Chapter 1127

Probably at the beginning of the year, before Lu Zhou had dug Chen Yang out of Yanda mathematics center, Professor Chen was studying Hodge's conjecture.

Lu Zhou still remembers that he studied his own hyperelliptic curve analysis method on the blackboard at that time, and used a very ingenious method to improve the mathematical tool originally designed for quasi Riemann conjecture, which was directly applied to the study of the algebraic topology of nonsingular complex algebraic clusters and the geometric correlation problem expressed by the polynomial equation defining the sub clusters.

At the beginning, it was because of this beautiful operation that Lu Zhou couldn't help but move his heart of loving talents and dug him from Yanda mathematics center to Jinling.

It's almost a year since now, and there's still no progress on Hodge's conjecture. In addition, we have been busy with the unified theory of algebraic geometry for a long time, so that the matter is almost forgotten by Lu Zhou.

"Go, go to my office and say."

Taking Chen Yang to his office, Lu Zhou went to the corner of the wall to help him drag a white board, and handed his marker to him.

Instead of wasting time on politeness, after taking over the pen, Chen Yang stood in front of the whiteboard and thought for a moment. First, he drew a circle on the whiteboard, then marked s beside it, and wrote a line of expression.

“…… For the compact surface s, the Gauss curvature K can be integrated on the whole surface. "

As he wrote, Chen Yang continued.

"As we all know, a surface doesn't have to have only one metric, so I tried to change the metric of S. After the measurement is changed, the corresponding Gauss curvature K will also change, but the integral value has nothing to do with the measurement of the surface, but only with the Euler indicator number x (s) of the surface. Using this property, we can -- "

look at the formula on the whiteboard, lift the eyebrows of the land boat slightly, and say with interest.

"Gauss bonnet formula?"

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

"Exactly."

After that, he wrote down the Gauss bonnet formula.

Seeing the finishing touch, the expression of interest on the face of the boat became more and more intense.

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

According to the properties of 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 Riemannian curvature. As for the integrand function, it is a very complex algebraic formula composed of curvature tensor, namely Gauss bonnet integrand function.

As for its integral on the whole manifold, it is determined by the Euler number x (m) of the manifold.

By using these properties, Hodge's theory can be extended to complete noncompact manifolds.

These profound mathematical meanings are obtained by Professor Chen Shengshen, that is, the mathematical connotation of the famous Gauss bonnet Chen formula.

In combination with Sir Atia's L2 cohomology method, we can continue along this line of thinking. I can't really prove this conjecture.

Of course, how to prove it needs further study.

Thinking of this, Lu Zhou nodded approvingly.

Second.

It's wonderful.

I don't know when, Chen Yang has stood behind a circle of people.

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

Looking at the formula on the whiteboard, Jimo's eyes glowed, and he whispered excitedly, "is this the legendary -"

seeing his younger martial brother talking, he Changwen frowned and murmured, "what is it, don't be critical."

Zimmer gave him a strange look.

"Guess Hodge! It's obvious. "

He Changwen: "..."

What's so obvious?!

But on closer inspection, it seems that it is.

Thinking of this, he Changwen cannot help comforting himself in his heart.

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

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

Obviously, he is only half of the way. There is no good idea about how to go.

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

"The idea seems interesting."

Looking back at Professor Perelman, Chen Yang was slightly shocked and said unexpectedly.

"When did you come?"

"About halfway through your writing Originally, I came to see Professor Lu, but I didn't expect to have any surprises here, "Perelman continued after a pause Can you write for me? "

Without any hesitation, Chen Yang resolutely let out the marker in his hand.After taking over the pen from Professor Chen, Perelman stood in front of the whiteboard and pondered for a moment. Then he left a few lines under his formula and continued to write.

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

“…… My suggestion is that for the later part of the proof, we can upgrade the compact manifold m problem to its general covering manifold and get the complete non compact manifold M. "

"According to attia's theorem, if we can prove that all but the middle L2 homology group are zero under the condition of section curvature..."

As he said this, he shook his pen slightly and wrote down a simple and beautiful formula.

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

at the moment of seeing this formula, Chen Yang's pupil slightly contracted.

The expression on his face showed a glimmer of enlightenment, he said, suppressing his excitement.

“…… We'll get proof of Hodge's conjecture! "

So here comes the question.

How can we prove that under the condition of section curvature, all of them are zero except the middle L2 homology group?

The conversation came to an abrupt end here.

After a brief excitement, the two fell into silence.

At last, I saw the land boat.

Notice two people look at oneself, from beginning to end a word didn't say land boat suddenly blinked, said with a smile.

"I think your ideas are all good Although I haven't studied the subject carefully, my intuition tells me that if I follow this path, I will surely get something. "

After a pause, he continued.

"This idea is very interesting. My suggestion is that you should study this topic together."

I always felt that the boat seemed to see something, but didn't make it clear.

Perelman frowned, hesitated.

"Are you not involved? This is a very interesting problem. "

It's more than fun.

Hodge's conjecture can be said to be the concentrated embodiment of abstract features in the development of modern mathematics. It studies the internal relations among the three branches of Mathematics: analysis, topology and algebraic geometry.

As for the difficulty, as a millennial problem, there is no doubt about it.

To Perelman's surprise, the boat didn't show much interest.

Lu Zhou: " Although I'm very interested, I have a lot of things waiting for me to deal with in imcrc. I'm afraid I haven't had more time recently to allocate to mathematics. "

On hearing the news, Perelman's face expressed regret.

"That's a pity."

"Although I'm afraid I can't spare time to help, I solemnly recommend Professor Chen to you," Lu Zhou said with a smile after patting Chen Yang on the shoulder. "He is an excellent scholar. I believe you know his ability too, so I won't boast much. In short, if you cooperate, I believe that we can solve this problem. "

Although he doubted this absolute statement, at a glance Professor Chen, Perelman didn't say anything, just nodded his head and seemed to recognize the partner.

Both of them are of the type that they don't talk much and don't communicate much.

After Lu zhouqing cleared his throat, he looked at Perelman and continued.

"In a word, is it OK for you 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. Don't care too much about her side. I really have something I want to do unfinished. I plan to be here Stay for a while, solve Hodge's conjecture and go back. "

Although it's surprising that Professor Perelman would choose to stay, this kind of good thing, of course, will not be rejected by the land boat, he said with a smile.

"Then you'd better live in the original apartment. I'll help you apply for an extension of the use time of the apartment."

Perelman nodded, thank you.

"Please."