Chapter 183 - 178: You’ll Be My Teaching Assistant
Tian Gang beckoned him over, then added, "For this class, you’ll be my teaching assistant."
As soon as he spoke, all the students below fixed their envious gazes on Li Dong.
’Academician Tian Gang’s teaching assistant?’
You have to understand, the position of Tian Gang’s teaching assistant was something that graduate students in the Yan University School of Mathematics would fight tooth and nail for, with no guarantee of getting it.
Tian Gang’s classes were different from the graduate courses of ordinary professors. The topics he taught were often directly linked to the international frontier, and every example in class was a bellwether for the current trends in the world of mathematics.
Being his teaching assistant meant you were qualified to stand by his side, recognized by him in front of all the faculty and students of the School of Mathematics.
Rumor had it that when Liu Ruochuan was a master’s student, he had been Tian Gang’s teaching assistant a few times. And now?
He was the Dean of the Yan University School of Mathematics and was about to become Huaxia’s first academician born in the post-80s generation.
Of course, this didn’t mean being a teaching assistant would automatically make you a dean or an academician, but the experience itself was an endorsement of your academic abilities by Tian Gang.
Bearing the gazes of over a hundred people in the lecture hall, Li Dong stood up from his seat.
Then he walked to Tian Gang’s side.
Tian Gang glanced at him, smiled, and then turned to the audience.
"Alright, let’s begin."
"In today’s class, I’d like to talk about the long-time existence problem of the Kähler-Ricci flow."
Tian Gang picked up a piece of chalk and wrote the first equation on the blackboard.
∂ω/∂t = -Ric(ω) + λω
"You should all be familiar with the fundamental equation of the Kähler-Ricci flow."
"But today, I don’t plan to start with the equation itself. I want to approach it from a different angle."
"Think about it. When we say a flow has ’long-time existence,’ what are we really concerned about?"
Tian Gang didn’t wait for the students to answer, continuing on his own.
"We aren’t concerned with whether the equation can be solved. Local existence is standard parabolic theory; Shi’s estimates can give you a short-time solution."
"What we’re really concerned about is whether the curvature will blow up during the flow’s evolution."
He turned and wrote another line on the blackboard.
"So the core question becomes: can you control the upper bound of the Riemann curvature tensor?"
"And to control the curvature, you have to control the degeneracy of the metric itself, which brings us back to a classic problem."
Tian Gang looked down at the audience.
"The C⁰ estimate of the Kähler potential."
At this point, he paused.
"Li Dong, help me write the list of references on the side board."
"Perelman’s paper on the entropy functional for the Ricci flow, the 2006 paper by Tian-Zhang, and Sesum’s 2005 curvature blow-up criterion."
Li Dong nodded, walked over to the side blackboard, picked up a piece of chalk, and began to write.
He had gone through all these papers while self-studying differential geometry. Although he hadn’t mastered them as thoroughly as he had his own field of number theory, he was clear on the basic framework.
As he wrote, he listened to Tian Gang continue his lecture.
"What did Perelman’s W-entropy functional, introduced in 2002, essentially accomplish?"
"He tied the evolution equation of the Ricci flow to a monotonicity formula."
"This means that as your flow progresses, there’s a quantity that is non-decreasing."
"This gives you an anchor point for a priori estimates."
At this point, Tian Gang’s speaking pace clearly quickened.
He began deriving the variational formula for Perelman’s μ-functional on the blackboard, writing line after line of equations at a rapid pace.
Li Dong stood to the side, initially just responsible for writing references and supplementary definitions on the side board.
But he soon noticed a problem.
Many of the graduate students in the audience could no longer keep up.
This was mainly because of Tian Gang’s distinctive lecturing style.
He rarely did any build-up, preferring to cut directly to the problem and intersperse intuitive explanations throughout his derivations.
For those who already had a solid background in the field, this style was an absolute treat—completely straightforward.
But for students whose foundations weren’t solid enough...
It was a disaster.
A few graduate students in the front row were still taking notes, but their expressions betrayed them. Their brows were furrowed, and it was clear they were struggling.
The graduate students in the middle rows had given up on taking notes entirely and had started taking pictures with their phones.
A familiar feeling washed over Li Dong.
It was the exact same feeling he got in the Cyan Dragon Study Group chat when listening to the experts discuss academic problems.
For those who could understand, the benefits were immense; every sentence was like opening a new window.
For those who couldn’t...
Heh.
That window was welded shut...
And in Academician Tian Gang’s class, many of these chosen ones from Yan University were no different from the bewildered Li Dong back in the group chat.
Uh, no, there was one difference. Li Dong could fleece the system, and they couldn’t.
When Tian Gang got to the proof of the μ-functional’s monotonicity, he needed to use a key lemma regarding the logarithmic Sobolev inequality.
He had just finished writing the statement of the lemma and hadn’t yet had a chance to say, "For the proof of this lemma, please refer to..."
Li Dong had already walked to the side board and written down the three core steps of the lemma’s proof.
Start with the asymptotic expansion of the heat kernel, proceed through Nash-Moser iteration, and finish the estimate using a cut-off function selection technique.
Tian Gang glanced back at the side board, slightly stunned.
Then he smiled.
"Yes, that’s the approach."
A few quick-witted graduate students in the audience also noticed this detail.
Li Dong had written it down noticeably before Tian Gang was about to get to that step.
A doctoral student whispered to the person next to him, "It seems like... he’s keeping up with Academician Tian’s train of thought."
"It’s more than just keeping up."
"He’s one step ahead."
The two of them exchanged a look, a single thought appearing in their minds.
’As expected of a monster who can publish as the first author in the *Mathematical Annals*.’
Tian Gang continued his lecture, but part of his attention remained on Li Dong.
For the first ten minutes, Li Dong had been a bit rusty; there were some concepts he needed to process in his mind before he could react.
But soon...
The value of two-fifths of a Newton began to show.
When Tian Gang spoke of connecting the Laplacian estimate of the Kähler potential with Moser iteration, Li Dong had already written the necessary interpolation inequality for the next step on the side board.
And when the lecture required Cheeger-Colding theory to explain the phenomenon of curvature concentration, Li Dong had already listed the key properties of Gromov-Hausdorff convergence beside it.
The more Tian Gang watched, the more he felt...
’Liu Ruochuan can’t teach him.’
’This young man’s learning ability and intuition have reached an unbelievable level.’
’He doesn’t need a supervisor. He needs someone who can point the way for him on multiple frontiers simultaneously.’
’And at Yan University...’
’The only one who can do that is probably me.’
...
「Time passed quickly.」
The nearly hour-long lecture was drawing to a close.
"Alright, that’s all I have for today on the long-time existence and curvature estimates of the Kähler-Ricci flow."
He looked down at the audience.
"If there’s anything you didn’t understand, you can ask me now."
The classroom fell into an awkward silence.
’Anything we didn’t understand?’
’You’d probably have to start over from the beginning...’
So, no one spoke up.
Tian Gang wasn’t surprised that no one asked any questions.
He turned his head and looked at Li Dong, who was standing beside him.
"What about you? Is there anything you didn’t understand?"
Li Dong thought for a moment.
To be honest, he had indeed struggled for the first ten-odd minutes, especially when Tian Gang mentioned Perelman’s W-entropy functional. His prior knowledge of that area was mainly at a conceptual level, with many gaps in the details.
But relying on his 0.4 Basic Attributes, combined with the fact that Tian Gang was a genuinely skilled lecturer, his speed of comprehension had accelerated.
By the second half, he had completely kept up with Tian Gang’s pace.
"I understood everything."
No one thought Li Dong was bragging, because his performance had already said it all.
Tian Gang nodded, a flash of satisfaction in his eyes.
"Good."
Then, his tone shifted.
"Oh, right."
"Could you do me a favor?"