther along, Jae-won found more reasonable problems suited to Electrical Engineering: integrated-circuit design and optimization, communication-signal-processing algorithms, embedded-system design. A student who had diligently attended classes might at least attempt these—though “at least” was relative to the Millennium Problems; for a brand-new freshman they remained formidable.
“Well? Which will you solve?” Expectation filled Professor Kamen’s face; every other professor’s gaze was fixed on Jae-won. It seemed they had even placed bets on his choice.
Jae-won considered for only a moment. “Hmm. This one looks interesting.”
After a mere ten seconds he pointed. A ripple of shock ran through the professors.
“Wait—what? That’s the Poincaré conjecture.” Professor Kamen’s expression was anything but reassuring. The Applied Mathematics professors had written the Millennium Problems purely for fun, never imagining Jae-won would choose one.
“Yes, it’s fine. I can erase the others, right? The proof may take a little space.”
Jae-won picked up an eraser and cleared the problems beneath the Poincaré conjecture to make room. Reactions varied: some saw arrogance, others thought a puppy of a freshman was overreaching. Not everyone was negative, however; the Applied Mathematics professor who had written the problem moved closer to watch.
Jae-won paid the reactions no mind. He closed his eyes briefly, as if in meditation, then opened them. Chalk in hand, he began to write without pause. The proof of the Poincaré conjecture had begun.
“Mmm!” The professors’ mood shifted within a single minute. Professor Conner of Applied Mathematics, standing right beside Jae-won, was not the sort to indulge empty boasts. Had the proof been nonsense, he would have stopped it at once. Instead, with every line Jae-won wrote, Conner let out an involuntary gasp. The professors who had been watching with folded arms now sensed something extraordinary and crowded forward.
They had no choice. The Poincaré conjecture stated that if every closed curve in three-dimensional space could be continuously shrunk to a point, then the space was homeomorphic to a three-sphere. It had once been one of the seven Millennium Problems, yet by the twenty-first century countless mathematicians had tackled the list until only the Riemann hypothesis remained. Human minds alone could not solve them; eventually even artificial intelligence had joined the fray. Of the seven, the Poincaré conjecture had been the first to fall—originally proved by Grigori Perelman, later simplified through further research.
The proof now flowing across the blackboard was the elegant method derived by the S-class AI GS-X after the technological singularity, ported into human-readable form via the Machine Psychology Module—an achievement of Jae-won’s own. Where Perelman’s proof had spanned fifty-seven A4 pages, GS-X’s version required only three. The module had been tested by proving the Poincaré conjecture itself. Thus Jae-won could write it out fluently without even entering his Memory Palace.
Proving a problem modern mathematics had failed to solve for decades was astonishing enough. The equations themselves were breathtakingly beautiful.