Probably at the beginning of the year, when Lu Zhou had not yet poached Chen Yang from the Mathematics Center of Yan University, Professor Chen was studying the Hodge conjecture.

Lu Zhou still remembers that at that time, he studied his own method of superelliptic curve analysis on the blackboard, and used a very ingenious method to improve this mathematical tool originally designed for the quasi-Riemann conjecture and directly apply it to the study of the algebraic topology of nonsingular complex algebraic clusters and the geometric correlation problems expressed by the polynomial equations that define the subclusters.

It was precisely because of this beautiful operation that Lu Zhou couldn't help but love his talent and dug him from the Yanda Mathematics Center to Jinling.

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

"Let's go, go to my office."

took Chen Yang to his office, Lu Zhou personally 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, who was standing in front of the whiteboard, thought for a moment, first drew a circle on the whiteboard, then marked S next to it, and wrote down a line of expressions.

“…… For a compact and edgeless surface S, its Gauss curvature K can be integrated over the entire surface. ”

While writing, Chen Yang continued.

"It's a well-known fact that a surface doesn't necessarily have just one measure, so I tried to change the S measure. After changing the metric, the corresponding Gauss curvature K also changes, but the integral value is not related to the metric of the surface, but only to the Euler indicative number X(S) of the surface.

Looking at the equation on the whiteboard, Lu Zhou's eyebrows were raised slightly, and he said with interest.

"Gauss-Bonnet formula?"

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

"Exactly."

After that, he wrote the Gauss-Bonnet formula.

Seeing this finishing touch, the look of interest on Lu Zhou's face became more and more intense.

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

According to the properties of the high-dimensional Riemannian manifold M, the Gauss curvature can be generalized to the cross-sectional curvature, and its value can be determined by the tensor of the Riemann curvature. As for the integrand, it is a very complex algebraic expression composed of curvature tensors - the Gauss-Bonnet integrand.

As for its integration over the entire manifold, it is determined by the Euler indicative number X(M) of the manifold.

Using these properties, it is possible to generalize Hodge's theory to complete non-tight manifolds.

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

Combined with Sir Attia's method of cohomology on L2, and continuing along this line of thought, I wonder if we can really prove this conjecture.

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

Thinking of this, Lu Zhou nodded approvingly.

Seconds.

It's wonderful.

I don't know when, there was already a circle of people standing behind Chen Yang.

As early as when he first started to board books, the people in the office noticed this.

Staring at the equation on the whiteboard, Zimmer's eyes lit up, and he whispered excitedly: "This, is this the legendary-"

Seeing that his junior brother was only halfway talking, He Changwen frowned and said in a low voice: "What the hell is it, don't sell it." ”

Zimmer gave him a strange look.

"Hodge guessed! It's obvious. ”

He Changwen: "......"

What's so obvious about this?!

But if you look closely, it seems that this is indeed the case.

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

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

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

Obviously, he has only come halfway to this line of thought, and he doesn't have a good idea of how to go next.

I don't know when I came to the office, Professor Perelman, who had been standing silently next to him, suddenly spoke.

"That line of thought seems interesting."

Looking back at Professor Perelman, Chen Yang was slightly stunned and said with some surprise.

"When did you come here?"

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

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

Taking the pen from Professor Chen's hand, Perelman, who stood in front of the whiteboard, pondered for a moment, then left a few lines blank under his equation and continued to write.

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

“…… My suggestion is that, for the purposes of the later part of the proof, we can elevate the tight manifold M problem to its universal cover manifold and get the complete non-tight manifold M. ”

"According to Attia's theorem, if we can prove that the cross-sectional curvature is zero, except for the middle L2 cohomology group......"

As he spoke, the pen in his hand shook slightly, and he quickly wrote down a line of concise and beautiful 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 contracted slightly.

The expression on his face instantly showed a hint of realization, and he said in a suppressed tone of excitement.

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

So here's the problem.

How can it be proved that, under the condition of cross-section curvature, the rest are zero except for the middle L2 cohomology group?

At this point, the conversation came to an abrupt end.

After a brief moment of excitement, the two fell silent.

In the end, they all looked at Lu Zhou.

Noticing that the two of them were looking at him, Lu Zhou, who had not said a word from beginning to end, suddenly blinked and said with a smile.

"I think you all have a good idea...... Although I haven't studied this subject in detail, my intuition tells me that if I follow this path, I will be able to gain something. ”

After a pause, he continued.

"It's a very interesting idea, and my suggestion is that you might as well work on this topic together."

I always felt that Lu Zhou seemed to see something, but he didn't make it clear.

Perelman frowned and asked hesitantly.

"Aren't you involved? That's an interesting puzzle. ”

It's more than interesting.

The Hodge conjecture can be said to be a concentrated embodiment of abstract features in the development of modern mathematics, which studies the internal relationship between the three major branches of mathematics: analysis, topology, and algebraic geometry.

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

To Perelman's surprise, Lu Zhou did not show a strong interest.

Lu Zhou: "...... I'm interested, but there's still a lot of work to be done on the IMCRC side, and I'm afraid I don't have any more time to allocate to the math side lately. ”

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

"That's a shame."

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

Although he was skeptical of this absolute statement, after glancing at Professor Chen, Perelman didn't say anything, just nodded, and seemed to approve of the partner.

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

After Lu Zhou cleared his throat, he 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've already called my mother, and she told me to do what I wanted to do without paying too much attention to her." I do have something I want to do that I don't want to accomplish and I'm going to ...... here Stay a little longer, solve the Hodge conjecture and go back. ”

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

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

Perelman nodded and thanked him.

"I'm sorry for you."