Chapter 1127: New ideas for Hodge conjecture

Probably at the beginning of the year, before Lu Zhou had dug Chen Yang from Yanda Mathematics Center, the professor Chen was studying Hodge's conjecture.
Lu Zhou still remembered that he was studying his own super elliptic curve analysis method on a blackboard, and used a very clever method to use this mathematical tool originally designed as a quasi-Riemann conjecture. After improvement, he applied it directly to Africa. Study on the algebraic topology of singular complex algebraic clusters and the geometric correlation problem expressed by the polynomial equations that define the subcluster.
It was precisely because of this beautiful operation that Lu Zhou couldn't help but love him and dug him from Yanda Math Center to Jinling.
It's been almost a year now, and the subject of Hodge's conjecture has not progressed in the slightest. In addition, the matter of the unified theory of algebraic geometry has been busy some time ago, so that Lu Zhou almost forgot about it.
"Go and say to my office."
Taking Chen Yang to his office, Lu Zhou personally went to the corner to drag him a white board, and handed his marker to his hand.
Instead of wasting time on guest sets, after receiving the pen, Chen Yang standing in front of the whiteboard thought for a moment. First, he drew a circle on the whiteboard, then marked S next to it, and wrote a line of expressions.
"... For a compact, edgeless surface S, the Gauss curvature K can be integrated over the entire surface."
As he wrote, 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 replacing the metric, the corresponding Gauss curvature K will also change, but the integral value will be the same as that of the surface. Metrics have nothing to do with, but only with Euler's explicit number X (S) on the surface. Using this property, we can— "
Looking at the calculations on the whiteboard, Lu Zhou raised his eyebrows gently and said with interest.
"Gauss-formula?"
The pen in his hand stopped, Chen Yang nodded and said.
"Exactly."
After all, he wrote the Gauss-formula.
Seeing the finishing touch, Lu Zhou's expression of interest grew stronger.
In fact, he probably already guessed what Chen Yang planned to do.
According to the properties of the high-dimensional Riemannian manifold M, the Gauss curvature can be generalized as 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 formula consisting of a curvature tensor—the Gauss-integrand.
As for its integral over the whole manifold, it is determined by the Euler's explicit number X (M) of this manifold.
With these properties, Hodge's theory can be generalized to complete non-compact manifolds.
These profound mathematical meanings were obtained by Professor Chen Shengshen, which is the mathematical connotation in the famous Gauss-Chen formula.
Combined with Sir Attia's L2 cohomology method, and continue to follow this line of thought, I can really prove this conjecture.
Of course, how to prove it, we need to study it in depth.
Thinking of this, Lu Zhou nodded approvingly.
Seconds.
It is wonderful.
I don't know when, there is a circle of people behind Chen Yang.
As early as when he started writing, people in the office noticed this.
Staring at the formula on the whiteboard, Ji Mo glowed with his eyes and whispered excitedly: "This, is it the legend-"
Seeing his teacher talk only halfway, He Changwen frowned, and whispered, "What the is it, don't sell it."
Ji Mo gave him a strange look.
"Hodge conjecture! Obviously."
He Changwen: "..."
What's so obvious about this? !!
However, if you look closely, it seems that this is indeed 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 too.
The pen on the white board stopped, and Chen Yang was lost in thought.
Obviously, he has only halfway to this idea, and there is no good idea of ​​how to go next.
Unknown when he came to the office, Professor Perelman, who was standing silent beside him, suddenly said.
"This idea looks interesting."
Looking back at Professor Perelman, Chen Yang stumbled slightly, and said with some surprise.
"When did you come here?"
"About halfway through what you wrote ... I was originally looking for Professor Lu. I didn't expect there would be any unexpected gains here," he paused, and Perelman continued. "Can you use me? ? "
Without any hesitation, Chen Yang resolutely gave up the marker in his hand.
Taking the pen from Professor Chen's hand, Perelman, standing in front of the whiteboard, pondered for a moment, and then left a few lines under his formula to continue writing.
"Since there is an existing unified theory of algebraic geometry that can be applied, I have omitted the proof of equation (3)."
"... My suggestion is that for the proof of the later part, we can raise the compact manifold M problem to its universal covering manifold and obtain a complete non-compact manifold M."
"According to Atiyah's theorem, if we can prove that the cross-section curvature is zero except for the middle L2 homology group ..."
As he said, the pen in his hand shook slightly, and quickly wrote down a line of simple and beautiful calculations.
[H ^ n (M) 6 ≠ {0}, and when q ≠ n, H ^ q (M) = {0}]
The moment he saw this formula, Chen Yang's pupils contracted slightly.
There was a moment of enlightenment on the expression on his face, he said in a depressing tone.
"... We will get proof of Hodge's conjecture!"
So here comes the problem.
How to prove that under the condition of section curvature, except for the middle L2 homology group, it is zero?
The conversation came to an abrupt end.
After a brief excitement, the two fell into silence.
Finally, they looked at Lu Zhou by accident.
Noting that the two looked at themselves, Lu Zhou who didn't say a word from beginning to end suddenly blinked and said with a smile.
"I think your ideas are good ... Although I haven't studied the subject carefully, I intuitively told me that following this path, 80% of them will 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 subject together."
I always felt that Lu Zhou could see something, but he didn't make it clear.
Perelman frowned, wondering.
"Are you not involved? This is an interesting problem."
More than interesting.
Hodge's conjecture can be said to be a concentrated embodiment of abstract features in the development of modern mathematics. The research focuses on the three major branches of mathematics: analysis, topology, and algebraic geometry.
As for the difficulty, as a millennial problem, UU reading www.uukanshu.com is undoubtedly unquestionable.
What surprised Perelman was that Lu Zhou didn't show much interest.
Lu Zhou: "... Although I am very interested, there are still a lot of things waiting for IMCRC to deal with. I'm afraid I haven't had more time recently and can be assigned to math."
Hearing the news, Perelman's face showed a regretful expression.
"That's too bad."
"Although I'm afraid I can't find time to help, I solemnly recommend Professor Chen to you," patted Chen Yang's shoulder and said with a smile, "He is an outstanding scholar. Regarding his ability, I believe You also understand, I will not brag. In short, if you cooperate, I believe that this problem can be solved. "
Although he was skeptical of this absolute statement, he glanced at Professor Chen and Perelman didn't say anything, just nodded his head, and it seemed that he recognized the partner.
Both were of the type that didn't talk much and didn't talk much.
Lu Zhou cleared his throat and watched Perelman continue.
"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. "My mother and I have already called. She asked me to do what I wanted to do. Don't worry too much about her. I really have nothing to do. Done, I'm going to be here ... I'll stay a little longer and settle Hodge's conjecture before going back. "
Although it was very surprising that Professor Perelman would choose to stay, Lu Zhou naturally would not refuse such a good thing, and immediately said with a smile.
"Then you stay in the original apartment, I will help you apply for an extension of the apartment."
Perelman nodded, thanking him.
"sorry to bother you."