Scholar’s Advanced Technological System

Probably at the beginning of the year, when Lu Zhou had not dug Chen Yang from the Yanda Math Center, Professor Chen was studying Hodge’s conjecture.

Lu Zhou still remembers that he studied his superelliptic curve analysis on the blackboard and used a very clever method to improve the mathematical tools originally designed for the Quasi-Riemann conjecture. The algebraic topology of singular complex algebraic clusters and the geometrical association problems expressed by the polynomial equations that define subclusters.

It was precisely because of this beautiful operation that Lu Zhou could not help but move the heart of love, and he dug him from the Yanda Math Center to Jinling.

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 came 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 expressions.

"... For a tightly bounded surface s, its gauss curvature k can be integrated over 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 s metric. After changing the metric, the corresponding gauss curvature k will also change, but the integral value is related to the surface. The metric is irrelevant, but only related to the surface's Euler number x(s). With this property, we can"

Looking at the calculations on the whiteboard, Lu Zhou's eyebrows gently lifted down and said with great interest.

"gauss-bon formula?"

The pen in his hand stopped, and Chen Yang said, he said.

"Exactly."

After that, he wrote the gauss-bon formula.

Seeing this finishing touch, the look of interest on Lu Zhou’s face became more and 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 very complex algebraic form composed of curvature tensors, namely the gauss-bon integrand.

As for its integral over the entire manifold, it is determined by the number of eules x(m) of this manifold.

Using these properties, the hodge theory can be extended to a complete non-compact manifold.

These profound mathematical meanings were obtained by Professor Chen Shengshen, the mathematical connotation of the famous gauss–bon–Chen formula.

Combined with Sir Attiya’s l2 coherence method, 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 applauded.

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

Seeing that his own younger brother spoke and only said half of it, He Changwen frowned his brow 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, Professor Perelman, who stood silent and stood silent, suddenly said.

"This idea seems a bit interesting."

Looking back at Professor Perelman, Chen Yang sneaked a little, and some accidentally said.

"When did you come over?"

"Probably when you wrote half of it... I originally came to Professor Lu. I didn't expect to have any unexpected gains here," paused. Perelman continued, "... can you use me? ?"

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

After receiving the pen from Professor Chen's hand, 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 equation (3)."

"... My suggestion is that for the proof of the latter part, we can raise the tight manifold m problem to its general cover manifold to get a complete non-compact manifold m."

"According to Att's theorem, if we can prove that the rest of the l2 cohomology group 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 hint of enlightenment, 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, I also looked at Lu Zhou in unison.

I noticed that the two men looked at themselves, and Lu Zhou suddenly said nothing from beginning to end, and said with a smile.

"I think your thoughts are good... Although I have not studied this subject carefully, but the intuition tells me to follow this path, 80% can be rewarded."

Suddenly, he continued.

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

I always feel that Lu Zhou seems to see something, but I have not explained 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 intrinsic connection between the three branches of mathematics, 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 imcrc. I am afraid that I have not had more time recently and can be assigned to mathematics."

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 Professor Chen," patted Chen Yang's shoulder, Lu Zhou said with a smile. "He is an excellent scholar, about his ability, I believe. You also know, I don't want to brag. In short, if you cooperate, I believe I can solve this problem."

Although he doubted this absolute statement, but looked at Professor Chen, Perelman did not say anything, just clicked, 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 the scorpion, 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. "My mother and I have already called. She asked me to do what I want 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 unexpected that Professor Perelman will choose to stay, this kind of good thing will not be rejected, and 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, thank you.

"sorry to bother you."