931 The Problem Of Manifold Classification (1/4)

Chapter 931 The problem of manifold classification (14)

"Ah Choo!"

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

"Who is thinking about me again..."

Sniffing and muttering to himself, Lu Zhou continued to look at the draft paper in front of him, and lightly tapped the ballpoint pen on the table.

"It's a bit difficult."

【...When n is greater than 2, two n-dimensional complexes completely intersect X^n(d), X^n(d') diffeomorphism, if and only if their Euler number, total degree and Pontrjagin class are all equal. 】

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

This statement may be a bit of a mouthful, but in fact, in differential topology, this is a fairly popular research direction, mainly studying the invariant properties of differential manifolds under diffeomorphism mapping.

However, what is a little troublesome is that although Lu Zhou has a lot of research on differential manifolds and topology, he doesn't have much research on the branch of mathematics called differential topology.

It can even be said that this is a completely new field for him.

However, there is nothing wrong with such a situation. After all, this Professor Chen Yang is studying the Hodge conjecture, which is originally two completely different problems from the Riemann conjecture he studied.

It is only because the hyperelliptic curve analysis method can be improved and used to study the generalization of the Cauchy-Riemann equation and the Riemann surface, which aroused his association with this problem...

"It's really embarrassing for me, Fat Tiger... Should I just leave it alone?"

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

Not to mention the comparison of who will come up with the result first, even if he doesn't pretend to be so forceful, it is not his style to give up when faced with difficulties.

Even if differential topology is not his research direction, with his understanding of differential manifolds and topology theory, it is not a big problem to quickly become familiar with the research points in this field.

What's more, just now the research on Riemann's conjecture has also fallen into a bottleneck. Instead of dying on a road that doesn't work, it's better to try to look up and see if there are no other noteworthy clues around.

If a bridge can be built between topology problems and complex analysis problems, maybe his hyperelliptic curve analysis method will be able to play a miraculous effect on the research of the Riemann zeta function...

"Xiao Ai, make me a cup of coffee, without sugar!"

Xiao Ai: [Master, instant coffee doesn’t need to add sugar? (°ー°〃)】

"... Long-winded, anyway, make me a cup of coffee."

[Okay, good master! (*)】

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

Putting aside the distracting thoughts in his heart, Lu Zhou focused all his attention on the draft paper in front of him.

Pushing open the buttoned pen cap with his thumb, he wrote the first line of text on the paper with the pen.

[Let f1, ..., fr be homogeneous polynomials about the complex coefficients of variables z0, ..., zn+r. This set of polynomials defines a complex algebraic variety 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. At this time, X is a 2n-dimensional smooth closed manifold... 】

The first step has been done.

Lu Zhou's eyes shone with excitement.

Although this feeling was very weak, he had an obvious hunch that he had found a correct path to the end of the maze.

"Take the 2n-dimensional smooth closed manifold as the entry point, and establish a relationship on the complex projective space..."

Once a breakthrough is opened in thinking, the rushing inspiration will come out like a flood, which cannot be stopped.

Lines of calculation formulas like musical notes poured down like an endless river in his pen, and soon the entire draft paper was filled with dense operators and numbers.

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

Xiao Ai, who was lying on the bookshelf, just quietly peeked at the side. Since Lu Zhou banned it from turning on the camera at will, only the camera on the drone can be used freely.

Although it hurts to see the master working so hard, but this kind of creative work alone, it really can't help much.

The sky outside the window gradually darkened.

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

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

As the sky outside the window gradually dimmed, not only did the originally smooth strokes gradually slow down, beads of sweat also gradually oozed from Lu Zhou's forehead.

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

The more he delved into this problem, the more he discovered how complicated the maze behind this problem was.

"Ahhh... how difficult it is!"

I don't know how many times he walked into a dead end, and finally he couldn't hold back. Lu Zhou turned into a violent brother in an instant, crumpled up the draft paper that could not be counted, and threw it into the wastebasket next to him. inside.

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

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

"But the strange thing is that it just can't prove that the proposition is wrong... No matter how many sets of parameters are brought in, the result that matches the proposition can be obtained."

Heaving a sigh of relief, Lu Zhou grabbed the fast-spinning pen, turned on the computer on the corner of the desk, logged into the literature database, and entered a few lines of keywords.

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 top-ranked papers.

"...It is impossible that such an interesting question has not been studied."

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

While muttering softly, Lu Zhou's eyes suddenly lit up at this moment, and he keenly captured a line of interesting titles from the vast database.

The author of the paper is Professor Kreck, who is well-known in the field of partial differential equations and topology, and this paper was published in the "Annals of Mathematics" in 1999, and the number of citations is also A little high.

According to the description in its abstract, the main content of this paper is to improve the Surgery theory and transform the problem of manifold classification into the calculation problem of certain side groups.

A look of interest gradually appeared in his eyes, and Lu Zhou quickly downloaded the paper and read the text of the paper line by line.

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

"that's it!"

"Let me just say, it's impossible that such an interesting question has never been studied before... At least someone must have studied a similar subject!"

"If the problem of manifold classification can be transformed into the calculation of some side groups... Oh my god, this is the last piece of the puzzle I'm missing!"

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

"Fortunately, it's just a little wrinkled and not torn off!"

Talking to himself, Lu Zhou held the draft paper and returned to the chair to sit down. He spread the draft paper on the table, and his eyes rekindled a raging fighting spirit.

"It's about to start!"

After saying that, the sound of brush strokes sounded again in the study.

But this time, it lasted until dawn before stopping...