Scholar’s Advanced Technological System

Probably at the beginning of the year, when Lu Zhou hadn’t dug Chen Yang from the Yanda mathematics center, the Chen Professor was studying Hodge’s conjecture.

Lu Zhou remembers that he studied his hyperelliptic curve analysis on the blackboard and used a very clever method to improve the mathematics tool originally designed for GFB Riemann’s Conjecture. The algebraic topology of singular complex algebraic clusters and the geometrical association problems expressed by the polynomial equations that define subclusters.

It was because of this beautiful operation that Lu Zhou couldn’t help but move the heart of love, and he dug him from the center of Yanda mathematics to the Nanjing side.

It’s been almost a year now, and there’s still no progress on the subject of Hodge’s conjecture. Plus, I’ve been busy with algebraic geometric unification theory some time ago, so that Lu Zhou is forgetting about it.

“Go, go to my office and say.”

With Chen Yang coming to his office, Lu Zhou personally went to the corner to help him drag a white board and handed his own marker to his hand.

Did not waste time on the guest set, after taking the pen, Chen Yang, who stood in front of the whiteboard, thought for a moment, first drawn a circle on the whiteboard, then marked S next to it, and wrote a line of expression.

“…For a tightly bounded surface S, its Gauss curvature K can be integral on the entire surface.”

While writing, Chen Yang continued.

“It is well known that a surface does not necessarily contain only one metric, so I tried to replace the metric of S. After changing the metric, the corresponding Gauss curvature K will also change, but the integral value is the same as the surface. The metric is irrelevant, but only related to the Euler’s indicative number X(S) of the surface. With this property, we can –“

Looking at the formula on the whiteboard, Lu Zhou’s eyebrows were gently lifted down and said with great interest.

“Gauss-Bonnet formula?”

The pen in his hand stopped, Chen Yang nodded said.

“Exactly.”

After that, he wrote the Gauss-Bonnet formula.

Seeing this finishing touch, the expression of interest on Lu Zhou’s face became more intense.

In fact, he probably already guessed what Chen Yang was going to do.

According to the properties of the high-dimensional Riemannian 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. As for its integrand, it is a complex algebraic form consisting of curvature tensors—the Gauss-Bonnet integrand function.

As for its integral on the entire manifold, it is determined by the Euler’s indicative number X(M) of this manifold.

Using these properties, Hodge theory can be extended to complete non-compact manifolds.

These profound mathematical meanings are obtained by the Chen Shengshen Professor, which is the mathematical meaning of the famous Gauss–Bonnet–Chen formula.

Combined with the L2 coherent method of Sir Attiya, I will continue to follow this line of thought. If I can’t do it, I can really prove this conjecture.

Of course, how to prove it, you need to study it in depth.

Thinking of this, Lu Zhou praised nodded.

Seconds.

It is wonderful.

I do not know when, Chen Yang has stood behind a circle.

As early as when he had just started the book, people in the office noticed this.

Staring at the formula on the whiteboard, Ji Mo’s eyes glowed, and the excited whisper said: “This is the legendary –”

Seeing that the Junior Brother spoke only halfway, He Changwen frowned and whispered: “In the end, it is awkward, don’t sell off.”

Ji Mo looked at him strangely.

“Hodge conjecture! Obviously.”

He Changwen: “…”

Where is this special? !

But look carefully, it seems to be the case.

Thinking of this, He Changwen could not help but comfort himself in his heart.

Well, if you look at it seriously, he should be able to see it.

The pen on the white board stopped, and Chen Yang fell into meditation.

Obviously, he only walked halfway through this idea, and there is no good idea about how to go.

I don’t know when to come to the office, standing next to the Perelman Professor, who has been silent, suddenly said.

“This idea seems a bit interesting.”

Looking back towards Perelman Professor, Chen Yang slightly stunned, and some accidentally said.

“When did you come over?”

“Probably when you wrote half of it… I was originally looking for a Ground Professor. I didn’t expect to have a windfall here,” paused. Perelman continued, “… can you use me? ?”

Without any hesitation, Chen Yang decisively gave up the marker in his hand.

Picking up the pen from Chen’s Professor, Perelman, standing in front of the whiteboard, pondered for a moment, then vacated a few lines under his formula and continued to write.

“Since there is a ready-made theory of algebraic geometry unification, I can omit the proof of the formula (3).”

“… My suggestion is that for the proof of the latter part, we can raise the tight manifold M problem to its universal cover manifold to get a complete non-compact manifold M.”

“According to Att’s theorem, if we can prove that the L2 cohomology group in the middle is zero except for the curvature of the section…”

Said, the pen in his hand shook a little, and quickly wrote a simple and beautiful calculation.

[H^n(M)6≠{0}, and when q≠n, H^q(M)={0}]

At the moment of seeing this line of calculations, Chen Yang’s pupils contracted slightly.

The expression on his face instantly showed a clear comprehension, and he suppressed the excitement.

“…we can get proof of Hodge’s conjecture!”

Then the problem is coming.

How to prove that under the condition of section curvature, except for the middle L2 homology group, the rest are zero?

The conversation suddenly stopped here.

After a brief excitement, the two men fell silent in silence.

Finally, they also looked towards Lu Zhou.

I noticed that the two people looked toward themselves, Lu Zhou suddenly said nothing with a word from the beginning to the end, said with a smile.

“I think your thoughts are good… Although I haven’t studied this subject carefully, I intuitively told me to follow this path and 80% can be rewarded.”

Paused, he continued.

“This idea is very interesting. My suggestion is that you should not study this topic together.”

I always felt that Lu Zhou seemed to see something, but did not explain it.

Perelman frowned, questioning lately.

“You don’t participate? This is a very interesting problem.”

It’s more interesting.

The Hodge conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the three branches of mathematics—the intrinsic connection between analysis, topology, and algebraic geometry.

As for the difficulty, as a thousand problems, it is naturally beyond doubt.

What makes Perelman surprised is that Lu Zhou did not show a strong interest.

Lu Zhou: “…Although I am very interested, there are still a lot of things waiting for me to deal with there. I am afraid I have no more time to assign to mathematics recently.”

Upon hearing this news, Perelman’s face showed a regretful expression.

“That’s really a shame.”

“Although I am afraid I can’t find time to help, I solemnly recommend Chen Professor,” patted Chen Yang’s shoulder, Lu Zhou said with a smile. “He is an excellent Scholar, about his abilities, I believe that you also understand, I will not brag. In short, if you cooperate, I believe that this problem will be solved.”

Although I doubted this absolute statement, but took a look at Chen Professor, Perelman did not say anything, just nodded, it seems to recognize the partner.

Both of them are of a type that has few words and there is not much communication.

After Lu Zhou cleared his throat, he watched Perelman continue.

“Speaking, are you okay to stay here? The unified theory of algebra and geometry has been completed.”

“No problem,” Perelman shook his head. “I have already called my mother. She asked me to do what I wanted to do. I don’t care too much about her. I really have nothing to do.” Finished, I am going to be here… Wait a little longer, and let Hodge guess it and go back.”

Although it is very surprising that Perelman Professor will choose to stay, this kind of good thing Lu Zhou naturally will not refuse, immediately said with a smile.

“Then you still live in the original apartment, I will help you apply for extending the use of the apartment.”

Perelman nodded, thanks.

“sorry to bother you.”