One morning, a few days before Christmas,
Cambridge’s weather turned fickle again. The sky, clear until the day before, suddenly clouded over, and heavy snow began to fall.
The sight of the campus blanketed in white snow was as beautiful as a scene from a fairy tale.
Clack.
Whitman, the dean of the mathematics department, set his coffee cup down in an empty spot amid the piles of documents. Then, out of habit, he logged onto arXiv.
It was to check whether any new papers had been posted overnight.
A list of newly registered papers appeared on the computer screen.
Scroll.
As he scrolled down, a familiar term made his hand stop.
“Smale mean value?”
It was a name he had not heard in a long time.
Stephen Smale. A mathematician from the generation above his own. A giant who had won two of the three most prestigious prizes in mathematics—the Fields Medal, the Abel Prize, and the Wolf Prize.
With a trembling hand, Whitman clicked the mouse.
When the paper opened, the title came clearly into view.
[On the Smale Mean Value Conjecture in Non-Uniform Measure Spaces]
It was a short, plain title.
Whitman immediately checked the author’s name. Then, startled, he rose from his seat.
“Yoo Seoha?”
He had wondered why Seoha had been hard to see lately; to think he had been working on this.
Moreover, the person listed as second author was no ordinary figure either.
Theodore Lanford,
A young mathematician from Stanford. A man who had drawn attention early on with papers connecting analysis and geometric theory. Whitman had even met him and spoken with him at conferences.
“What on earth happened?”
The idea that he had written a paper together with an undergraduate from MIT was difficult to understand by common sense.
“Don’t tell me they solved an unsolved problem together?”
Whitman adjusted his glasses and pulled the monitor closer.
His question was answered before long.
In the preface, Theodore revealed his motivation for participating.
[My role in this paper was minimal. I merely organized materials and provided some small assistance in rendering the author’s mathematical language easier to understand; the proof itself belongs entirely to the author.]
The contents were neatly arranged.
Seoha’s characteristic unkind explanations and leaps in logic were nowhere to be seen.
After reading a few lines, Whitman slowly drew in a breath.
“This part bears Theodore’s touch.”
The paper was astonishingly complete.
Its logical structure was smooth, its citations clean, and there was no hesitation in its choice of words.
Concise formulas and a dry prose style—the quality was excellent enough to appear in a major journal immediately.
But what mattered was the content.
He picked up a pen and slowly began reading through the paper.
One hour, two hours…
“Ah!”
His stiff expression gradually brightened, and when he reached the process of flattening the surface, he let out an exclamation of admiration.
And when Seoha finally derived the equilibrium point, he rejoiced as though it were his own achievement.
“Hahaha! I knew this would happen.”
Seoha had not approached Smale’s problem in the traditional way.
He had reborn the old concept of the mean value theorem into a new language: the definition of space.
“To redefine density… It’s a method no one but Seoha would even have attempted.”
Until now, no mathematician had presented such a solution.
Seoha’s paper retraced the reasons for failure one by one, as if explaining the cause of an old illness.
He recalled the words Smale had left behind decades ago when he had given up on this problem.
“The mean value must exist everywhere. We simply do not know how to approach it. I hope someone in the future finds the way.”
He scrolled down to the very last page.
The conclusion was only three paragraphs.
And yet, within those brief paragraphs, everything the speaker wished to say was contained.
“Stephen. It seems I’ve grown quite old as well.”
Was this what history was?
He leaned back against his chair and looked up at the sky. A thrilling shiver ran down Whitman’s back.
The power to solve, with a single line of intuition, a domain that existing approaches had failed to reach.
It was a sensation he had long forgotten since the appearance of Elijah Cronen, and Whitman felt that the moment when one generation gave way to another had arrived once more.
Whitman rose from his seat.
The snowfall was growing heavier.
Outside the window, students passed by through the snow, their bodies tightly hunched. This sight had not changed in the past, nor in the fifty years since.
“It will probably be the same fifty years from now…”
Even so, mathematics continued to advance steadily.
He was grateful that he was in a position to witness the scene of that progress firsthand.
* * *
The next morning dawned.
Even amid the busyness of final exams, a strange air flowed through the mathematics department.
Students peered at their phones and murmured among themselves.
“Did you see it? That new paper.”
“You mean the Smale problem, right? When did he even do that? He’s seriously amazing.”
“In our department, undergraduates solve unsolved problems.”
A day had passed since the paper was uploaded to arXiv.
In that short time, researchers all over the world were already sharing the link to the paper.
Hundreds of comments appeared 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. Seoha Yoo is still an undergraduate, so what kind of student is this?
He takes a horrifically complicated theorem, folds it geometrically, and then unravels it again through probability. It made me wonder whether what he was doing was even the mathematics I learned.
We’ll have to see after verification, but it was a brilliant proof that shook the assumptions and definitions themselves.
└Were there any errors?
└None that I could see. The derivation itself is art.
└How did Theodore end up collaborating with Seoha Yoo?
└They were fighting tooth and nail on Logia. Maybe it turned into an offline brawl?
└I’m a chemistry major, so I don’t know this student. Is he famous?
└Goodness, Seoha is the very future of our department. Starting with the dean, you have no idea how dearly everyone treasures him…
By night, Europe’s mathematicians had awakened.
The same posts circulated on forums in Berlin, Paris, and Zurich.
The mean value theorem in nonuniform spaces.
If it passed verification, it would be an event that rewrote analysis textbooks for the first time in half a century.
* * *
The next day,
Snow was still falling from the Cambridge sky.
Whitman had convened an emergency faculty meeting the day before.
The mathematics professors sat around the dean’s office, drinking tea.
They looked tired, as though they had been working for a long time, but the atmosphere in the room was strangely excited.
“What are the reactions like?”
At Whitman’s question, one professor tapped on his laptop.
“I just checked the European forums as well. The German mathematical community says they’ll begin an independent verification.”
Whitman laughed in a low voice.
“How diligent of them.”
On the table lay printed copies of the paper.
Yesterday, all the mathematics professors had gathered and carried out a preliminary verification of Seoha’s theory.
The result: no abnormalities.
“Dean, should we formally request verification? The American Mathematical Society, the AMS, has been contacting us.
The editors-in-chief of major journals are getting impatient too. The department office staff say they want to unplug the phones.”
It was not a problem known to the general public, but mathematically, it held great significance.
A problem so stagnant that even scholars had forgotten it because there had been no progress, not only toward an answer but even in methods of approach. Seoha had dragged it back to the center of academia after half a century.
“Since the second author is from Stanford, there’s movement on their end too. They’re planning to invite Seoha and Theodore Lanford to hold a lecture.”
The room stirred for a moment.
“Why would they invite our school’s student over there?”
“I heard Lanford only refined the structure of the writing. Isn’t the core idea entirely Seoha’s?”
“As long as his name is on the paper with him, he has a responsibility to explain it. To outside eyes, the two are already seen as a team.”
Whitman quietly raised a hand, stopping the commotion.
“Exactly which places are saying they will verify it?”
One of the professors answered.
“In Europe, Heidelberg University. In Asia, the University of Tokyo says it will hold an unofficial seminar. The others seem likely to move after seeing the response.”
Whitman thought for a moment, then struck the desk and rose.
“Good. Whether it’s verification or discussion, they can do it all here. At MIT.”
The professors’ faces stiffened all at once.
“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 ridiculous.”
“That’s right. Wouldn’t it be better to reconsider? If an error we failed to find is discovered, all of MIT will suffer a great embarrassment.”
Whitman slowly looked around at the professors.
“Everyone, since when has MIT been a school that only makes safe choices?”
Silence fell over the dean’s office.
“Twenty years ago, at the Topological Quantum Field Theory seminar, everyone said it was impossible.
But the framework we built then ultimately became the standard in the international academic community.”
One professor quietly raised his hand.
“At that time, the theory was just beginning to take shape. So even if we took a risk, we had the justification that ‘we are leading the trend.’
But isn’t this an old, unresolved problem? I don’t see any reason to rush.”
It was a calm and rational judgment.
The professors nodded as if in agreement.
But Whitman did not bend his will.
“If it’s justification you want, there is plenty.”
“What would that be?”
Whitman answered in a tone full of conviction.
“Where is the mathematics of a new era born? I want to leave behind proof of that.”
Silence flowed.
“Hmm…”
“Certainly, Seoha does have that quality.”
“A paradigm shift…”
Everyone understood the meaning of what he was saying.
“There is something special about Seoha’s mathematics. He is not bound by the authority of classical mathematics. He places limits nowhere.
To conservative scholars, this may feel somewhat foreign. But if the mathematics of a new era exists, I suspect it will take this form.”
Each professor wore an expression filled with thought.
But before long, they made up their minds and nodded. One of them raised a hand.
“I would like to look a little further ahead. This is only a hypothetical, but…
If, in the future, mathematics enters a new stage because of Seoha, and if the starting point is here, then shouldn’t we leave a proper record?”
“I agree as well.”
“We cannot let another school have him give the first lecture.”
Their opinions gathered in one place.
Whitman nodded in satisfaction.
“Then we will hold a colloquium. There will be no restrictions on qualifications for attendance. Anyone who wishes to come may come.
However, distinguished figures who can lend it credibility must attend.”
Once the decision was made, matters moved rapidly.
The department office became busy.
Harvard, Princeton, Oxford, Heidelberg, Tokyo, Beijing, Zurich—research institutes and universities in various places received invitations titled “MIT Colloquium.”
The news spread quickly in Korea as well.
Professors from Seoul National University, POSTECH, and KAIST shared the invitation in their group chat.
-MIT is holding a colloquium, and the topic is the Smale problem?
-You’re late to the news. It’s been causing an uproar at conferences lately.
-The speaker is Seoha? Amazing. I can’t believe they’re putting an undergraduate forward as the speaker.
-Ah… It would have been nice if he had come to our university. Then an international colloquium like this might have been held in Korea.
-It can’t be helped. We’re a country with fewer resources for mathematical research, so we can’t support him as much as they can there.
Sujeong was looking at her phone.
Compared to a year ago, her appearance had changed beyond recognition.
Her pure and delicate beauty drew the gazes of students passing by.
“Hey, it’s the goddess of the mathematics department.”
Someone whispered as they passed.
On the screen was an invitation to the colloquium, one she had obtained with difficulty through a chain of connections.
Sujeong’s gaze stopped in one place.
[Speaker : Seoha, Yoo]
[Affiliation : Department of Mathematics, Massachusetts Institute of Technology(MIT)]
Tremble, tremble.
The hand holding her phone shook.
“He told me he was going to KAIST!”
When she heard the news of Seoha’s university choice, she was already a student at KAIST.