Chapter 5 - The Plate Problem
Chapter 5 - The Plate Problem
Siwoo woke up early.
At first, this life felt unfamiliar.
Back when he lived in a hillside village in Busan, he could always hear sounds even at dawn.
The sound of motorcycles passing by, the sound of market trucks, the sound of a TV through the wall, the sound of someone walking up and down the stairs. But the American dawn was quiet—so quiet that sometimes it felt as if the whole world were asleep.
Siwoo got out of bed and looked out the window.
Under the still-dark sky, the residential area stretched out long.
Siwoo stood still for a moment, washed his face, and went downstairs.
In the kitchen, his mother was already cooking rice, and the smell of soybean paste was spreading.
Even in America, the smell of the breakfast table was similar to Busan.
Hyejin saw Siwoo and said immediately.
"You're up early. The alarm hasn't gone off yet, right?"
"Yes."
Hyejin let out a laugh mixed with a sigh.
"Your body is more accurate than a clock."
At the dining table, Taesik was clutching a coffee cup.
His eyes were half-closed, and he still looked like he hadn't fully woken up.
"Appa is still fighting with American time."
Siwoo smiled quietly, and Taesik put down his coffee and asked.
"Going for a run again today?"
"Yes."
"Don't run too hard."
Hyejin added immediately from behind.
"And don't catch a cold right after sweating."
"I know."
"Drink some water, too."
"Yes..."
"Don't overdo it."
"..."
Taesik chuckled.
"Your nagging is the same even in America."
Hyejin said without even looking back.
"You should listen to it once in a while, too."
"Why me again?"
Even in America, Omma's nagging and Appa's grumbling hadn't changed.
The air on the walking path behind the house was cold, and Siwoo started to run slowly.
The interval of his feet touching the ground, the length of his breath, his heartbeat, his body temperature—everything was quietly organized in his head.
If it were before, he would have just run, but now it was different. Exercise was also a calculation.
He ran at a constant speed, not so much that he would overexert himself, not so much that his breathing would become ragged, and not so much that his body would overheat, while thinking of the name that had been circling in his head for a few days.
The Hodge Conjecture, one of the seven great mathematical problems, algebraic geometry and topology, complex manifolds, algebraic cycles.
When he first saw it, it was curiosity.
Why hasn't it been solved yet?
Where is it blocked?
If people have been clinging to it for decades, there must be a reason.
But as he started reading, the feeling changed.
It was complex, incredibly complex.
But at the same time, it looked natural.
It looked like countless lines were tangled, but from a distance, it seemed to be connected in a single flow.
Even while running, Siwoo drew the structure in his head.
Complex manifolds didn't feel like symbols on paper, but like space, and algebraic cycles looked not like lines, but like a skeleton.
Cohomology was hard to explain in words, but to Siwoo, it approached him like a way of reading the traces of empty space.
What others saw as formulas first, Siwoo saw as shapes first, and that shape turned into a formula.
After finishing his exercise, Siwoo returned home, and after showering and sitting at the table, Hyejin brought out the food.
Taesik was flipping through the newspaper and asked.
"What are you looking at again today?"
"Math."
"Something difficult again?"
"Yes."
"That Hodge something or other?"
Siwoo nodded.
"The Hodge Conjecture."
Taesik tried to follow the name but failed.
"Ho... Hojji."
Hyejin burst into laughter.
"Sounds like the name of a dish."
Siwoo also laughed a little, and Taesik picked up his chopsticks with a wronged face.
"No, why is the name so hard?"
"Because it's a difficult problem."
"That doesn't mean the name has to be hard too."
Hyejin said while serving him soup.
"Try explaining it simply."
Siwoo thought for a moment while eating.
Explaining it simply was too difficult, and just as he was wondering how to explain it, the plate on the table caught his eye.
On the white plate, rolled omelets were placed neatly.
"Looking at it from the outside, it's just a plate."
Siwoo lifted the plate slightly.
"It feels like proving what kind of structure is hidden inside."
Taesik nodded very seriously.
"Ah..."
"Did you understand?"
"It's the Plate Problem."
Siwoo paused for a moment.
"It's not that simple, though."
"Still, Appa understands that well."
Hyejin said with a smile.
"From now on, in our house, it's the Plate Problem."
"Even Omma?"
"It's better than Hodge or Hojji."
At 8:00 AM, Siwoo went up to his room, and a familiar piece of paper was stuck on the door.
-Don't come in until 10 o'clock.
And below that was a note Hyejin had newly attached.
-Make sure to drink water.
Siwoo chuckled.
There was even a small drawing of a cup under the note.
The drawing was a bit crooked, but the meaning was clear.
He placed a water cup on the desk, sat in the chair, and took out a stopwatch.
1 hour 59 minutes 59 seconds.
Siwoo took a deep breath, placing his finger on the button.
Today, he wouldn't go to the end. Two hours, no more than that.
"Start."
The stopwatch moved, and the senses in his head began to change.
The world slowed down, and sounds became clearer.
-Algebraic geometry
-Topology
-Complex manifolds
-Hodge decomposition
-Algebraic cycles
-Existing proof attempts
Paths that were blocked even though they weren't counterexamples. Countless formulas and annotations were spread out on the desk, and Siwoo picked up a pencil.
He wrote formulas on the paper. He wasn't calculating; he was looking at the structure.
Complex spaces moved three-dimensionally in his head, like a giant maze.
Normal people find their way inside a maze, but Siwoo thought with the feeling of looking down at the entire maze from above.
Where it was blocked, where it connected, where it wasn't a path to begin with—he saw those all at once.
30 minutes, Siwoo pushed the first paper to the side.
The existing approach made too many assumptions first.
If assumptions increase, the path becomes precise but heavy at the same time.
The weight bothered Siwoo.
A good structure should be light.
1 hour, his hands became faster, and the formulas seemed to decrease, but the meaning actually grew larger.
Unnecessary branches were cut off, and the remaining lines gathered into one center.
What that center was, wasn't clear yet, but he had the feeling that he was getting closer.
1 hour 20 minutes, Siwoo's hand stopped.
"...Wait."
Something was strange. The existing approaches were all too divided.
Geometry was geometry, algebra was algebra, topology was topology; they pretended to talk to each other, but in reality, they were knocking on the same wall with different grammars.
Siwoo put down his pencil and closed his eyes.
The space opened up again in his head, and complex shapes appeared and disappeared.
Some structures looked like geometry, some like algebra, some like topology.
But from a distance, they were all pointing to the same thing.
"Why did they divide it?"
It becomes difficult if you look at it divided.
What if you look at it as one structure from the beginning? Maybe...?
Siwoo took out a new paper and wrote briefly at the top.
-Plate Problem
This time, the formula was different.
It wasn't the order used in existing papers, and he didn't put geometric conditions first.
He didn't push algebraic expressions first either.
He didn't rely on topological interpretation.
He placed all three at the same time and looked at the parts that didn't match, not the parts that overlapped.
The mismatch, the remaining part, the unexplained residue—that residue might actually be the core.
Siwoo's eyes deepened a little, and his fingertips became faster.
Siwoo kept being conscious of the time, and there was a stopwatch on one side of his field of vision.
"Up to here."
Siwoo wanted to push everything to the end.
He felt like he would see more if he went a little further, but he was aware of it now.
"A little more" is dangerous. Showing it and enduring it are different.
Siwoo organized what he saw today on the last paper.
Existing approaches are separated.
Possibility that the residue between separated structures is the core.
View the inner structure of the plate not as the outer surface, but as the remaining traces.
Next time, rearrange based on the residue.
Beep, beep, beep, exactly two hours had passed, and Siwoo put down his pencil.
He still wanted to do more, and lines were continuing to connect in his head, but Siwoo took his hand off.
"Done."
When he opened the door, Hyejin was waiting at the end of the hallway.
"Are you done?"
"Yes, I'm done."
"Don't do any more today."
Siwoo thought for a moment.
"What about reading?"
"No."
"What about papers?"
"That's also reading."
"Then what about thinking?"
Hyejin narrowed her eyes.
"I can't stop you from thinking."
"Yes."
"Don't think deeply on purpose."
Siwoo made a slightly wronged face.
"That's the hardest part."
"Do it anyway, you hear?"
"Yes..."
Lunch was set on the table downstairs, and Taesik asked as soon as Siwoo came down.
"Did you solve the Plate Problem a bit?"
"A little, but the plate is more complex than I thought."
Taesik nodded.
"Expensive plates are usually complex."
Hyejin said with a dumbfounded face.
"What are you talking about?"
"No, you know, like pottery."
Siwoo laughed softly while eating.
"The metaphor wasn't bad."
Taesik's face brightened.
"See? Appa helped."
Hyejin shook her head.
"You only help in weird ways."
The atmosphere at the table was light for the first time in a while.
In the afternoon, a portion of Siwoo's summary notes was delivered to Professor Jonathan Reed.
It wasn't sent by Siwoo himself; Andrew Miller, who was checking the daily volume, made a copy and sent it.
Jonathan initially intended to read it lightly.
He thought it would be a child's memo, an unrefined idea, an interesting approach, but his expression changed after turning the first page.
"Plate Problem?"
He laughed as if it were ridiculous.
It sounded like a joke just by the title, but the laughter disappeared from the second line.
-Existing approaches are separated.
-Possibility that the residue between separated structures is the core.
Jonathan leaned forward.
The memo was rough, and it couldn't be called a proof.
It wasn't even a summary, but there was a direction. And that direction was pointing toward the parts that existing papers repeatedly missed.
He called Andrew right away.
"Did he write this today?"
"Yes."
"How long did it take?"
"Within the two-hour limit."
Jonathan went silent.
"Professor?"
"...Andrew."
"Yes."
"Don't give this child any other assignments for the time being."
"Are you saying we should only let him look at the Hodge Conjecture?"
"No. I mean we shouldn't get in his way."
"Is it that significant?"
"I don't know yet."
Jonathan said honestly.
"But at least this isn't just a scribble."
That night, before going to sleep, Siwoo wrote down today's conclusion on one side of his notebook again.
-Plate Problem
-Not the surface, but the remaining traces
-Empty space between divided disciplines
-Next time, look at that empty space first