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Chapter 241: Chapter 232: Robert Langlands

While Li Dong was spending a few days mulling over how to ”relax”...

「Across the ocean. Eckhart Hall, University of Chicago, Illinois.」

This old Gothic Revival building, constructed in 1930, had stood on the University of Chicago campus for nearly a century. The entire second floor was the mathematics library, while the math department’s offices began on the third floor.

In a second-floor hallway.

Frank Callegari, a renowned University of Chicago mathematician specializing in number theory and the Langlands Program, leaned against a window, a half-finished cup of coffee in his hand.

Standing opposite him was Wu Baozhu, a top scholar who had won the Fields Medal for proving the fundamental lemma for automorphic forms.

”You can’t blame the department for everything.”

Wu Baozhu said slowly.

”There are only so many spots on the doctoral admissions committee each year. Someone has to wait their turn for a seat.”

”But my student this year is really good.”

Frank retorted, unconvinced.

”He works on Galois deformations, with a background from Berkeley.”

”Then just nominate him again next year.”

”Next year? By next year, he’ll have been poached by MIT.”

Wu Baozhu just smiled and didn’t reply.

Frank snorted, drained his coffee, and turned back to his office, which was adjacent to the library wall.

The door slammed shut with a BANG.

Wu Baozhu stood where he was, shaking his head.

’Frank’s been a bit hot-headed lately.’

He turned and went back to his own office across the hall.

...

In his office.

Frank tossed the empty cup onto his desk, pulled a hand-crank coffee grinder from a cabinet, and slowly ground a new batch of beans.

His mood gradually calmed with the turning of the wooden handle.

’I’m not going to think about it anymore.’

’I’ll deal with all that department crap some other day.’

He sat down at his computer with the freshly ground coffee and habitually opened his academic email.

As an associate editor for the *Mathematical Annals*, he had to process a batch of manuscripts every day.

But in reality, not many submissions were distributed to him.

The editorial office screened one round, the chief editors went through another, and what was left to land in his inbox were almost exclusively submissions from authors with established academic reputations in his field of research.

Today, there was only one new email in his inbox.

Frank’s gaze swept over the title.

[A Criterion of Zeros for the Local-Global Compatibility of GL(n) Automorphic Representations and its Generalization to the General Case]

He read the title carefully.

A name immediately popped into his head.

—Li Dong.

That young man from the East who had recently become incredibly famous in the mathematics world.

That dimensionality reduction algorithm that still hadn’t been made public.

And that paper, *Proof of Montgomery’s Conjecture on Association for the Interval |α|∈[0, 4]*.

It had pushed a theoretical boundary that had been stuck for half a century from 1 all the way to 4 in a single stroke.

Frank let out a long breath.

He recalled the seminar at Yan University’s Sunshine Hall over a month ago.

He had originally planned to go.

He’d even booked his flight, but the department had saddled him with a last-minute admissions committee review. He couldn’t get away and ended up not being able to go.

Later, he found a recording of the seminar and watched the entire thing alone in his office.

After the half-hour recap of the Montgomery conjecture proof, he thought the young man was truly impressive.

But what really made his heart race was the part that came after, when Li Dong pulled a middle school teacher onto the stage.

That ”layer-by-layer recursive filtered embedding” written by Yang Shengguo, and the few correction terms Li Dong added so casually...

He himself had been walking down that same path for nearly three years.

He had seen that door, too.

He just hadn’t decided whether to push it open.

He never expected that young man from Huaxia to bring his middle school teacher...

...and in the space of a single afternoon, push that door wide open in front of the world’s top number theorists.

And after pushing it open, Li Dong had even smiled and said:

”This is just a side project.”

”What I really want to do is GL(n).”

”From the general case to its complete generalization.”

When Frank saw that part, he’d spilled coffee all over his keyboard.

He had high expectations for Li Dong.

But he wasn’t entirely convinced.

For over a hundred years, generations of mathematicians had beaten their heads bloody against the wall of GL(n)’s local-global compatibility.

’Could a young man really do it?’

...

’But how long has it been?’

Not even two months.

And now this paper was lying quietly in his inbox.

The anger in Frank’s chest over the admissions issue instantly vanished.

He pulled his chair closer and clicked on the attachment.

The PDF loaded.

The author field.

[Li Dong (Yan University), Arthur Penrose, Yang Shengguo (Jiangcheng Seventh Middle School)]

???

’Penrose?’

’That’s my old friend. Why would his name be on this paper?’

’And without an affiliation?’

Frank was puzzled, but that wasn’t his concern. The most important thing now was to read the paper.

But he was also taken aback when he saw [Yang Shengguo (Jiangcheng Seventh Middle School)].

Were there submissions to the *Mathematical Annals* from middle school teachers? Certainly.

But...

None had ever made it to his desk.

This was a first.

He chuckled.

Then he turned the page.

...Seventy-six pages.

Frank prepared to spend the entire afternoon on it.

He read on, page by page, line by line.

Very slowly.

Several times, he flipped back to compare his current reading with a lemma in a previous Chapter.

He muttered to himself unconsciously.

”Using the convergence interval of the pair correlation function F_π(α) to constrain the upper bound of the ramification index in reverse... this reverse step... it actually works...”

”The compatibility of the ε-factors... he didn’t use Deligne’s classical construction here. He just let the zero-point statistics swallow it whole...”

”In the totally ramified case, this critical width of 4/n... it wasn’t just pulled out of thin air. It was reverse-engineered from the second moment of F_π(α)...”

”Using 10²³ zero samples as a computable probe for the local Langlands correspondence...”

The more he read, the slower he went, and the more engrossed he became.

...

At some point, there were two knocks on the door.

Wu Baozhu poked his head in.

”Frank, want to go downstairs for a bite?”

Frank, his eyes glued to the screen, waved a hand near his keyboard.

”...”

Wu Baozhu stood at the door and watched him for a moment.

’Is he still mad about the admissions thing?’

’How can he be so stubborn?’

He shook his head, gently closed the door, and left.

...

At the same time Frank was buried in the paper...

「On the East Coast of the United States, in Princeton, New Jersey.」

In a north-facing office at the Institute for Advanced Study.

This had once been Albert Einstein’s office.

As the proposer of the ”Langlands Program” and one of the most influential titans of modern mathematics, Robert Langlands was sitting in the chair he had used for over thirty years.

He was nearly ninety years old now, and his back wasn’t what it used to be; he needed a small cushion for support.

But he still maintained a habit from his youth.

He came to the office every day.

Sometimes he would write a little, sometimes he would read new drafts from his colleagues, and sometimes he would just open the window and look down at the old path that Godel and Einstein had walked.

Today, he wasn’t reading anything.

A video was playing on his laptop screen.

The scene was Yan University’s Sunshine Hall.

A young man from the East stood before a whiteboard, and next to him was a middle-aged man half a head shorter.

The whiteboard was already covered, section by section, with writing.

Langlands gently pressed the pause button.

Then he leaned back in his chair and closed his eyes.

He had lost count of how many times he had watched this video since one of his students sent it to him over a month ago.

He had lived long enough.

Long enough to see the ”conjecture” he had casually jotted down in a letter decades ago be filled in, piece by piece, by his peers.

’It was filled in beautifully...’

But no matter how beautiful it was, it was still hollow, because its foundation—universality—was not yet complete.

He himself couldn’t do it, either.

’He often wondered if he had enough time left to see that foundation be laid.’

Now, that young man from across the ocean was standing on stage, saying:

”I’ll lay the foundation.”

Langlands opened his eyes, looking at that face, which still held a trace of boyishness.

He slowly smiled.

And whispered.

”...How much longer will it take?”

”Child.”