Scholar’s Advanced Technological System

At about the beginning of the year, before Ark had dug Chen Yang out of Yan University Mathematics Center, the professor was studying Hodge's assumptions.

The Ark remembers that he studied his own hyperelliptical curve analysis on the blackboard at the time, and used a very ingenious method to use this mathematical tool, originally designed for quasi-Liman speculation, to improve it and apply it directly to the study of the algebraic topology of non-bizarre multiplexed algebraic clusters and the geometric correlation issues expressed in the polynomial equations defining their subclusters.

It was precisely because of this beautiful operation that the Ark could not help but love the heart and dug him from Yan Da Mathematical Center to Jinling.

Almost a year has passed now, and the subject of Hodge's speculation has not progressed at all, plus the fact that he has been busy with the theory of algebraic geometry unification for some time now, so that the Ark has almost forgotten about it.

“Go to my office and say it. ”

Chen Yang came to his office, and Ark went to the corner of the wall to help him drag a white board, and handed his marker pen to him.

After taking the pen, Chen Yang, who stood in front of the whiteboard, thought for a moment, first drew a circle randomly on the whiteboard, then marked the S next to it, and wrote down a line of expressions.

“… for a firmly infinite curved surface S, its Gauss curvature K can be integrated over the entire curved surface. ”

As he wrote, Chen Yang continued.

“It is well known that a curved surface does not necessarily contain only one measure, so I tried to replace the measure of S. After changing the measure, the corresponding Gauss curvature K will also change, but the integral value is not related to the measure of the curve, but only to the Euler exemplary number X (S) of the curve, which we can -”

Looking at the formula on the white board, the ark eyebrows lifted gently and said with interest.

“Gauss-Bonnet formula? ”

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

“Exactly. ”

Go ahead, he wrote the Gauss-Bonnet formula.

Seeing this drawing of dragon dots, the face of the ark became more interested in the divine color.

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

The Gauss curvature can be promoted as a cross-sectional curvature, depending on the nature of the Gaoville Lehman stream M, and its value can be determined by the tensor of the Lehman curvature. As for its accumulated function, it is a complex algebraic expression consisting of the curvature tensor - the Gauss-Bonnet accumulated function.

As for its integral across the stream, it is determined by the Euler exemplary number X (M) of this stream.

By taking advantage of these properties, the HODGE theory can be extended to complete non-tight currents.

These profound mathematical meanings were obtained by Professor Chen Province, the famous Gauss-Bonnet-Chen formula for mathematical content.

Combined with Sir Attia's L2 upward synchronization method, continue along this line, perhaps this conjecture can be proven.

Of course, just how to prove it in detail needs to be studied further.

Think of it here, the ark praises the nod.

Seconds.

It's wonderful.

Somehow, Chen Yang has stood behind a circle of people.

People in the office noticed this as early as the beginning of his book.

Staring at the formula on the white board, Seamus glanced and whispered excitedly: “Is this the legend?”

He Changwen frowned and whispered: “What is it? Don't sell Guan. ”

Dimmer looked at him strangely.

“Hodge, guess! Obviously. ”

He Changwen: “…”

Where the hell is this obvious?!

But if you look closely, it seems so.

Thinking about it, He Changwen couldn't help but comfort himself in his heart.

Well, he should be able to see it, too, if he's serious.

The pen on the white board stopped and Chen Yang fell into contemplation.

Obviously, he was only halfway there, and he didn't have a good idea how to go.

When he came to the office, standing beside him, Professor Perelman, who had been silent, suddenly spoke.

“This idea seems a little interesting. ”

Looking back at Professor Perelman, Chen Yang stunned slightly, some unexpectedly said.

“When did you get here? ”

“Around half the time you wrote... I was supposed to come to Professor Lu, I didn't expect there to be any unexpected gains here,” he paused, Perelman continued, "... could you give me a pen? ”

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

After taking the pen from Professor Chan's hand, Perelman, standing in front of the white board, pondered for a moment and then left a few lines empty under his formula to continue writing.

“Since there is a readily available theory of algebraic geometry uniformity that can be applied, the proof of Formula (3) has been omitted. ”

“… my suggestion is that for the later part of the proof, we can raise the tight flow M issue to its universal overlay flow form to get a complete non-tight flow M. ”

“According to Attia's theorem, if we can prove that all but the middle L2 tune group is zero under the conditions of cross-sectional curvature...”

As he spoke, his pen shook slightly and quickly wrote down a simple and beautiful formula.

[H ^ n (M) 6 ¹ {0}, and when q ¹ n, H ^ q (M) = {0}]

The moment I saw this line of formula, Chen Yang's pupils contracted slightly.

The look on his face instantly revealed a glimmer of enlightenment, and he said in an agitated tone.

“… we can get proof of Hodge's guess! ”

So here comes the problem.

How can it be demonstrated that all but the L2 tuning group in the middle is zero under the conditions of cross-sectional curvature?

The dialogue suddenly stopped here.

After a brief excitement, the two men fell into silence without agreement.

Finally, he looked at the ark in unison.

Noting that the two men looked at themselves, the ark suddenly blinked without a word from beginning to end, smiling and saying.

“I think your thoughts are all good… Although I have not studied the subject carefully, my instincts tell me that following this path will probably yield something. ”

After a pause, he continued.

“This idea is very interesting, and my suggestion is that you should study this topic together. ”

It always felt like the ark seemed to see something, but didn't make it clear.

Perelman frowned and hesitated.

“Aren't you involved? This is an interesting challenge. ”

It's more than fun.

Hodge's guess is arguably a concentrated expression of the abstract features of modern mathematical development, studying the intrinsic links between the three branches of mathematics - analysis, topology, and algebraic geometry.

As for difficulty, as a millennium dilemma, it is undoubtedly true.

Perelman was surprised that the Ark did not show a strong interest.

Ark: “… although I am very interested, there are still a lot of things waiting for me at IMCRC, and I'm afraid I haven't had more time lately to allocate to the math side. ”

Upon hearing the news, Perelman's face showed a regrettable expression.

“That's too bad. ”

“I'm afraid I can't take the time to help, but I solemnly recommend Professor Chen to you," he patted Chen Yang's shoulder, Ark Zhou said with a smile, "He is an excellent scholar, I'm sure you know about his abilities, I won't brag too much. In short, with your cooperation, I am sure that this problem will be resolved. ”

While sceptical of this absolute assertion, one glance at Professor Chen and Perelman did not say anything, just nodded and seemed to acknowledge the partner.

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

After clearing his voice, he watched Perelman continue.

“By the way, are you sure you want to stay here? The unified theory of algebra and geometry has been completed. ”

“No problem," Perelman shook his head, "my mother and I had already called, and she asked me to do what I wanted, not to worry too much about her side. I do have things I want to do that are not done, and I intend to stay here… for a little while and settle Hodge's assumptions before going back. ”

Although it was surprising that Professor Perelman had chosen to stay, such a good thing would naturally not be rejected by the Ark, he said with a smile.

“Then you should stay in the original apartment. I'll help you apply for an extension of the apartment. ”

Perelman nodded, thanking him.

“Please. ”