Sujeong was sitting in the audience.
Thump, thump.
“Huh? What’s wrong with me?”
The thought of seeing him again after so long made her heart race for no reason.
Sujeong gripped the program pamphlet in her hand tightly.
“I’m being stupid. I just came to watch a presentation.”
She tried to scold herself, but her heart would not calm easily.
The lights dimmed, and Seoha walked out.
Without realizing it, Sujeong drew in a deep breath.
“He’s changed so much.”
In the span of a year, he had grown almost beyond recognition. He was much taller now, and his shoulders had broadened.
In the past, whenever he was nervous, he would hunch his shoulders up tight, but now, even in front of such a large audience, he did not seem intimidated.
His once gentle features had become much more defined. There was still a boyish smile left on his face, but the atmosphere around him had changed greatly.
“Still, he’s not exactly an idol, is he?”
Having watched Seoha since he was young, Sujeong found it difficult to understand the media’s fuss.
Sujeong looked around.
“This is no joke.”
The author of the complex analysis textbook she was studying at school was sitting in the front row.
Seoha had become important enough to gather people like them for his lecture.
The host took the microphone.
“Today, we will hear a new solution to a problem that has remained unsolved for decades: Smale’s problem.
Please welcome the author of the paper. Seoha Yu!”
Clap, clap, clap!
A gentle sound of applause rang out.
Sujeong clapped hard as well.
As if he were still a child, Seoha bowed his head politely toward the audience, then went up onto the podium.
When he gave the signal, the lights dimmed and the projector began to shine.
“The mean value theorem is a concept that has been used for a very long time. But Stephen Smale wondered whether, when that theorem was moved into the world of complex numbers, a mean value could still exist.
To his question, I intend to answer: yes, it exists.”
The manuscript he had spent days discussing and writing with Theo.
To Seoha, who had thought all he needed to do was prove it neatly, Theo had said,
“A considerable number of the people coming won’t understand your formulas. You have a duty to explain it in a way they can understand, too.”
The two of them had spent the entire holiday agonizing over how to structure the lecture.
A shape appeared on the screen.
Against a black background, a single curve was drawn in white lines, and two points appeared upon it.
The one on the left was blue, the one on the right red.
“There are two points. A starting point and an ending point.
A function moves and changes between them, and at times loses its direction.”
When Seoha gestured, the curve moved smoothly.
Whoosh—
A sound effect rang out.
The shapes swayed as if pushed by waves, then began moving around frantically, changing places.
Small exclamations escaped from the audience.
“Smale’s question was this.
Even within this complex movement, does a mean exist somewhere?”
When Seoha stopped speaking, the curve on the screen halted.
The two points stopped moving as well.
“The problem was this.
This curve we see is, in truth, not flat. To the eye, it appears to be an ordinary line, but in the world of complex numbers, this line is bent countless times. As if one more invisible dimension were hidden inside it.”
When Seoha gestured, the curve slowly moved and began changing three-dimensionally.
Not a line, but a surface. Not a surface, but a curved surface. The curve twisted and moved. Then, after writhing violently, it finally surged out beyond the screen.
“Oh my!”
“Huh?”
“Jesus!”
Startled members of the audience let out short cries here and there.
One professor, as if amused, laughed and pushed up his glasses.
“This is the true form of the world we call a plane. Though it may look smooth on the outside, inside it exist countless layers that are bent and curved.
This is also why Smale’s mean value remained an open problem.”
The professors seated in the front row nodded with their arms crossed.
This was something no one had accomplished in decades.
“I tried converting this into a plane.”
At Seoha’s words, the air in the auditorium changed subtly.
As if the main point began here, the mathematicians’ eyes gleamed.
“In this process, twenty-one auxiliary lemmas were necessary. Each curved surface twists under different conditions, and each of them had to be unraveled one by one.”
With Seoha’s gesture, the screen moved again.
A vast structure appeared, in which countless curved surfaces overlapped.
There were thin gaps between the surfaces, and within each gap flowed irregular streams of color.
“The first theorem.
Maintain the contact conditions of the curved surfaces at a constant state.”
As soon as he finished speaking, the uneven contact surfaces aligned cleanly.
“That guy…”
Having realized Seoha’s intention, Kronen laughed heartily.
“The second theorem.
The removal of nonlinear distortion that occurs near the extrema of the curve.”
The folded curved surface unfolded. And the bent lines inside it straightened evenly and connected.
The audience stared blankly at the screen, forgetting even to breathe.
“The third.
Unravel the topological twists that occur on the complex plane.”
The scale grew larger and larger.
This time, the entire screen rotated. As if a complicated knot were being untied, the surfaces returned to their proper places and began aligning themselves.
The fourth, the fifth…
Each time a theorem was completed, the result was implemented on the screen in real time according to Seoha’s explanation.
As if by magic, every time Seoha’s hand moved, the world inside the screen transformed.
“The seventh.
Intrinsic displacement to minimize discontinuities.”
Thousands of curves appeared on the screen at once.
When Seoha raised his hand, they began to move. Like flowing water, they naturally found their own places.
The chaotic curves flowed organically, as if they had gained life.
Seoha’s theorems continued.
Everyone was drawn into the screen as though bewitched.
Following Seoha’s gestures, the lines and surfaces that had been surging in disorder began to take shape.
And the gaps at the boundaries began to fill.
After this was repeated twenty times, the structure had become something entirely different from what it had been at first.
“The final theorem.
Connect the local variations of all curved surfaces into a single plane.”
Papapat!
“Wow…”
“Oh my God!”
Someone sprang up from their seat.
They were witnessing a mystery.
The sight of a theorem once considered impossible within the complex plane nearing completion.
The countless tangled spaces converged into a single surface.
The world within the screen was now a perfect plane.
Just as they had first seen, a single white line connected a blue point and a red point.
Between them, a single green point bloomed.
“Now we can find the mean point. At this green point, the positional relationship between the red point and the blue point becomes completely equal.
This is the mean value I found—the balance that appears at the end of countless discontinuities and distortions.”
For a while, the mathematicians could not say anything, as if moved.
Clap, clap, clap!
Applause spread.
The small sound gradually grew louder, until at last it swept over the entire auditorium like a wave.
“Ah!”
“That’s impossible. He really solved it?”
Sighs of disbelief burst out.
Scholars looked at one another and whispered something in excited voices.
Elijah Kronen stroked his chin with a strange smile.
“How does it feel? Seeing the fish that got away.”
Whitman shrugged and gave an irritating smile.
“It was not a fish, but a whale. From the beginning, it was beyond the vessel I could contain. I believe Seoha’s lecture today will have a great impact on our fellow mathematicians going forward.”
“In what way?”
“A kind of revolution. A revolution of perspective. The mathematical world has been far too rigid until now.
I think a more creative and freer atmosphere will form from here on.”
Whitman nodded.
“In truth, I too am learning a great deal from watching Seoha.”
“At your age?”
“Age has nothing to do with scholarship. I am still improving.
Until now, the mathematical world has focused too much on simplifying problems. But Seoha did the opposite. He acknowledged complexity and resolved it one piece at a time.”
Kronen shook his head.
“That is not something just anyone can do.
What Seoha did is, in truth, close to a miracle. It is like having tens, hundreds of threads tangled into a complete mess, and picking out the single strand that leads to the correct answer.
It was a proof that could only be possible with terrifyingly sharp intuition supporting it.”
Seoha was taking questions from the audience.
Professors dug into the gaps in his theory, pressing him on conditions that had not been specified concretely in the paper.
But Seoha was smiling brightly, as if he did not feel burdened at all. Kronen felt he knew why.
“Because they’re not trying to fight.”
He thought there was a problem with the term “thesis defense” itself.
If one wins the defense, does the theory become fact?
Kronen shook his head.
It was not divided into victory and defeat. It had either already been revealed, or it had not.
Seoha purely longed for truth.
That was why he could enjoy even the question-and-answer process.
One professor raised his hand.
“Do you believe your planarization process is complete? Could discontinuities not still remain?”
Seoha thought for a moment, then nodded.
“I can’t say there’s no possibility of that. If that time comes, a new auxiliary lemma may be necessary.”
Another professor asked,
“Then would it not be difficult to view this theory as a completed law?”
Seoha shook his head.
“The systems and perspectives humans handle may expand.
But even within them, the law I established will not change. Rather, through that process, my theory will continue to be proven and serve as a foundation for other theories.”
Seoha held not the slightest doubt toward the theory he had established.
The professor who had asked the question let out a great exclamation of admiration.
From the back, a professor began to applaud.
Soon, the applause spread around him. Before the colloquium, there had indeed been those who underestimated Seoha. Not because they were arrogant, but because of Seoha’s age.
But in this moment, no one was conscious of his age.
Seoha bowed deeply.
The applause erupted again.
This time, it was not cheering, but applause filled with respect for the young mathematician.
* * *
Sujeong quietly left the venue.
Her steps felt heavy.
At first, she had planned to find Seoha after the presentation ended and greet him.
She had wanted to ask him.
How he could have left without saying a word to her, and whether things had been difficult for him all this time.
But when she actually stood before the door, her feet would not move.
Mathematicians whose names even she knew were applauding him. Seoha had now become a person of an entirely different world.
She had wanted to brag.
That she had ranked first in her year this semester, that she had received a proposal for corporate collaboration despite being a freshman.
And even that she had shocked everyone by rapidly raising her Logia ranking.
Like before, she had wanted to say with a confident face, “I’ll be your strength.”
“This much isn’t enough.”
To become someone who could support him, she first needed the right to stand beside him.
Sujeong clenched her fist tightly.