Chapter 930: The problem of manifold classification (1/4)

"A sneeze!"
The villa of Zhongshan International, sitting in his study, was holding a pen on the desk, writing something at the desk, and sneezing without warning.
"Who is thinking about me again ..."
After sucking his nose, he muttered to himself, Lu Zhou continued to look at the draft paper in front of him, and the ball-point pen on his hand lightly clicked on the table.
"A bit difficult."
[... When n is greater than 2, two n-dimensional complexes completely intersect xn (d), xn (d ′) differential homeomorphisms, if and only if their euler number, full degree, and pontrjag class are equal. 】
Strictly speaking, this is not a complex analysis problem or a traditional partial differential equation problem, but a very interesting classification problem for smooth manifolds.
This argument may be a bit verbal, but in fact it is a relatively popular research direction in differential topology, mainly studying the invariant nature of differential manifolds under differential homeomorphism mapping.
However, it is troublesome that although Lu Zhou has studied both differential manifolds and topology, there is not much research on the mathematical branch of differential topology.
It can even be said that this is a completely new field for him.
However, there is nothing wrong with this situation. After all, this professor Chen Yang is studying the Hodge conjecture, and the Riemann conjecture he studied is originally two completely different problems.
It is only because the hyper-elliptic curve analysis method can be improved to study the Cauchy-Riemann equation and the generalized surface of the Riemann surface, which led to his association with this problem ...
"It's really embarrassing for me to be a fat tiger ... would you just leave it alone?"
After pondering for a while, Lu Zhou shook his head, and finally kicked the idea out of his mind.
Not to mention that it is better than someone to get the result first. Even if it is not installed, it is not his style to give up in the face of difficulties.
Even if differential topology is not his research direction, with his understanding of differential manifolds and topological theory, it is not too much of a problem to quickly become familiar with the research points in this field.
What's more, the research on Riemann's conjecture is now stuck in a bottleneck. Instead of dying on a road that is inaccessible, it is better to try to look up and look around and no other clues worth noting.
If a bridge can be built between topological problems and complex analysis problems, maybe his hyperelliptic curve analysis method can play a wonderful effect on the study of the Riemann zeta function ...
"Ai, help me make a cup of coffee, no sugar!"
Xiao Ai [Master, instant coffee does not need to be sweetened? (° ー ° 〃)】
"... Well, help me get a cup of coffee anyway."
[Okay, good master! (???)】
The drone floated from the bookshelf and whispered to the outside of the study.
Putting aside his thoughts, Lu Zhou focused all his attention on the draft paper in front of him.
With the thumb cap open, he wrote the first line of text on the paper with a pen.
[Let f1, ..., fr be homogeneous polynomials with complex coefficients of the variables 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. At this time, x is a 2n-dimensional smooth closed manifold ... 】
The first step is done.
Lu Zhou's eyes flashed with excitement.
Although this feeling was weak, he had a clear hunch that he had found a correct way to the end of the maze.
"Take a 2n-dimensional smooth closed manifold as the starting point to establish an association in the complex projective space ..."
Once a breakthrough is opened in thinking, the inspiration from the Pentium is gushing like a flood, and it is impossible to stop it.
The line-by-line calculations poured down like flowing rivers in his pen, and soon the entire draft was filled with dense operators and numbers.
Time passed minute by minute, and more and more manuscripts piled on the table.
Xiao Ai, lying on the bookshelf, peeped beside him so quietly. Since Luzhou banned it from randomly opening the camera, only the camera on the drone can use it freely.
Although it's a bit distressing to see the owner so hard, but this creative work alone can't really help.
The sky outside the window gradually darkened.
The boat lying on the table, besides reaching out to turn on the table lamp at the corner of the table, had no other action besides calculating.
However, the situation does not seem to be moving in a more optimistic direction.
As the sky outside gradually darkened, not only did the original smooth strokes gradually slow down, but the beads of sweat gradually leaked out of Lu Zhou's forehead.
The situation does not seem to be as optimistic as he imagined.
The more he delves into this problem, the more he finds out how complicated the maze is.
"Ahhhh ... it's so hard!"
I don't know if it was the first time I walked into a dead end, and finally the impatient Lu Zhou instantly turned into a irritable brother, crumpled up half of the draft paper that could not be counted, and slammed it into the nearby basket in.
"No! If the proposition is true, I won't think of it for so long ..."
The ball-point pen in his hand turned faster and faster, and Lu Zhou's frowning eyes stared at the original problem, his lips opened and closed thinking.
"But weirdly, it just can't prove that the proposition is wrong ... no matter how many sets of parameters are taken in, you can get results that agree with the proposition."
Taking a long breath, Lu Zhou grasped the pen that moved quickly, opened the computer at the corner of the table, logged into the literature database, and entered a few lines of keywords.
Rows of retrieved papers quickly appeared in front of him.
After selecting a few tags and sorting by the number of citations, Lu Zhou quickly picked out several top-ranked papers.
"... It's impossible that no one has studied such an interesting question ~ EbookFREE.me ~ At least you should find something similar."
With a light reading in his mouth, at this moment Lu Zhou's eyes suddenly lighted, and he keenly captured a line of interesting titles from the vast database.
The author of the paper is Professor Kreck, a famous Daniel in the field of partial differential equations and topology. This paper is a paper published by him in the "Journal of Mathematics" in 1999. A little high.
According to the description in the summary, the main content of this paper is to improve the theory of surgery and to transform the problem of manifold classification into the calculation of some matching groups.
A look of interest gradually appeared in his eyes, and Lu Zhou quickly downloaded the paper and looked at it line by line.
After about ten minutes, when he carefully looked at the end of the thesis, the expression of interest in his eyes suddenly turned into excitement.
"that's it!"
"I'll just say that such an interesting problem must not have been studied by no one before ... at least someone must have studied a similar subject!"
"If you can translate the problem of manifold classification into the calculation of some matching groups ... Maje, this is the last piece of puzzle I am missing!"
He got up from the chair in excitement, walked to the land boat beside the wastebasket, and quickly pulled out the draft paper that had been crumpled by him.
"Fortunately, it was just a little wrinkled and not shredded!"
To himself, Lu Zhou squeezed the draft paper back to the chair and sat down, spreading the draft paper in his hand on the table, his eyes rekindled the fierce fighting spirit.
"It's about to start!"
Having said that, the sound of cymbal strokes sounded again in the study.
And this time, it didn't stop until dawn ...
.