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Chapter 198: Chapter 194: That’s Enough, Teacher Yang

Yang Shengguo showed no sign of stage fright.

He first gave a slight bow to the audience, then picked up a marker and began to speak.

”The method I’m using is a layer-by-layer recursive filtered embedding.”

”The starting point of this method is to treat the filtration structure of the Hodge-Tate decomposition as a set of natural stratified coordinate systems. Then, we embed the parameterized path of the p-adic integral into the filtered subspace of each layer.”

”When e_v = 2, we only need to handle the first-order filtration. The universal obstruction happens to fall on the zero-layer and is canceled out by the first-order weights.”

”And now, what we need to do is generalize this logic to e_v = 3.”

He glanced down at the grim-faced Zhou Shenzhi and the expressionless Jiang Yubai in the audience.

It was as if he was telling them...

Watch closely!

”First, let’s start with the simplest case.”

”The local-global compatibility from GL(1) to GL(2) hinges on the GUE convergence of the automorphic L-function’s zero-pair correlation function, F_π(α), within the interval |α|∈[0, 1]. This convergence is completely consistent with the zero-distribution statistics of the Riemann ζ-function for GL(1).”

”This is a step I believe everyone is familiar with.”

”But I want to extract something more fundamental from this step.”

Yang Shengguo turned and wrote the first formula on the whiteboard.

F_1: Fil^0(ρ_π, v) → Fil^1(ρ_π, v)

The moment he wrote this formula, most of the audience fell silent.

At first, they couldn’t quite understand it.

Because Yang Shengguo’s system of notation was different from that in conventional Algebraic Number Theory literature.

He wasn’t using the standard notation from Fontaine’s theory; instead, he used a more intuitive method of expression.

This set of notations was like a dialect only he could speak. The grammar was correct, but you needed time to get used to it.

The professors in the audience exchanged glances.

Someone whispered to the person next to them.

”Did he create this filtered embedding notation himself?”

”It seems so, but look at the direction of the arrow...”

”Wait, did he directly encode the Hodge-Tate weights into the kernel of the embedding map?”

”If that’s the case...”

The discussion in the audience gradually died down.

But they all, intentionally or not, glanced in the direction of Jiang Yubai and Zhou Shenzhi.

Yang Shengguo ignored the audience’s reaction and continued writing.

Below that embedding map, he wrote out a recursive relation.

What this recursive relation meant was that after you complete the filtered embedding for the k-th layer, the embedding condition for the (k+1)-th layer doesn’t need to be constructed from scratch. It can be directly inherited from the results of the k-th layer via a natural ”lifting operator.”

And the construction of this lifting operator precisely utilized the compatibility conditions of the Hodge-Tate weights between different filtered layers.

No orthogonality was needed.

Only layer-by-layer compatibility.

This was exactly what Li Dong had said earlier.

...

In the audience.

Zhou Shenzhi stared intently at the whiteboard.

He could still follow the first few lines of formulas.

After all, he had indeed worked on the computational part of that p-adic integral path deformation scheme back in the day. Though the underlying logic wasn’t his, he was familiar with the surface-level technical details.

But when Yang Shengguo wrote out that recursive relation...

A buzz went through Zhou Shenzhi’s mind.

The problem he had been stuck on for over three years.

Yang Shengguo had solved it with a single recursive relation.

The color drained from Zhou Shenzhi’s face, inch by inch.

He had to admit one thing: he was truly no match for his junior...

...

But Jiang Yubai, sitting next to Zhou Shenzhi, found his eyes growing brighter and brighter.

When Yang Shengguo wrote out the explicit construction of the lifting operator, Jiang Yubai’s breathing quickened noticeably.

Not only did he understand it, but he felt as though his own line of thinking had been completely unblocked.

The construction of that lifting operator gave him a perspective he had never considered.

If he were to combine this layer-by-layer recursive framework with the Galois cohomology techniques he had accumulated over many years...

He didn’t even need to see what Yang Shengguo wrote next.

He could already see the entire path forward.

He just desperately wished Yang Shengguo would stop now...

...

Onstage.

Yang Shengguo wrote faster and faster.

He found it strange himself.

He hadn’t felt this way in over a decade.

Back when he was a master’s student at Capital University, this was the most active his mind had ever been.

No.

It was even better than back then.

Every thought in his mind was exceptionally clear, every step of the derivation impeccable.

Things that had been buried deep in his Memory for over a decade were now pouring out ceaselessly.

And it wasn’t just recollection.

It was a new understanding.

As for the reason...

Naturally, the passive aura of the Inheritance of the Torch was at work.

Yang Shengguo’s Logic attribute was at least 0.1 or higher.

And Inheritance of the Torch granted a floating bonus of 10% to 20% to companions whose Basic Attributes reached 0.1, with the specific percentage depending on their research synergy.

Old Yang had taught Li Dong for nearly three years.

Synergy?

It was maxed out, of course.

...

But Yang Shengguo had been away from the front lines of academia for too long, after all.

When he reached the second layer of the recursive embedding, his marker suddenly paused.

He looked at a line of formulas on the whiteboard, his brow slightly furrowed.

The second-layer lifting operator required a functorial property of the p-adic period map in crystalline cohomology.

He remembered this property, but the details were hazy.

He stopped.

Jiang Yubai breathed a small sigh of relief.

’He’s stuck...’

But just then, Li Dong, who was standing beside him, spoke up unhurriedly.

”Teacher Yang, for the functorial property of the second-layer lifting operator, you can directly use the equivariant version of Faltings’s p-adic comparison theorem.”

He walked to the whiteboard and wrote a line next to where Yang Shengguo had stopped.

T_crys → T_dR ⊗ B_crys^(φ=1)

”This isomorphism is natural in the equivariant category, so the functoriality of the lifting operator is automatically satisfied at every layer. No extra verification is needed.”

Yang Shengguo glanced at it and immediately understood.

”Right... use the equivariant version from Faltings...”

He picked up his marker again and continued writing.

Li Dong stood beside him, smiling as he watched his teacher ”schooling” the professors in the audience.

Time passed just like that.

Ten minutes.

Half an hour.

An hour.

Two hours.

...

Whiteboard after whiteboard was filled. Staff members continuously brought in new ones from a side door, numbering the full ones and setting them aside.

No one noticed the time.

In the middle of it all, Yang Shengguo got stuck twice more.

The first time was while handling the cross-terms of the second-order Hodge-Tate weights, when he forgot a key corollary of Berger’s equivalence theorem for (φ, Γ)-modules and filtered modules.

Li Dong prompted him, ”Teacher Yang, Berger’s equivalence theorem holds for crystalline representations. The cross-terms can be resolved directly in the category of (φ, Γ)-modules without needing to return to filtered modules.”

The second time was during the final convergence estimation stage, where Yang Shengguo’s control over the order of the remainder term was slightly off.

Li Dong walked up, added a line of correction under the inequality he had written, and tightened the remainder term from O(p⁻²) down to O(p⁻⁵/²).

Each time, Yang Shengguo understood immediately, picking up the thread without the slightest hesitation.

...

Just then, a staff member quietly approached Tian Gang.

”Academician Tian, the time...”

Before he could finish, Tian Gang shot him a look.

”Don’t worry about the time. If anyone has a problem with it, tell them to come see me.”

The staff member nodded and turned to leave.

Some people in the hall did get up and start to leave one by one.

But before leaving, without exception, they would grab a colleague and say:

”Save my seat. I’m just going to grab a bite to eat and I’ll be right back.”

No one was willing to truly leave.

...

Another half hour passed.

In the audience, Zhou Shenzhi’s face was now deathly pale.

His notebook lay open, but he had only taken notes on less than the first third of the presentation.

He could no longer understand the parts that followed.

Zhou Shenzhi’s hands were trembling slightly.

He was suddenly struck by a powerful feeling.

’What have I even been doing for the past ten-plus years?’

...

As for Jiang Yubai, the light in his eyes had long been extinguished.

Because if these two continued, they would finish his research project for him.

This project.

He had worked on it for over a decade.

If successful, it was meant to be the core achievement he would use to contend for the Wolf Prize.

And now?

It was being completed, step by step, in front of the whole world, by a high school math teacher and a university freshman.

Just as Jiang Yubai was agonizing over this, Li Dong suddenly spoke.

”Teacher Yang.”

Yang Shengguo, who was writing with great momentum, turned his head when Li Dong called out to him.

”Hm?”

He looked at Li Dong, his face full of confusion.

Li Dong stood beside him, smiled, and shook his head.

”That’s enough, Teacher Yang.”