Translator: Dreamscribe
One morning, a few days before Christmas,
Cambridge's weather turned fickle again. The sky, clear until the day before, suddenly clouded over and thick snow began to fall.
The sight of white snow settling over the campus was as beautiful as a scene from a fairy tale.
Clack.
Harold Whitman, dean of the mathematics department, set his coffee cup down in an empty spot between piles of documents. Then, out of habit, he logged into the archive.
He was checking to see if any new papers had been uploaded overnight.
A list of newly registered papers appeared on the computer screen.
Whirr.
As he scrolled down, a familiar term stopped his hand.
'Smale's mean value?'
It was a name he hadn't heard in a long time.
Stephen Smale. A mathematician one generation above him. A giant who had received two of the three most prestigious awards in mathematics: the Fields Medal, the Abel Prize, and the Wolf Prize.
Whitman clicked the mouse with trembling hands.
When the paper opened, the title stood out clearly.
[On the Smale Mean Value Conjecture in Non-Uniform Measure Spaces]
It was a short, understated title.
Whitman immediately checked the author's name. Then he shot up from his seat in shock.
"Yu Seo-ha?"
He had been thinking the student hadn't been seen around much lately, and it turned out he had been working on something like this.
Moreover, the person listed as second author was no ordinary name either.
Theodore Langford.
A young mathematician from Stanford. He had drawn early attention for his papers bridging analysis and geometric theory. Whitman had even met him at a conference and exchanged words.
"How on earth did this happen?"
The idea that Langford had co-authored a paper with an MIT undergraduate was difficult to comprehend.
"Don't tell me they solved an open problem together?"
Whitman adjusted his glasses and pulled the monitor closer.
His questions were answered before long.
Theodore had stated his motivation in the preface.
[My role in this paper is minimal. I merely organized the materials and offered some assistance in making the author's mathematical language more accessible. I wish to make clear that the proof itself belongs entirely to the author.]
The content was neatly organized.
Seo-ha's characteristic unfriendly explanations and leaps of logic were nowhere to be found.
Whitman read a few lines and slowly drew in a breath.
‘This part must have been Theodore’s touch.’
The paper was astonishingly complete.
The logical structure was smooth, the citations were clean, and there was no hesitation in the choice of words.
Concise formulas and a dry writing style; the quality was excellent enough to be published in a major journal right away.
But what mattered was the content.
He picked up a pen and began reading the paper slowly.
One hour, two hours...
"Ah!"
His rigid expression gradually brightened, and he let out an exclamation of admiration at the process of flattening the curved surface.
And when Seo-ha finally derived the equilibrium point, he rejoiced as if it were his own achievement.
"Hahaha! I knew it would come to this."
Seo-ha had not approached the Smale problem in the traditional way.
He had reborn the old concept of the mean value theorem into a new language: the definition of space itself.
"Redefining density like that, it's an approach no one but Seo-ha would even attempt."
No mathematician had ever proposed such a solution until now.
Seo-ha's paper retraced the reasons for failure one by one, as if diagnosing the cause of a long-standing illness.
Whitman recalled what Smale had said decades ago when he gave up on this problem.
"The mean value probably exists everywhere. We just don't know how to reach it. I hope someone in the future will find the way."
He scrolled to the very end of the page.
The conclusion was only three paragraphs.
Yet within those short paragraphs, everything the author wished to say was contained.
"Stephen. It seems I've gotten quite old myself."
Was this what history looked like?
He leaned back in his chair and looked up at the sky. A thrilling shiver ran down Whitman's spine.
The power to resolve, with a single line of intuition, a domain that prior approaches could never reach.
A sensation he had long forgotten since the emergence of Elijah Cronen; Whitman felt that the moment when one generation gives way to the next had arrived once again.
Whitman rose from his seat.
The snowfall was growing heavier.
Students outside the window passed by hunched against the snow. This scene had not changed in the past, nor had it changed fifty years later.
'It'll probably be the same fifty years from now, too...'
And yet, mathematics steadily advanced.
He was grateful to be in a position where he could personally witness that progress firsthand.
* * *
The next morning dawned.
Even amid the busy final exams, a strange energy was flowing through the mathematics department.
Students peered at their phones and murmured among themselves.
"Did you see it? That new paper that went up."
"You mean the Smale problem? When did he even do that? He's seriously incredible."
"In our department, undergrads solve open problems."
A day had passed since the paper was uploaded to the archive.
In that short span of time, researchers around the world were already sharing links to the paper.
Hundreds of comments had been posted on online forums where mathematicians gathered.
[It appears the Smale problem has been solved at MIT]
-I received the paper last night and read it, and I had chills the entire time. Yu Seo-ha is still an undergraduate. How does a student like this even exist?
He pulls in horrifically complex theorems, folds them geometrically, and then unravels them through probability. I found myself wondering whether what he does is even the same mathematics I learned.
It will need to go through verification, of course, but it was a brilliant proof that shook the very assumptions and definitions.
└Were there any errors?
└None that I could see. The derivation process itself is a work of art.
└How did Theodore end up collaborating with Yu Seo-ha?
└Weren’t they fighting like crazy in LOGIA? Could it be they met in person?
└I'm a chemistry major, so I don't know this student. Is he famous?
└Aigoo, Seo-ha is the future of our department. From the dean on down, everyone treasures him...
When night fell, the European mathematicians woke up.
The same posts were circulating on forums in Berlin, Paris, and Zurich.
The mean value theorem in non-uniform spaces.
If it passed verification, it would be an event that rewrites analysis textbooks for the first time in half a century.
* * *
The next day,
Snow was still falling from the Cambridge sky.
Whitman had called an emergency faculty meeting the day before.
The mathematics professors sat around the dean's office, sharing tea.
They looked tired, as though they had put in a long stretch of work, but the atmosphere in the room was strangely buoyant.
“What’s the reaction so far?”
At Whitman's question, one professor tapped on a laptop.
"I just checked the European forums as well. The German mathematics community says they're launching an independent verification."
Whitman chuckled quietly.
"They don't waste any time, do they."
Printed copies of the paper lay on the table.
The day before, all the mathematics professors had gathered and conducted a preliminary verification of Seo-ha's theory.
The result: no issues found.
"Dean, shall we make an official verification request? The American Mathematical Society has been reaching out.
The editors-in-chief of major journals are getting anxious too. The department office staff say they want to rip the phone cords out of the wall."
It was not a problem well-known to the public, but it held immense significance in mathematics.
A problem so stagnant that even scholars had forgotten about it, with no progress in approach, let alone a solution. Seo-ha had dragged it back to the center of the academic world after half a century.
"The second author is from Stanford, so there's movement on that end too. They're planning to invite Seo-ha and Theodore Langford for a lecture."
The room stirred for a moment.
"Why would they invite our student there?"
"I heard Langford only refined the structure of the writing. The core ideas all came from Seo-ha, didn't they?"
"Since he put his name on it, he has a responsibility to explain the paper. Outside observers already see them as one team."
Whitman quietly raised his hand to silence the commotion.
"Where exactly are the institutions that want to conduct verification?"
One of the professors answered.
"In Europe, Heidelberg University, and in Asia, the University of Tokyo says they're holding unofficial seminars. The others will likely wait and react after seeing the initial responses."
Whitman thought for a moment, then slapped the desk and stood up.
"Very well. Whether it's verification or discussion, we can do it all right here. Here at MIT."
The professors' faces stiffened in unison.
“Dean, isn’t that too hasty a decision?”
"Official verification hasn't even begun yet. If we step forward now, we could end up looking foolish."
"That's right. Perhaps you should reconsider? If an error we missed is discovered, MIT as a whole will suffer a tremendous embarrassment."
Whitman slowly looked around at the professors.
"Everyone, since when has MIT been a school that only makes safe choices?"
Silence filled the dean's office.
"Twenty years ago, during the Topological Quantum Field Theory seminar, everyone said it was impossible.
But the theorem we produced then went on to become the international standard."
One of the professors quietly raised a hand.
"Back then, the theory was just beginning to take shape. So even with the risk, there was justification in the sense of 'we're leading the trend.'
But this is a long-standing open problem, isn't it? I don't see any reason to rush."
It was a cool-headed and rational judgment.
The professors nodded in agreement.
But Whitman did not yield.
"There is no shortage of justification."
"And what would that be?"
Whitman answered with conviction in his voice.
"Where is the mathematics of a new era being born? I want to leave that evidence behind."
Silence followed.
"Hmm..."
“There is definitely something special about Seo-ha.”
"A paradigm shift..."
Everyone understood what he meant.
"There is something special about Seo-ha's mathematics. He is not bound by the authority of classical mathematics. He sets no limits anywhere.
Conservative scholars may find this somewhat foreign, but if mathematics of a new era does exist, I suspect it would take a form such as this."
Each professor wore a contemplative expression.
But soon they steeled their resolve and nodded. One of them raised a hand.
"I want to look further ahead. This is purely hypothetical, but...
If, in the future, Seo-ha ushers mathematics into a new stage, and if this place is where it all began, shouldn't we be the ones to leave that record?"
"I agree."
"We can't let another school host the first lecture."
The opinions converged into one.
Whitman nodded with satisfaction.
"Then, we will hold a Colloquium, a public lecture. There will be no restrictions on who may attend. Anyone who wishes to come is welcome.
However, distinguished figures who can lend it credibility must be in attendance."
Once the decision was made, things moved at breakneck speed.
The department office became a hive of activity.
Invitations titled "MIT Colloquium" arrived at research institutes and universities everywhere: Harvard, Princeton, Oxford, Heidelberg, Tokyo, Beijing, Zurich, and more.
The news spread quickly in Korea as well.
Professors at Seoul National University, POSTECH, and KAIST shared the invitation in their group chats.
-MIT is holding a colloquium, and the topic is the Smale problem?
-You're late to the news. It's been all the buzz at conferences lately.
-The speaker is Seo-ha? That's remarkable. I can't believe they're putting an undergraduate forward as the speaker.
-Ah... I wish he had come to our university. Then an international colloquium like this could have been held in Korea.
-It can't be helped. We're a country with few resources for mathematics research, so we can't offer support the way they can.
Su-jeong was looking at her phone.
Compared to a year ago, she had changed beyond recognition.
Her refined beauty caught the eyes of passing students.
"Hey, that's the math department goddess."
Someone whispered as they walked by.
On the screen was a colloquium invitation she had painstakingly obtained through several connections.
Su-jeong's gaze lingered on one spot.
[Speaker : Yu Seo-ha]
[Affiliation : Department of Mathematics, Massachusetts Institute of Technology(MIT)]
Tremble tremble.
Her hand holding the phone shook.
"He told me he was going to KAIST!"
By the time she heard the news of Seo-ha's enrollment, she was already a student at KAIST.
*****
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