The View from the Observatory
The global mathematics community was set ablaze.
In particular, mathematicians who were familiar with the Atiyah-Singer Index Theorem but had never actually applied it in practice felt as though they were looking at God.
However, those majoring in algebraic geometry, topology, and differential geometry—who had studied Atiyah-Singer deeply—were a bit slower to react.
They had all immediately grabbed their pens to verify if what they had just witnessed was indeed correct.
Yet, even without picking up a pen, the masters who perfectly understood Atiyah-Singer had been staring in sheer astonishment long before the final equation was written on the whiteboard.
They had intuitively sensed that the intermediate theorems were all building toward that ultimate equation.
On the screen, the boy smiled brightly as he spoke.
"Um, actually, I only learned about this theorem later, so I had to force the connection a little bit. But it still connects, right?"
Edward Kang, standing beside the boy, also wore a shell-shocked expression.
It was clearly not a staged performance.
Edward Kang was not an academic who could act that well.
"Ah... I should explain it to you..."
The boy furrowed his brow slightly and tapped the basic formula of the Atiyah-Singer Index Theorem in the center of the whiteboard. Then, he scratched his head.
"I don't know many mathematical terms yet, so please understand even if I'm not very precise! And I might not be able to answer properly if you ask me things later. I don't really know that much."
Several professors, who had been waiting to join the debate after gauging the atmosphere following Renoir Dubois's turn, hastily removed their names from the video conference waiting list.
When a child cheerfully admitted his own ignorance—though he seemed anything but ignorant—right from the start, how could this even be called a debate?
The moment anyone lobbed a sharp, critical question at that child in this public forum, they would undoubtedly be ostracized by the mathematical community for a long time.
In fact, perhaps from the moment the boy first appeared, this debate was an impossibility.
With their names safely removed, the professors settled back to listen to the boy with far lighter hearts.
Sigma sequences filled the upper left of the whiteboard.
Summing infinitely over all non-zero integer vectors k, then summing and truncating only those where the magnitude of k is less than or equal to N, then summing over the n components within that, with the coefficients corresponding to each k and j varying entirely depending on p under three layers of sigmas...
For a hardworking undergraduate or graduate student, the process was incredibly tedious and lengthy, but it was ultimately like putting together a jigsaw puzzle.
In other words, it was a path that quite a few people could follow.
"Uh, so, everyone should have followed along up to this point. And here, I set up an observatory that wasn't originally there."
"You can rest for a bit, or you can go to the top and enjoy the path you've walked so far! And when you go back down to the ground and follow the path, there's another observatory."
"Like this, once you visit both observatories and get your admission tickets, you don't have to walk on the ground anymore. The observatories are connected to each other. It's like a cable car. The next observatory is the same, and the one after that is the same too. Once you've visited all seven observatories, you can travel through the sky from then on!"
"And then, you can reach this destination!"
From the Hamiltonian setup, the emergence of the Small Denominator Problem, controlling the small denominators with the Diophantine condition, iterative canonical transformations, proof of convergence, proof of the invariant torus, to the emergence of topological invariants.
And finally, an incredibly elegant and clean temple erected at the very end.
If only those observatories and that noble temple had stood there in isolation, most people would have given up on climbing even the first observatory.
Or perhaps, knowing only of its existence, they wouldn't have even dreamed of approaching it.
Renoir Dubois adjusted his slipping glasses and nodded.
He was well aware that his face was being broadcasted.
Therefore, this nod was a sign of absolute acknowledgment.
It was shocking enough that an eleven-year-old perfectly understood this difficult theorem.
But he was even more deeply shaken by the fact that the boy had substituted the theorem into his own equations, rendering it into a tangible, graspable form.
As a result, as long as one simply followed this child's path, anyone could easily comprehend that notoriously difficult theorem. The emotion he felt now was beyond mere awe.
The KAM Theorem deals with orbital stability on a torus, while the Atiyah-Singer Index Theorem topologically calculates the index of a differential operator on a manifold.
Since a torus is also a manifold and Hamiltonian mechanics utilizes differential operators, the connection itself made mathematical sense.
But no one had ever thought to connect them.
The KAM Theorem was the language of analysis and dynamical systems, while Atiyah-Singer was the language of topology and algebraic geometry.
Mathematicians who knew both fields deeply were extremely rare.
Furthermore, the KAM Theorem was a bottom-up theory, while Atiyah-Singer was a top-down theory.
They were looking at the same phenomenon, but from completely opposite directions.
To think that what I dismissed as the most vulgar and what I worshipped as the most noble were actually pointing to the exact same thing...
He had occasionally taught this theorem to his students.
But seeing his students ultimately fail to grasp it, he had thought it couldn't be helped.
Believing that was just the nature of the theorem.
Thinking of it as a temple that only the chosen few could ever reach.
Yet, the very method he had despised his entire life had become the most natural path into the temple he considered the greatest.
Renoir took off his glasses and rubbed his face with his hands.
Sigh, just who on earth was I trying to argue with?
He felt like crawling into a hole.
There was only one person to blame.
Edward! You absolute bastard!
Just then, the small child on the screen bowed politely.
"Thank you so much for listening to my story."
He lifted his head.
"I'm still lacking, so I barely managed to get this far. But I believe the professors, who know so much more than I do, will make it even more wonderful! Uh, please do! I want to see it too!"
Renoir's eyes locked with the child's on the screen.
The boy couldn't see him, but he saw the boy perfectly.
Unfortunately, his was the only face currently displayed on the waiting list screen.
"Ha... dumping the hard work on me, are you..."
Renoir let out a hollow laugh and nodded.
Though his counterpart couldn't see it, he did it clearly enough for every mathematician watching the screen to notice.
This meant today's debate was effectively over.
There wasn't a single flaw, not just in the mathematics and logic, but even in his attitude.
Frankly, what mathematician would dare debate this child now?
Now that he had conceded, no one would truly challenge the boy.
Renoir moved his mouse and clicked the microphone button to request to speak.
He wanted to express his gratitude to the benefactor who had forcibly expanded his mathematical universe.
Edward Kang, who had been standing dazed on the platform, rushed toward his laptop at someone's shout.
And the boy sat down in front of the laptop as well.
Renoir smiled warmly and turned on his microphone.
"Bonjour, Cheol-jun."
"Hello!"
Renoir on the screen was smiling warmly.
"I am Renoir Dubois, and I teach mathematics at the École Normale Supérieure in Paris, France."
Cheol-jun quickly bowed his head.
"I am Lee Cheol-jun, a fifth-grader in Class 5, Grade 5 at Hanyoung Elementary School!"
"Those observatories you showed us just now. I have lived my entire life only at their peaks. Without knowing what kind of soil lay below, or how steep the stairs were, I merely babbled on, believing that the view from the top was the only thing of beauty."
"But thanks to you, today I learned how to stand on the ground for the first time. I also realized how much the elegance I tried to protect was actually turning away the footsteps of so many others. Thank you. You have taught me once again why mathematics must be an art."
"Wow! Thank you so much!"
"I will gladly take on the request you made earlier. No, please leave it to me. Upon the path that anyone can walk, I shall build and show you a temple made of the most beautiful marble I know."
"Really? That sounds so wonderful!"
"I look forward to meeting you in person soon."
"Yes! Me too!"
Renoir smiled warmly and bowed, and Cheol-jun jumped up from his chair to bow deeply in return.
In doing so, the screen broadcasting Cheol-jun's face went blank for a moment.
As soon as Renoir exited the conference screen, someone else repeatedly clicked the button to request to speak.
Since it was someone who hadn't pre-registered on the waiting list, Edward Kang verified their identity.
Then, Edward Kang's face briefly slipped out of the frame.
But seeing his shoulders shake slightly, it seemed he was laughing.
Soon, a face appeared on the video conference screen.
It was an East Asian man with a flushed red face.
"I am Kenichi Sato, lead researcher at the National Astronomical Observatory of Japan in Mitaka, Tokyo."
Cheol-jun quickly stood up and greeted him.
"Hello. I am Lee Cheol-jun."
"First of all, I found your presentation just now very interesting."
"Thank you!"
"However, I have a few doubts. Those equations just now—did you really understand and create them all by yourself?"
"Um, could you please say that again? I didn't quite catch..."
"Ahem. I am asking if you created those equations yourself."
As Sato spoke slowly and deliberately, Cheol-jun smiled brightly.
"I came up with the initial idea, but the professors at Seohan University helped me with the rest!"
"Ha. The rest?"
"Well, thinking about it, I think the professors helped me with about 80% of it."
Sato's eyes gleamed.
It was already a foregone conclusion that his paper would be refuted.
But he had to leave a scratch, even if only like this.
He needed to disrupt the mathematicians' blind faith to shift the focus of attention.
Doubt was a monster that grew rapidly once created.
An issue could always be buried with a bigger issue.
"So they basically made almost everything for you! And you just memorized all the equations written there?"
"Well, it's true the professors made them for me, but I just wrote the equations as I saw them."
"As you saw them? What does that mean?"
"The path the stars travel!"
"Ha, what on earth does that... Hmph. Then, even if we substitute other orbital values instead of those orbital values, you should be able to solve it on the spot, correct?"
"Um, that would take quite a bit of time..."
"I knew—"
"But if it's the observed values for SBR-714, I think I can do it right away."
"Wait... you can?"
Cheol-jun smiled brightly.
"Yes!"
"Th-then, right ascension is 19h 25m 12.43s, declination is -24° 15' 03.2''..."
Cheol-jun walked energetically toward the whiteboard.
And thus, this day was permanently etched into history as the most humiliating day for the Japanese astronomical community.
California, Pasadena. NASA's Jet Propulsion Laboratory (JPL).
In a corner of the Mission Design and Navigation Section (MDNS), the department responsible for calculating and designing the trajectories of planetary probes.
Researcher Sarah Reyes, who was tapping away at her keyboard, giggled.
Sarah tapped her keyboard again.
"With whom?"
Sarah startled and turned her head.
"Whew, you sure have a lot of free time. What's that? A YouTube stream?"
"Actually, it's a math forum..."
"Sigh, Sarah. Is this really the time to be watching that?"
"B-but it's a forum calculating stellar system coordinates..."
"Calculating what?"
MDNS Department Head Anderson furrowed his brow.
"What is this? It's just a kid."
"He's not just a kid..."
"Then what?"
"That kid just solved the Many-Body Problem using the KAM Theorem."
"The KAM Theorem? Are you pulling my leg?"
"Do I look crazy?"
"Hmm. Right. You'd have to be crazy to try and pull my leg. Explain in detail."
"Well, you see..."
After listening to the entire explanation, Anderson cocked an eyebrow.
With Fields Medalist Edward Kang and even Renoir Dubois, a giant of the European mathematical community, making appearances, it was hard to dismiss this as mere gossip.
"Hmm, eleven years old?"
"Yes!"
"And he's Korean?"
"Yes!"
Anderson pondered for a moment before speaking again.
"Gather all the information on the boy and bring it to me. I'll look into it thoroughly and contact the OSE (Office of STEM Engagement)."
Sarah Reyes smiled brightly and saluted.
"Yes, sir!"