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Chapter 213: Chapter 205: He Can’t Solve This Problem

Zhou Qifeng looked at the expression in Li Dong’s eyes and recalled what Dean Gao had said.

"If he likes physics, he should enjoy this..."

’Looks like he really does love physics.’

Then he shot Li Dong another glare.

"What are you looking at? Just do it."

Li Dong lowered his head again and reread the problem.

This problem was difficult.

He couldn’t solve it right now.

The first part required using the Fujikawa method for path integral measure transformation. He had only encountered some of the ideas while lurking in group chats, but he had never systematically studied the rigorous mathematical treatment of the fermionic path integral measure.

The second part was even more brutal.

The Atiyah-Singer index theorem.

This thing spanned the three major fields of differential geometry, Algebraic Topology, and Functional Analysis. It was one of the most profound achievements in the twentieth-century world of mathematics.

Although he had already accumulated considerable knowledge in mathematics, he had never properly tackled the tools involved in the index theorem, such as fiber bundle theory, the Chern-Weil homomorphism, or the heat kernel expansion method.

The third part looked the "simplest," but in reality, it required a deep understanding of the renormalization structure of Quantum Field Theory to clearly explain why the chiral anomaly is one-loop exact.

’I can’t do it.’

’At least, not today...’

’However...’

He could see the path.

Just as he had seen the isomorphic mapping between macromechanics and thermodynamic phase transitions when working on his Qiu Prize project, he could now see the underlying structure of this problem.

The essence of the chiral anomaly was not a dynamical problem, but a topological one.

The non-trivial transformation of the path integral measure essentially reflected the global topological properties of the connection on the fiber bundle—the second Chern class.

And the Atiyah-Singer index theorem was the very bridge connecting the analysis to the Chern number.

He could see this path.

But many bricks were missing along the way.

He needed to find these bricks in the library.

Li Dong looked up and said to Zhou Qifeng,

"Professor Zhou, you don’t expect me to solve this right now, do you?"

"How about you give me two days?"

Zhou Qifeng’s expression returned to calm.

He had noticed a light in the kid’s eyes the moment he saw the problem.

’Gao Yuanlin’s judgment... was most likely correct.’

"That’s fine. A week is fine too."

"When the time comes, just organize your line of thought and email it to me. It doesn’t have to be perfect. Write down as much as you have."

Li Dong agreed, then picked up Yang Shengguo’s temporary campus card and related materials, saying,

"Professor Zhou, Director Liu, I’ll be leaving then."

"Go on."

Li Dong pushed the door open and left.

The moment the door swung shut,

Zhou Qifeng let out a long sigh of relief.

’The act is finally over.’

Director Liu watched this scene from the side, a little curious.

"Professor Zhou, that problem... is it very difficult?"

Zhou Qifeng didn’t answer directly.

He pulled another A4 sheet of paper from a different compartment in his drawer.

This paper was covered in derivations—the result of Zhou Qifeng himself spending a whole afternoon trying to work through it after receiving the problem.

Of course, he hadn’t finished it either. It wasn’t a problem that could be solved in a few days. If he wanted to solve it completely, it would probably take him a week or two as well.

He handed the paper to Director Liu.

"Take a look."

Director Liu took it.

His expression froze just from looking at the first part...

He was a postdoc, after all. Even if his field wasn’t a perfect match, he still had a fundamental grasp of physics and mathematics.

But he struggled to even understand the first step of what was written on the paper.

From the second step onward, he was completely lost.

"Professor Zhou... what level is this problem exactly?"

Zhou Qifeng closed his eyes and rubbed his temples.

"How should I put it."

He thought for a moment...

"To solve this problem completely, you first have to master the entire theory of fiber bundles and connections in differential geometry."

"Then, you need to understand the construction of the Chern-Weil homomorphism and characteristic classes in Algebraic Topology, and be proficient in the spectral theory of elliptic operators and the heat kernel expansion method in Functional Analysis."

"Finally, you must have a complete mastery of path integral quantization and renormalization theory in Quantum Field Theory."

"These four fields—each one is an independent branch of mathematics or physics. A graduate student could spend a year or two just trying to get through any single one of them."

"And what this problem requires is to master and connect them all."

Director Liu swallowed hard.

"So... do you think Li Dong can solve it?"

Zhou Qifeng laughed.

"He can’t solve this problem. I just want to see his line of thought."

’Solve it? What a joke. Even if you, Li Dong, are some kind of born prodigy, it still takes time to accumulate knowledge, doesn’t it?’

...

As for the Li Dong who Professor Zhou claimed couldn’t solve the problem, he first made a call to Yang Shengguo.

"Teacher Yang, where are you?"

Yang Shengguo’s voice came from the other end of the line, sounding quite excited.

"I’m outside the west gate of the university. I just got here by subway. What’s up?"

"Come over and get something."

Ten minutes later, the two met at the west gate of Yan University.

Li Dong handed an envelope to Yang Shengguo.

"Your temporary campus card and VPN account. From now on, you won’t need me to let you in and out of the university. You can also come and go from the library freely and download any electronic documents you want."

Yang Shengguo snatched the envelope. He had been waiting for this day for a long time.

"For real?"

"For real. The paperwork was just approved."

Yang Shengguo had once been a man full of passion for research.

But then he ended up standing at a lectern for over a decade.

If Li Dong hadn’t dragged him onto the stage of Sunshine Hall, he probably would have never touched an academic paper again in his life.

And now...

The resources of Yan University lay quietly in the palm of his hand.

Yang Shengguo didn’t say much.

He patted Li Dong’s shoulder, tucked the envelope away, and started walking onto campus.

His destination—the library.

Watching Old Yang’s impatient back, Li Dong smiled and shook his head.

He didn’t follow him.

Instead, he decided to go to the cafeteria to grab a bite first, because...

He had a hunch that he wouldn’t be able to have a proper meal today.

...

「Yan University Cafeteria」

Li Dong casually got a bowl of braised pork over rice, carried it to a corner, and began organizing the problem’s structure in his mind as he ate.

’He didn’t want just a simple line of thought—wasn’t that something you could get with a single glance?’

’So, he wanted it all!’

In the center of his Memory Palace’s grand hall, the A4 paper lay waiting.

Li Dong circled the problem.

’What is the core difficulty of the first part of this problem?’

’The transformation of the path integral measure.’

’This requires expanding the fermion field in terms of the eigenfunction system of the Dirac operator, and then calculating the Jacobi determinant of the expansion coefficients under the chiral transformation.’

... (Omitted)

’So, Li Dong needs to study the heat kernel expansion method.’

’Right now, he only has a relative concept of it, but hasn’t studied it systematically.’

’Then there’s the second problem.’

’The Atiyah-Singer index theorem.’

’The proof of this theorem itself is super complicated. Of course, the problem doesn’t require Li Dong to prove it, only to connect the result of the first part with the index theorem.’

’But to make that connection, you first have to understand what the index theorem actually says.’

’And to understand the index theorem, you need a theoretical foundation in fiber bundles, connections, and characteristic classes (especially Chern classes).’

’He has a certain foundation in these mathematical concepts, but it’s far from enough.’

’So he needs to quickly catch up on fiber bundle and Chern-Weil theory.’

’As for the last problem...’

’The Adler-Bardeen theorem.’

’This is Li Dong’s weakest area.’

’Because most of the Quantum Field Theory knowledge he had encountered before came from Zhou Qifeng...’

Zhou Qifeng: ???

’So now he urgently needs to tackle Peskin & Schroeder.’

Inside his Memory Palace, Li Dong organized these three steps into a clear learning path.

’Then it’s time to get to work!’

He wolfed down the rest of his rice in a few bites and headed for the cafeteria entrance.

He completely failed to notice Qi Yu behind him, who was attempting to eat spicy rabbit.

"Hoo, hoo... So spicy!"

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