Around the beginning of the year, before Lu Zhou poached Chen Yang from the Yan University mathematics center, Professor Chen was already researching the Hodge conjecture.
Lu Zhou still remembered that Chen Yang was researching his hyperelliptic curve analysis method on the blackboard. He used a very clever method to apply this mathematical tool, which was originally designed for the Riemann conjecture, to the algebraic topology of non-singular complex algebraic clusters, as well as the polynomial equation definition of its subclusters.
Because of this, Lu Zhou couldn't help but want to poach Chen Yang from the Yan University mathematics center to Jinling.
It had been almost a year since then, and there was still no progress on the Hodge conjecture. Plus, Lu Zhou had been busy with the unified theory of algebraic geometry, so he almost forgot about this matter.
"Let's go to my office."
Lu Zhou brought Chen Yang to his office. He personally went to the corner and pulled out a whiteboard. He then handed Chen Yang a marker.
Chen Yang didn't waste time on pleasantries. He stood in front of the whiteboard and thought for a moment. First, he drew a circle on the whiteboard, then marked it with an S, and wrote down an expression.
"… For a compact boundless surface S, its Gauss curvature K can be integrated on the entire surface."
While writing, Chen Yang continued speaking.
"As we all know, a surface doesn't only have one metric, so I tried to change the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value has nothing to do with the metric of the surface. Instead, it is related to the Euler characteristic X (S) of the surface. Using this property, we can — "
Lu Zhou looked at the calculations on the whiteboard and raised his eyebrows as he spoke with interest.
"Gauss-Bonnet formula?"
Chen Yang stopped writing and nodded.
"Exactly."
Then, he wrote down the Gauss-Bonnet formula.
Seeing this finishing touch, the interest on Lu Zhou's face grew stronger.
In fact, he had already guessed what Chen Yang was going to do.
According to the properties of high-dimensional Riemann manifold M, the Gauss curvature can be generalized to the section curvature, and its value can be determined by the tensor of the Riemann curvature. The integrand was a complex algebraic expression composed of curvature tensors — the Gauss-Bonnet integrand.
The integral over the entire manifold was determined by the Euler characteristic X (M) of the manifold.
Using these properties, Hodge's theory could be extended to complete noncompact manifolds.
These profound mathematical meanings were obtained by Professor Shiing-Shen Chern, which was the mathematical meaning of the famous Gauss – Bonnet – Chen formula.
Combining it with Sir Atiyah's L2 cohomology method and continuing along this line of thought, they might actually be able to prove this conjecture.
Of course, as for how to prove it, he still needed to do some in-depth research.
Lu Zhou nodded in approval.
Amazing.
This is amazing.
At some point, a group of people had gathered behind Chen Yang.
The moment he started writing on the whiteboard, the people in the office noticed him.
Staring at the calculations on the whiteboard, Li Mo's eyes lit up, and he whispered excitedly, "Is this the legendary —"
Seeing that his Junior Brother was only halfway through his sentence, He Changwen furrowed his brows and whispered, "What is it? Don't keep us in suspense."
Ji Mo looked at him strangely.
"The Hodge conjecture! It's pretty obvious. "
He Changwen: "…"
How is it obvious?!
But looking closely, it does seem to be the case.
Thinking of this, He Changwen couldn't help but comfort himself.
Yeah, if I look closely, I should be able to see it too.
Chen Yang stopped writing on the whiteboard, and he began to think.
Obviously, he was only halfway through this line of thought, and he didn't have a good idea of what to do next.
Professor Perelman, who had been standing in the office for a long time, suddenly spoke.
"This line of thought seems interesting."
Chen Yang turned around and looked at Professor Perelman. He paused for a second and spoke.
"When did you get here?"
"Around halfway through … I came here to see Professor Lu, I didn't expect to find him here." Perelman paused for a second and said, "Can I use the pen?"
Without any hesitation, Chen Yang handed over the marker in his hand.
Perelman took the pen from Professor Chen and stood in front of the whiteboard. He pondered for a second and began to write a few lines below his equations.
"Since there's a unified theory of algebraic geometry, I won't need to prove formula (3).
"My suggestion is, for the next part of the proof, we can upgrade the compact manifold M to its general covering manifold, obtaining a complete non-compact manifold M.
"According to Atiyah's theorem, if we can prove that the L2 homology group in the middle is zero under the condition of cross-section curvature …"
The pen in his hand trembled, and he quickly wrote down a simple and beautiful line of equations.
[H ^ n (M) 6 ≠ {0}, and when q ≠ n, H ^ q (M) = {0}]
The second Chen Yang saw this line of equations, his pupils contracted.
He had a look of realization on his face, and he spoke excitedly.
"… We will be able to prove the Hodge conjecture!"
But here's the problem.
How do I prove that under the condition of cross-section curvature, the L2 homology group in the middle is zero?
The conversation came to an abrupt end.
After a brief moment of excitement, the two fell into silence at the same time.
Finally, they all looked at Lu Zhou at the same time.
Lu Zhou noticed that the two of them were looking at him. He didn't say a word, but he suddenly blinked and spoke with a smile.
"I think your ideas are pretty good … Even though I haven't studied this topic carefully, my intuition tells me that if we follow this path, we will most likely be able to find something."
He paused for a second and spoke.
"This idea is very interesting. My suggestion is, why don't you guys research this topic together?"
Chen Yang felt like Lu Zhou knew something, but he didn't say it clearly.
Perelman frowned and spoke hesitantly.
"You're not participating? This is a very interesting problem. "
It's more than interesting.
The Hodge conjecture could be said to be the embodiment of the abstract characteristics of modern mathematics. It studied the internal relationship between the three major branches of mathematics — analysis, topology, and algebraic geometry.
As for the difficulty, it was a Millennium Prize Problem, so it was obvious.
What surprised Perelman was that Lu Zhou didn't seem very interested.
Lu Zhou: "… Even though I'm interested, I still have a bunch of things to deal with at the ILHCRC. I'm afraid I don't have much time to do mathematics."
Perelman showed a regretful expression when he heard the news.
"That's a shame."
"Even though I don't have time to help, I recommend Professor Chen to you." Lu Zhou patted Chen Yang on the shoulder and said, "He's an excellent scholar. I'm sure you know what he's capable of, so I won't brag too much. Anyways, if you guys cooperate, I believe you can solve this problem. "
Even though Perelman was skeptical, he looked at Professor Chen and didn't say anything. He only nodded. It seemed like he recognized Chen Yang as his partner.
Both of them didn't talk much, so they didn't communicate much.
Lu Zhou cleared his throat and looked at Perelman.
"Speaking of which, is it okay for you to stay here? The unified theory of algebra and geometry has been completed. "
"No problem." Perelman shook his head and said, "I've already called my mother. She told me to do what I want to do and not worry about her. I still have things I want to do, so I plan on … staying here for a while until I solve the Hodge conjecture. "
Even though Lu Zhou was surprised that Professor Perelman chose to stay, he obviously wouldn't refuse such a good opportunity.
He smiled and said, "Then you should stay in your old apartment. I'll help you apply for an extension."
Perelman nodded and thanked him.
"Thank you."
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