Translator: Dreamscribe
“I’m home.”
“Welcome back. You haven’t eaten yet, have you?”
“No, I’m hungry, Mom.”
Whoooosh.
Seo-ha set down his bag and washed his hands.
“Oppa! Did you have a tea party at the lab again today?”
The moment he sat down at the table, Seo-eun’s eyes sparkled as she asked. Cecilia had given Seo-eun some tea leaves as a gift. Thanks to that, a new routine had been added to the time he spent with his little sister.
“Yeah, we drink tea every day.”
Maybe it was Cecilia’s influence, but lately the lab’s beverage trend had been shifting from coffee to black tea.
“Let’s eat.”
Mi-young brought out the food.
A freshly boiled soup, an assortment of seasoned vegetables, and golden-grilled mackerel were laid out on the table.
‘Ah, this is nice.’
It was ordinary and warm.
Seo-ha picked up his spoon and took a mouthful of soup. The warmth flowing down his throat felt as though it were soothing his overexcited brain cells.
“The tea leaves Cecilia gave me, I drank them with my friends the other day, and everyone said they were so good! I told them about the tea party, and they all said they were jealous of me!”
Perhaps she was happy her brother had come home early, because Seo-eun chattered away nonstop. The impression of a tea party hosted by a real British noble must have been quite striking, Seo-ha thought.
“Thanks for the meal!”
After dinner, he helped Seo-eun with her studies.
His little sister’s math skills were improving by the day. Seo-eun liked to prove problems in her own way.
“Oppa, here’s where the circle and the line meet. So the coordinates are....”
Maybe it was because they’d played with shapes together since she was little? She had a preference for finding answers by drawing graphs or figures when solving algebraic equations.
“Right. Well done.”
Seo-ha looked at Seo-eun with warm eyes.
The moment she reached an answer on her own, without him needing to explain. Seo-ha loved watching his little sister do that.
Study time was over, and now it was time for himself.
Trudge, trudge.
Seo-ha climbed the stairs and grasped the door handle.
Click.
The instant the door closed, his expression sank into something cold.
“Haah, haah.”
He stood with his back against the door, breathing roughly. Then he hurriedly pressed the button on the CD player resting on his desk. A gentle performance of winter began to flow out.
‘Calm down.’
Ducky, who had been quiet for a while, kept trying to break free of his control these days.
He wanted to ride in along Seo-ha’s optic nerve right this instant and dive onto the plane of the Zeta Function. The infinite Zeros laid out upon the Critical Line, he wanted to fling himself into that perfect lattice of equations and swim through it.
The reason was easy to guess. It was probably because, somewhere in his head, Seo-ha could now see the rough path leading all the way to a proof of the Riemann Hypothesis. Just as one cannot expect patience from a beast with tempting prey before its eyes, this was an unavoidable phenomenon.
Grind.
Seo-ha clenched his teeth.
‘Not yet.’
Ducky’s rampages had left deep trauma on his family.
Besides, even if he set him loose right now, there was no guarantee he could sprint straight to the correct answer in one go. There was no telling how many days it might take, and no certainty that he would come back properly either.
“Ducky, wait. The problem isn’t going to run away. It’s a problem that’s waited nearly two hundred years. There’s no reason it can’t wait a few more months, right?”
Seo-ha closed his eyes and muttered to himself.
Tomorrow too, he would sit at the table his mother had set and eat. He had to stand before that door while keeping up his daily life, helping Seo-eun with her studies. It couldn’t be getting dragged toward the problem like a moth to a flame; it had to be him opening the door himself.
Ducky’s voice came to him, as if mocking this resolve.
‘Looks to me like you want to solve it even more than I do? Don’t resist. Just throw yourself at it. We’d be the first humans in history to witness the moment all the Zeros fall into line under a single rule. How ecstatic would that be?’
Seo-ha glared at the doll sitting in the corner.
A beaver wearing a duck mask, ridiculous to look at. It glinted its eyes as if seeing straight into the abyss of Seo-ha’s soul.
‘Family? Daily life? Those things are lighter than dust before this immense truth. Let’s write the proof for months on end, until our finger joints are crushed. Compared to the rapture we’ll taste at the end of it, everything else is trivial, isn’t it?’
Swish.
Thwack.
Seo-ha snatched up his pillow, took aim at the beaver, and hurled it mercilessly.
“You’re right about one thing. That I want to prove this just as much as you do. But I’m going to do it my way.”
His throat was dry. He felt stifled.
The agonizing hunger of a man who has held his breath underwater for several minutes, in the instant just before bursting up to the surface.
Seo-ha staggered over and sat down at his desk. Then he opened his notebook.
Scratch scratch.
Seo-ha’s pupils dilated strangely.
Beyond his eyes, instead of the room of reality, there stretched an infinitely extending plane. The moment the tip of Seo-ha’s pen touched the paper, the rows of Zeros began to expand into three dimensions.
And the lines rising up off the paper twined together, unfolding into a vast topological space. The dream that the masters of geometry, Alexander Grothendieck and Alain Connes, could not achieve even after devoting their entire lives, was being newly created in Seo-ha’s hands.
Noncommutative Geometry and Arithmetic Geometry.
Seo-ha’s pen crossed the sea lying between the two continents. A bridge was laid down following the pen, and at the center a great continent rose up.
Noncommutative Algebra, Hilbert Space, Spectral Theory.
Mathematical theories born in quantum mechanics met Seo-ha’s intuition and were reorganized into a new language.
* * *
The next day, the lab.
Seo-ha was eating a doughnut. It clearly wasn’t his first one; piled up beside him were the wrappers of chocolate bars and cookies.
Theo glanced sidelong at the pile.
“Seo-ha, nothing happened last night?”
Seo-ha turned his head, taking a bite of the doughnut.
“My body feels like it wants sugar. I can’t eat this stuff at home.”
Three chocolate bars, two bags of cookies, and now the doughnut in his hand. The usual Seo-ha would have been fine all morning on a single cup of tea.
Theo’s gaze turned to the blackboard. Fourteen lines of equations were written there. But he couldn’t grasp at a glance what those brief lines meant.
Seo-ha put on a sheepish expression.
When he’d opened his eyes that morning, his body had felt heavy, as if he’d run a marathon the day before. His fingertips were numb, and behind his temples there was a humming throb.
“Son! Are you okay?”
At the breakfast table, Mi-young had studied Seo-ha’s complexion, then hurried over and felt his forehead.
“I think you have a fever?”
“N-no, Mom! The heating must have just been too strong.”
Seo-eun tilted her head.
“Huh? But I was cold?”
Seo-ha laughed and waved his hand.
The team members came in one by one.
“Good morning!”
“Did you see outside? The weather’s so nice today!”
“I brought sandwiches today.... Hm?”
“Sri! Your desk is a mess again. I told you how many times to throw those parts away....”
Cecilia came over carrying two teacups.
Clink.
She set down on Seo-ha’s desk a teacup brimming with honey and milk. Her gaze was fixed on the blackboard.
“...When did you write this?”
“I came in early this morning.”
Cecilia strode up to the blackboard. Theo was already beside her, staring at it intently. She set down her teacup too, folded her arms, and began to read the fourteen lines of equations.
The atmosphere was anything but ordinary.
Su-jeong, Sri, and Ben joined the lineup one after another.
The notation of Étale Cohomology, Grothendieck’s language. And yet it was defined over Alain Connes’s noncommutative space. Cecilia’s eyes narrowed. These two things didn’t mix. At least not in the mathematics she knew. That was why she had come here fully prepared.
“Seo-ha, can you explain it?”
At Cecilia’s request, Seo-ha nodded.
His long explanation followed.
Clatter.
By the time they had refilled their teacups three times, the team members were finally able to understand the concept.
“It’s not a completed theorem yet.”
“I imagine not.”
Theo stroked his chin. The thinking of several great masters was distilled into the equations.
‘Incredible. But....’
The idea of using a Projective Module to carry the role of a Vector Bundle into the noncommutative world, and defining cohomology over that, was beautiful, but this theory required a great many Lemmas.
Seo-ha had surely known intuitively that this hypothesis held. But gaining the recognition of the academic community was another matter. Every condition had to be without flaw.
“Poincaré Duality, finite-dimensionality, compatibility with base change. Each of these is a problem demanding its own independent proof. And probably a few more besides these.”
Seo-ha nodded.
Cecilia’s eyes wavered violently. Only a few days ago had the gateway leading to the Demon Lord’s castle finally opened. What had leapt out from there was a monster, one of the Four Heavenly Kings who governed space.
“Kuhahahaha! You wretched little vermin! Vanish into the dust of spacetime!”
It was as if she could hear his voice in her ears.
A legendary demon that even the heroes of old, Hilbert, Grothendieck, and Alain Connes, had failed to surpass. She had braced herself for years of fierce battle, but her party had the greatest hero of all time.
In her eyes, she could see the scene of the battlefield.
The hero strode forward and broke from the formation alone. And then...
“Seo-ha, wait!”
Too late.
Seo-ha’s sword moved without hesitation.
The first stroke.
Noncommutative Algebra.
Kaboom.
The instant the hero drew his sword, a shockwave swept across the land. The fields of the Commutative (Commutative * the property where the order of operations does not change the answer), where the heroes of old had fought, split apart, and a new terrain rose up.
A warped earth where the result changes according to the order of multiplication, a land the monster had never once set foot upon.
“How dare you, you little brat!”
The monster roared in fury.
The next blade.
A six-line theorem drawing out a dual structure from the spectrum of the Dirac Operator.
“Gweeeek!”
The monster shrieked.
The hero’s sword changed direction. An equation that placed Noncommutative Algebra over a Hilbert Space and restored the geometric structure.
A single key inequality pierced through the entire structure. Just like finding the exact gap between the scales on the giant monster’s neck and stabbing it.
Six more blades followed.
Fourteen strokes in all hacked the monster to pieces.
It was overwhelming force, pure and simple.
“Aaaaargh!”
At last the final line was written.
A powerful blow that cleaved the thick neck in two!
Its head snapped back, and the monster’s enormous body pitched forward.
“...Did we get it?”
The Sri of her imagination poked his body out and asked.
Cecilia jolted in shock, whirled around, and glared at him. The real Sri, who had been quietly looking at the equations, shuddered for no reason at the chill in her gaze.
Rumble.
It wasn’t over.
The monster’s body seeped down beneath the earth, and a complex equation was drawn. Its body scattered into dozens of small fragments. And each shard, cloaked in the shell of a Lemma, crept into every corner of the blackboard.
“Second phase.”
Cecilia murmured quietly to herself.
Commutator estimates, trace-class convergence, injectivity of module homomorphisms, existence of a Spectral Gap, compatibility of index pairs, and many other theorems besides.
He had summoned his minions.
The Heavenly King was resting, his shattered main body hidden behind the summoning circle.
“There’s a lot of them.”
“There are. But if we work hard from here on, it won’t take too long. First, let me take the trickiest one....”
At Cecilia’s words, Seo-ha picked up a piece of chalk, looking apologetic. But she raised her hand and stopped him.
“No! From here on, it’s us.”
Theo looked at her and nodded.
Seo-ha’s Hyper-focus Mode had terrible fuel efficiency. The two of them thought along different lines, yet their intentions were oddly aligned.
With a grim expression, Cecilia glared at the list of Lemmas.
While he rested, his comrades would take care of the pawns.
After all, that was the standard formula for a hero’s party.
*****
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