Chapter 1128: A New Idea for Hodge's Conjecture

Probably at the beginning of the year, before Lu Zhou poached Chen Yang from Yanda Mathematics Center, this Professor Chen was studying Hodge's conjecture.

Lu Zhou still remembered that he was studying his hyperelliptic curve analysis method on the blackboard, and used a very ingenious method to improve the mathematical tool originally designed for the quasi-Riemann conjecture and directly apply it to the non- On the algebraic topology of singular complex algebraic varieties, and the study of geometric correlation problems expressed by polynomial equations that define subvarieties.

At the beginning, it was precisely because of this beautiful operation that Lu Zhou couldn't help but be moved by his love for talents, and he was dug from Yanda Mathematics Center to Jinling.

It has been almost a year now, and there is still no progress on the topic of Hodge's conjecture. In addition, he has been busy with the unified theory of algebraic geometry some time ago, so that Lu Zhou almost forgot about it.

"Go, go to my office and talk."

Taking Chen Yang to his office, Lu Zhou went to the corner to help him drag a whiteboard, and handed his marker pen to his hand.

Without wasting time on politeness, after taking the pen, Chen Yang standing in front of the whiteboard thought for a moment, first drew a circle on the whiteboard, then marked S beside it, and wrote down a line of expression.

"...for a compact and boundless surface S, its Gaussian curvature K can be integrated over the entire surface."

While writing, Chen Yang continued.

"It is well known that a surface does not necessarily have 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 that of the surface Metric-independent, but only related to the Euler characteristic number X(S) of the surface, using this property, we can—”

Looking at the formula on the whiteboard, Lu Zhou raised his eyebrows slightly and said with interest.

"Gauss-Bonnet formula?"

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

"Exactly."

After all, he wrote down the Gauss-Bonnet formula.

Seeing this finishing touch, Lu Zhou's face became more and more interested.

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

According to the properties of the high-dimensional Riemannian manifold M, the Gauss curvature can be extended to the section curvature, and its value can be determined by the tensor of the Riemannian curvature. As for its integrand, it is a very complex algebraic formula composed of curvature tensors—that is, the Gauss-Bonnet integrand.

As for its integral on the whole manifold, it is determined by the Euler characteristic number X(M) of this manifold.

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

These profound mathematical meanings were obtained by Professor Chern Shiing-shen, which is the mathematical connotation in the famous Gauss–Bonnet–Chen formula.

Combined with Sir Attia's L2 cohomology method, continue to follow this line of thought, maybe you can really prove this conjecture.

Of course, how to prove it specifically still needs to be studied in depth.

Thinking of this, Lu Zhou nodded approvingly.

seconds.

It's wonderful.

At some point, there was already a circle of people standing behind Chen Yang.

As early as when he just started writing on the blackboard, the people in the office noticed this side.

Staring at the formula on the whiteboard, Jimo's eyes glowed, and he whispered excitedly: "This, is it the legendary—"

Seeing that his junior brother only spoke half of what he had said, He Changwen frowned, and said in a low voice, "What the hell is it? Don't be fooled."

Ji Mo gave him a strange look.

"Hodge's guess! It's obvious."

He Changwen: "..."

How is this so obvious? !

But on closer inspection, it does seem to be the case.

Thinking of this, He Changwen couldn't help comforting himself in his heart.

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

The pen on the whiteboard stopped, and Chen Yang fell into deep thought.

Obviously, he has only reached half of this line of thought, and he still has no good idea of ​​how to proceed.

In the office who came to the office at some unknown time, Professor Perelman, who had been standing silently by the side, suddenly spoke.

"This line of thought seems interesting."

Looking back at Professor Perelman, Chen Yang was slightly taken aback, and said in surprise.

"When did you come here?"

"About halfway through your writing... Originally, I came to see Professor Lu, but I didn't expect to find unexpected gains here." After a pause, Perelman continued, "...Can you use my pen for me? ?”

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

Taking the pen from Professor Chen, Perelman, who stood in front of the whiteboard, pondered for a moment, then blanked a few lines under his formula, and continued to write.

"Since there is a ready-made unified theory of algebraic geometry that can be used, I will omit the proof of formula (3)."

"...My suggestion is that for the proof in the following part, we can raise the compact manifold M problem to its general covering manifold, and obtain a complete non-compact manifold M."

"According to Attia's theorem, if we can prove under the condition of section curvature that all the others are zero except for the L2 homology group in the middle..."

As he spoke, the pen in his hand shook slightly, and he quickly wrote down a line of simple and elegant calculations.

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

The moment he saw this line of calculation, Chen Yang's pupils shrank slightly.

A hint of enlightenment appeared on the expression on his face in an instant, and he said in a suppressed excited tone.

"...We can get the proof of Hodge's conjecture!"

So here comes the problem.

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

The conversation came to an abrupt end here.

After a short period of excitement, the two fell silent at the same time.

Finally, they looked at Lu Zhou in unison.

Noticing that the two looked at him, Lu Zhou, who hadn't said a word from the beginning to the end, suddenly blinked and said with a smile.

"I think your ideas are good... Although I haven't studied this topic carefully, my intuition tells me that if I follow this path, I will probably be able to gain something."

After a pause, he continued.

"This idea is very interesting. My suggestion is that you might as well study this topic together."

I always felt that Lu Zhou seemed to see something, but he didn't understand what he said.

Perelman frowned, and asked Chi Chi.

"Aren't you going to participate? That's an interesting puzzle."

More than interesting.

Hodge's conjecture can be said to be a concentrated expression of the abstract features in the development of modern mathematics. It studies the internal connections among the three branches of mathematics-analysis, topology, and algebraic geometry.

As for the difficulty, as a millennium problem, it is naturally beyond doubt.

What surprised Perelman was 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 at IMCRC. I am afraid that I don't have more time recently, and I can allocate it to mathematics."

Hearing the news, Perelman had a regretful expression on his face.

"That's a real pity."

"Although I'm afraid I won't be able to spare time to help, I solemnly recommend Professor Chen to you," Lu Zhou patted Chen Yang on the shoulder and said with a smile, "He is an excellent scholar. Regarding his ability, I believe You also understand, so I won't brag. In short, if you cooperate, I believe that this problem will be solved."

Although he expressed doubts about this absolute statement, he glanced at Professor Chen, and Perelman didn't say anything. He just nodded, apparently approving the partner.

Both of them are the type who don't talk much, and don't communicate much.

After clearing his throat, Lu Zhou looked at Perelman and continued.

"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, "I have already called my mother, and she told me to do what I want, and don't pay too much attention to her side. I really have something else I want to do. Done, I plan to stay here... for a while, and go back after solving Hodge's conjecture."

Although it was surprising that Professor Perelman would choose to stay, Lu Zhou naturally would not refuse such a good thing, and said with a smile immediately.

"Then you should still live in the original apartment, and I will help you apply for an extension of the use of the apartment."

Perelman nodded and thanked.

"sorry to bother you."