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Chapter 172: Chapter 167: The Worth of 2/5 of a Newton (Part 2)

”Sure, but I might not understand it.”

”I’m only a freshman, after all.”

The graduate students in the audience were thinking: ’What the...?’

Guan Yi also felt that what he was doing was perhaps inappropriate.

But he had a gut feeling.

’This person... he can definitely help me grasp that flash of inspiration.’

So, he gritted his teeth and spoke anyway.

”Teacher Li Dong, my research topic is the long-time existence of Ricci flow on high-dimensional compact manifolds.”

...

He explained the idea that had just flashed through his mind.

”When I was listening to your lecture on controlling the remainder term of the correlation function using zeros, I had an idea...”

”Your technique of using a weighted average to suppress higher-order oscillation terms layer by layer... could it possibly be transplanted into the energy estimation framework for curvature flow?”

”If the time integral of the Weyl tensor could be truncated with a similar weighted window function, and then the contribution of the higher-order terms could be suppressed into subordinate terms through layer-by-layer iteration... is it possible that this path could work?”

As his voice fell, the hall fell silent for a few seconds.

Most of the graduate students had no idea what Guan Yi was talking about.

Lin Xue, sitting beside him, looked even more bewildered.

’A handsome guy with brains is truly handsome...’

Li Dong stood at the podium, having heard Guan Yi’s entire question.

He frowned.

To be honest, he was familiar with the field of Ricci flow, but he hadn’t studied it in depth.

He had read Perelman’s papers on using Ricci flow to prove the Poincaré Conjecture, and he had also seen Hamilton’s early work on the short-time existence of Ricci flow.

But the decay problem of the Weyl tensor in high-dimensional cases?

He had never encountered that before.

However, what did 0.4 in Basic Attributes represent?

It represented two-fifths of a Newton.

And what did Newton represent? Go read the side stories!

Two-fifths of a Newton might not be enough for him to grasp the entirety of an unfamiliar field at a glance, but it was enough for him to grasp the skeleton of a problem in a short amount of time.

The prerequisite was...

he had to first understand what the skeleton looked like.

So Li Dong looked at Guan Yi and asked a question that, in Guan Yi’s view, was somewhat unexpected.

”Can I ask you a few questions first?”

Guan Yi didn’t understand what he meant, but he nodded anyway.

”Yes.”

So, Li Dong began to ask.

”This coupling of the Weyl and Ricci tensors you mentioned, where exactly does it appear in the evolution equation? Is it in the quadratic expansion of Rm * Rm, or is it what’s left after the divergence term is canceled out?”

Guan Yi was taken aback.

It was a very basic question.

For him, it was something one should have understood within the first month of studying Ricci flow.

But he still answered honestly.

”It’s the Weyl-Weyl cross-term, after the quadratic expansion of Rm * Rm.”

”This term doesn’t exist in three dimensions. In four dimensions and above, it creates a nonlinear coupling with the evolution equation of the Ricci tensor, which the standard Shi’s estimate can’t suppress.”

Li Dong nodded and asked another question.

”When you were using integral norms before, what did you use for the energy functional? The L² norm, or a weighted Sobolev norm?”

”L²,” Guan Yi said. ”I’ve tried weighted ones too, but the choice of weight function is very sensitive. The slightest deviation makes the lower-order terms blow up.”

Li Dong didn’t ask any more questions.

He closed his mouth and began to think.

Hundreds of pairs of eyes in the audience were fixed on him.

Guan Yi was actually starting to lose hope.

From the two questions Li Dong had asked, it was clear that his knowledge in the field of Ricci flow was not deep.

This was perfectly normal.

After all, he was only a freshman. His main field was analytic number theory; it was impossible for him to be an expert in every branch of mathematics.

Guan Yi was about to give Li Dong a way to back out gracefully.

”It’s alright, Teacher Li Dong. This isn’t your field anyway. I was just randomly...”

”Wait a moment.”

Li Dong looked up.

His 0.4 Logic attribute was firing on all cylinders.

”You just said the choice of weight function is very sensitive, and the lower-order terms tend to blow up.”

”So, have you ever tried... instead of starting from the energy functional, starting from the asymptotic expansion of the heat kernel?”

Guan Yi froze.

”The heat kernel?”

”Yes,” Li Dong said. ”You want to control the long-time decay of the Weyl tensor. In essence, you need to control the dissipation rate of the curvature tensor under the Ricci flow.”

”Instead of trying to brute-force the Weyl-Weyl cross-term, you should try a different perspective.”

”You see, the Ricci flow itself is a generalization of the heat equation. Its fundamental solution is the heat kernel.”

”If you write the L² norm of the Weyl tensor in the form of a heat kernel convolution, its long-time decay behavior can be described by the asymptotic expansion of the heat kernel.”

...

He paused here for a moment.

”As for your idea about the weighted window function, the direction is correct. But you shouldn’t use it directly to truncate the time integral of the Weyl tensor.”

”What you should be using it for is to control the higher-order remainder terms of the heat kernel’s asymptotic expansion.”

”Because after expanding the heat kernel to a finite order, the decay of the remainder term is essentially the same problem as the Weyl tensor decay you encountered before.”

”However, after truncating via weighted averaging, the oscillation of the remainder term can be absorbed layer by layer. You’ll need to verify the technical details of this step yourself, but the broad framework should be sound.”

After Li Dong finished speaking, Guan Yi stood there, completely stunned.

It wasn’t because he understood every single word Li Dong had said.

It was because—

the part he did understand was precisely that fleeting inspiration he had lost.

The asymptotic expansion of the heat kernel.

Transforming the global energy estimate into a recurrence of local geometric invariants.

...

He saw it.

Although the details still required a great deal of verification and calculation, the direction... the direction was right.

Guan Yi opened his mouth but couldn’t speak.

His hands were trembling.

Sitting in the front row, Professors Cai Tianxin and Xu Hongwei exchanged a glance.

Their eyes were filled with disbelief.

They had both seen it.

When Li Dong was asking those basic questions just now, he clearly didn’t know much about the field of Ricci flow.

But after clarifying those most basic concepts, he had immediately provided a structural line of thought.

And this line of thought... it wasn’t just some casual remark.

Starting from the asymptotic expansion of the heat kernel, transforming the energy estimate into a recurrence of local geometric invariants...

If this framework really worked, the problem that had stumped Guan Yi for half a year might be solved within a few months.

Did such people really exist in the world?

The poet Cai Tianxin took a soft breath, and six words appeared in his mind:

Quick on the uptake, first to the truth.

Xu Hongwei’s expression was even more complicated.

He was even wondering if it was possible to recruit Li Dong to Zhejiang University.

However, just as he was wondering how to bring it up, Li Dong was already smiling and speaking to Guan Yi.

”The question you raised was very insightful.”

His tone was as casual as if they were just chatting.

”By the way, have you considered coming to Yan University for your graduate studies?”

”Our very own Professor Tian Gang at Yan University is in the field of differential geometry. His work on Ricci flow and Kähler geometry is world-class. If you were to join his group, your research on this topic would progress much faster.”

Xu Hongwei: ”...”

Cai Tianxin whispered to Xu Hongwei, ”Old Xu, your student is about to be poached.”

Xu Hongwei’s face was expressionless, but a vein on his temple throbbed twice.

’The nerve of this guy.’

’Does he think I’m not here?’

Guan Yi, however, didn’t notice the subtle tension between the two professors.

”I... I’m already confirmed for the direct Ph.D. program. My supervisor is Professor Xu Hongwei.”

He subconsciously glanced back at Xu Hongwei in the front row, then turned back to face Li Dong.

”But... thank you, Teacher Li Dong.”

This ”Teacher” was much more natural than the one he had said earlier.

Li Dong: ...

’Damn it, your supervisor is right here, and you ask me instead of him?? What the hell, I thought he wasn’t even present!’

Li Dong awkwardly pulled out his phone.

”So... want to add each other on WeChat?”

The vein on Xu Hongwei’s temple throbbed again.

...

「At the same time.」

In the gatehouse of Jiangcheng Seventh Middle School, the old security guard was opening a package.

It was a journal.

Printed on the cover was a line of English: Annals of Mathematics.

”Teacher Yang! The journal you ordered is here!”

the old guard shouted at the top of his lungs.

On the second floor of the academic building, a middle-aged man holding a thermos leaned halfway out of a window.

”Coming.”