Chapter 275 - 259: Willing

Chapter 275: Chapter 259: Willing

Li Dong didn’t speak. He was waiting for Professor Wang to continue.

Professor Wang spoke slowly.

"I’ve spent my whole life wrestling with automorphic forms."

"I struggled in my early years, but my luck turned out okay later on."

"In this field, I’ve managed to produce a little something, ’A Weak Form of Langlands Functoriality for GL₂ with Ramification ≤ 2’."

"And also the étale decomposition of Hecke algebras at prime places, which I did with a collaborator from the Netherlands."

"Both at home and abroad, my papers are fairly well-recognized."

He said it matter-of-factly.

But in Li Dong’s mind, the label he had for "Professor Wang" was already being quietly rewritten.

He was no longer just some generic "old professor."

Instead, he was a veteran who had truly fought on the front lines of the Langlands Program for decades.

"A few years ago..." Professor Wang continued, "Jiang Yubai from Capital University actually had someone reach out to me twice."

"At the time, he was pushing his work on GL₂ with ramification ≤ 3 and wanted me to go over and work with him on the corresponding extension for the functoriality side."

Li Dong raised an eyebrow.

He really hadn’t known about that.

"At the time... I didn’t go."

Professor Wang chuckled.

Listening, Li Dong silently nodded to himself. ’If he’d gone with Jiang Yubai, Professor Wang would probably be finished by now.’

Professor Wang continued.

"Sigh. But back to the matter at hand."

"After your GL(n) paper came out, I sat down and read through it more than a dozen times myself."

"The line of research I used to pursue is now just a piece of scrap paper."

"But there’s no need to mourn over it. Old things are meant to make way for the new, anyway."

"But the problem is..."

He looked at Li Dong.

"The next stop you sketched out on that blueprint—the Li Clan’s Conjecture."

"I can’t get there."

"I can see the entrance, but I can’t take the step."

The seminar room fell silent.

Li Dong didn’t answer right away.

’Deep down, he’d always known one thing for certain.’

’His universality paper had solidified the foundations for the entire Langlands Program.’

’But just because the foundation was solid didn’t mean the road ahead would be easy.’

’From his paper, to the Li Clan’s Conjecture, and then to the true completion of the Langlands Program...’

’There was still a cloud in between.’

’He himself could only see the silhouette of the grand structure behind that cloud.’

’As for the vast majority of scholars in the field...’

’They couldn’t even see the silhouette of that structure.’

’What were they doing right now?’

’They were still looking up, searching for the cloud.’

Professor Wang’s voice sounded again.

"On my own, I can’t get there."

Li Dong still said nothing.

He was thinking about something he had never seriously considered before.

’He had always been the type to chew on a problem by himself.’

’Although he had Penrose and Yang Shengguo for the universality paper, that was a spur-of-the-moment thing, not a permanent team.’

’As for the Langlands Program, he had long ago decided in his heart that he would be the one to complete it.’

’He had attribute bonuses.’

’He had a Memory Palace.’

’But even so...’

’No matter how fast he ran, a day still only had twenty-four hours.’

’He still had to join Professor Wu Kai’s group at the School of Chemistry.’

’The Fields Medal countdown in the group chat was still ticking away: two years and eleven months.’

’He really couldn’t be in two places at once.’

’But if he had a group of veterans behind him, helping him raise the roadbed...’

’...he could run much faster.’

’More importantly...’

’He could point them in a direction and let others start walking.’

’Then, when he had time, when his attributes had leveled up, or when a "primitive derivation of the Li Clan’s Conjecture" just happened to drop in the group chat one day...’

’...he could follow up, and things would go much faster.’

The more Li Dong thought about it, the more he felt this was the right approach.

He had never put together a research group himself.

Back when he, Penrose, and Yang Shengguo hammered out the universality paper, they never even used the term "research group." It was just a whiteboard, a seminar room, and the three of them.

’What kind of research group was that?’

But this time was different.

This time, he didn’t just want a single proof.

He wanted an entire path, and he had no idea how many Fields Medal winners might emerge from it...

Li Dong looked up at Professor Wang.

"Professor Wang."

"Regarding this path, let me tell you about the starting point I see."

Li Dong said, his tone very serious.

"Please don’t take this the wrong way."

"I don’t have a complete roadmap either."

"I can only tell you that I believe this path has two entrances."

"The first is the localized generalization of base change functoriality for GL(n)."

"I actually hinted at this direction in the appendix on conjectures at the end of my paper, but I didn’t elaborate on it at the time."

"That’s because it wasn’t the main thrust of that paper."

"The core of it can be summed up in one sentence."

Li Dong said, enunciating every word.

"On both sides of the zero-pair correlation equation, you have the Hecke projections of two automorphic representations at a local place, and their difference can be controlled by an accumulated point curvature."

Professor Wang’s eyes instantly lit up.

The line he had jotted down at the classroom door just now was only two technical terms away from the sentence Li Dong had just spoken.

"The second is from the opposite direction, coming down from the zero equivalence criterion for general reductive algebraic groups."

"The difficulty of this path isn’t in the technique, but in the concept."

"Previously, everyone assumed functoriality was a matter of algebra."

"My view is that it’s actually, first and foremost, a matter of analysis..."

"This line of reasoning is the origin of my Li Clan’s Conjecture."

"If someone could translate the phrase ’the zero-point statistical structure is the same’ into a computable and verifiable criterion."

"Even a weak version..."

"...then this path would truly be open for people to walk."

Li Dong laid out the starting points of the two paths.

Of course, it was all still vague. For some parts, he himself only had a rough idea, not even close to a complete draft.

But just these two starting points were enough.

Professor Wang didn’t move; he was still deep in thought.

After a minute or two, he finally spoke.

"Li Dong..."

"These two starting points of yours..."

"Especially the first one."

"The étale decomposition work our group did before... it can connect to that."

"And..."

He paused abruptly.

He realized that in the time it took Li Dong to say those few sentences, the dense fog in his mind about "what do I do next" had mostly dissipated.

He couldn’t help but think of that term again.

The master effect.

...

Professor Wang looked at Li Dong.

’This person really wasn’t even twenty years old.’

He sighed.

In the world of mathematics, it wasn’t that he hadn’t seen geniuses before.

Jiang Yubai had been a genius in his youth, too.

His collaborator from the Netherlands had supposedly gotten a group of Princeton professors to write him letters of recommendation when he was only twenty-two.

But this kid, Li Dong...

’Is he a genius?’

’No!’

’He’s a master!’

When Professor Wang looked at Li Dong again, his gaze held a hint of reverence that even he himself hadn’t noticed.

Just then, Li Dong spoke up very suddenly.

"Professor Wang."

"I’m preparing to establish a research group."

Professor Wang was taken aback.

"Huh?"

He hadn’t expected Li Dong to suddenly bring this up.

Li Dong ignored his confusion and continued.

"I want to see the Langlands Program to its completion."

"It’s too slow for me to do it alone."

"Would you..."

"...be willing to join?"

The seminar room fell silent once more.

Professor Wang’s mouth hung open, speechless for a long moment.

’Join Li Dong’s research group?’

’In other words...’

’He, Wang Zhigang, a man nearing sixty, was being asked to become a "subordinate" to this college freshman.’

His first, instinctive reaction was to refuse.

It wasn’t because Li Dong wasn’t qualified.

It was because he himself felt a little...

...unwilling to lose face.

Although Li Dong was now among the world’s top mathematicians and was, in terms of qualifications, more than capable of being a leader.

But he was, after all, not even twenty years old—younger than his own children. It would sound bad if word got out.

So, he was still prepared to refuse.

But just at that moment...

...the thought of "the splitting of H_v in the neighborhood of an accumulation point" popped into his head again.

That line of thought, which had been illuminated by a single sentence from Li Dong in the classroom, was now crystal clear before his eyes.

He suddenly thought...

’If he left Li Dong’s side...’

’...would he ever have another moment of inspiration like this in his lifetime?’

He knew it better than anyone.

The one thing he lacked was never intelligence.

It was knowing "where to go next."

And Li Dong could not only tell him where to go, but just by being near him, he could think more clearly himself.

Professor Wang thought again of the paper that old man Langlands himself had personally commented on.

He thought of the Li Clan’s Conjecture.

’Compared to things like these...’

’...my own bit of pride is worth jack shit.’

After all, at the end of the sixteenth century, Tycho Brahe was already Europe’s foremost master of astronomical observation, over fifty years old and world-renowned.

But the moment he met Kepler in Prague—a man more than twenty years his junior, so poor he couldn’t even afford firewood for the winter...

...he decided to hand over all the precise observational data that he had spent his entire life painstakingly collecting.

In a letter to a friend, he wrote a line that would be quoted time and time again.

"I have observed my entire life."

"But only he can see the ’why’ behind the star charts."

That year, Tycho Brahe was fifty-four.

Kepler was twenty-nine.

The transition from Tycho Brahe to Kepler was separated by a single decision.

A decision that cast aside all pride.

Professor Wang sat in his chair in the seminar room.

He suddenly felt that the choice Tycho Brahe had made back then was one that he, an old man from the twenty-first century, could also make this very afternoon.

He looked up at Li Dong.

"I’m willing."