"Ahh

In the villa of Zhongshan International, sitting in his study, Lu Zhou, who was writing something in front of his desk with a pen, sneezed without warning.

"Who's thinking about me......"

Snorting and muttering to himself, Lu Zhou continued to look at the scratch paper in front of him, and the ballpoint pen in his hand lightly lit on the table.

"It's a little difficult."

【…… When n is greater than 2, two n-dimensional complex complete intersections X^n(d), X^n(d') differential homomorphism, if and only if their Euler number, full order, and Pontrjagin class are equal. 】

Strictly speaking, this is not a complex analysis problem, nor is it a partial differential equation problem in the traditional sense, but an interesting classification problem about smooth manifolds.

This may be a bit of a mouthful, but in fact, in differential topology, it is a fairly popular research direction, mainly to study the invariant properties of differential manifolds under differential homeomorphic mapping.

However, some trouble is that although Lu Zhou has studied differential manifolds and topology, he has not studied much of the mathematical branch of differential topology.

It could even be said that this was a completely new field for him.

However, it is understandable that this situation will occur, after all, this Professor Chen Yang studied the Hodge conjecture, and the Riemann conjecture he studied were originally two completely different problems.

It was only because the hyperelliptic curve analysis method happened to be improved and used to study the Cauchy-Riemann equation and the generalized surface of the Riemann surface that it triggered his association with this problem......

"It's really embarrassing for me to be a fat tiger...... Why don't you just leave it alone? ”

After thinking for a while, Lu Zhou shook his head, and finally drove this uncompetitive thought out of his brain.

Not to mention that it's better to compare who gets this result first, even if he doesn't pretend to be forced to do so, it's not his style to give up in the face of problems.

Even if differential topology is not his research direction, with his understanding of differential manifolds and topology theory, it is not too much of a problem to quickly familiarize himself with the research points in this field.

What's more, the research on the Riemann conjecture has also reached a bottleneck, and instead of stumbling on a road that doesn't work, it is better to try to look up and see if there are any other noteworthy clues.

If he can bridge the gap between topology and complex analysis, his hyperelliptic curve analysis method may be able to work wonders in the study of Riemann zeta functions......

"Xiao Ai, help me make a cup of coffee, unsweetened!"

Xiao Ai: [Master, instant coffee doesn't need to be sweetened? (°ー°〃)】

“…… Anyway, help me make me a cup of coffee. ”

【Good, good host!】 (??? *)】

The drone floated up from the bookshelf and whined outside the study.

Putting aside the distractions in his heart, Lu Zhou focused all his attention on the scratch paper in front of him.

With his thumb open the clasped cap, he used the pen to write the first lines of text on the paper.

[Let f1,......,fr be a homogeneous polynomial with respect to the complex coefficients of the variable z0,......,zn+r. This set of polynomials defines a complex algebraic cluster X...... in the complex projective space CP^(n+r)

[At this time, X is called a complex complete intersection, if the complex dimension of X is n, when X is a smooth manifold, it is called a smooth complex complete intersection, and X is a smooth closed manifold of 2n dimensions......]

The first step is done.

Lu Zhou's eyes flickered with excitement.

Although the feeling was faint, he had a clear hunch that he had found the right path to the end of the labyrinth.

"Using a 2n-dimensional smooth closed manifold as the starting point to establish a correlation ...... in the complex projection space"

Once a breakthrough is opened in the mind, the rushing inspiration will pour out like a flood, and it cannot be stopped.

Lines of notes poured down his pen like a rushing river, and soon a whole sheet of scratch paper was filled with a dense mass of operators and numbers.

As the minutes passed, more and more manuscript papers were piled up on the table.

Xiao Ai, who was lying on the bookshelf, just peeked quietly beside him. Since Lu Zhou forbade it to turn on the camera casually, it can only use this camera freely on the drone.

Although it is a little distressing to see the owner working so hard, but this kind of creative work alone, it really can't help much.

Outside the window, it was getting dark.

Lu Zhou, who was lying in front of the table, had no other actions other than calculating except for reaching out and turning on the lamp in the corner of the table.

However, the situation does not seem to be moving in a more optimistic direction.

As the sky outside the window gradually darkened, not only did the originally smooth strokes gradually slow down, but beads of sweat gradually seeped out of Lu Zhou's forehead.

The situation does not seem to be as rosy as he imagined.

The more he delved into the problem, the more he realized how complex the labyrinth was hidden behind it.

"Ah......hh It's so hard! ”

I don't know how many times I walked into a dead end, and finally Lu Zhou, who couldn't hold his breath, instantly turned into an irritable brother, crumpled the scratch paper that was no longer counted halfway, and fell into the paper basket next to him.

"That's not right! If the proposition is true, I can't think of it after thinking about it for so long......"

The ballpoint pen in his hand turned faster and faster, and Lu Zhou, who was frowning, stared at the original question, his lips gently opened and closed.

But strangely, it can't prove that the proposition is wrong...... No matter how many sets of parameters you bring in, you can get a result that matches the proposition. ”

With a sigh of relief, Lu Zhou pinched the pen that was spinning quickly, opened the computer placed in the corner of the desk, logged in to the literature database, and entered a few lines of keywords.

Rows and rows of retrieved papers soon appeared in front of him.

After selecting a few tags and sorting them by the number of citations, Lu Zhou quickly picked out a few of the top papers.

“…… It's impossible that no one has studied such an interesting question. ”

"At least you should be able to find something similar."

Reading softly in his mouth, at this moment, Lu Zhou's eyes suddenly lit up, and he keenly captured an interesting line of titles from the vast database.

The author of the paper is Professor Kreck, a well-known expert in the field of partial differential equations and topology, and this paper is a paper published in the Annals of Mathematics in 1999, and the number of citations is a bit high.

According to the description in the abstract, the main content of this paper is to improve the theory of Surgery and to transform the problem of manifold classification into a computational problem for certain ligand groups.

A look of interest gradually appeared in his eyes, and Lu Zhou quickly downloaded the paper, and looked at it line by line along the body of the paper.

After about ten minutes, when he carefully read the paper at the end, the look of interest in his eyes suddenly turned into excitement.

"That's it!"

"Well, I'll just say, it's impossible that no one has studied such an interesting question before...... At least someone must have studied something similar! ”

"If the problem of manifold classification could be translated into the computation of certain ligand groups...... Damn, this is the last piece of the puzzle I'm missing! ”

Excitedly got up from his chair, walked to Lu Zhou next to the wastebasket, and quickly turned out the scratch paper that he had crumpled up earlier.

"Fortunately, it's just a little wrinkled, and it hasn't been torn off!"

Talking to himself, Lu Zhou pinched the scratch paper and returned to the chair and sat down, spreading the scratch paper in his hand on the table, his eyes rekindled with a raging fighting spirit.

"It's time to start!"

After that, the sound of brushstrokes sounded again in the study.

And this time, it lasted until dawn......