"Achoo!"
Lu Zhou was sitting in his study room at Zhongshan International. He was writing something at his desk when he suddenly sneezed.
"Who's thinking about me this time …"
Lu Zhou sniffed and muttered to himself. He then looked at the draft paper in front of him and tapped the pen in his hand on the table.
"This is a bit difficult."
[… When n is greater than 2, two n-dimensional complex intersects X ^ n (d), X ^ n (d ') are diffeomorphisms if and only if their Euler number, total degree, and Pontrjagin class are equal.]
Strictly speaking, this wasn't a complex analysis problem, nor was it a traditional partial differential equation problem. It was an interesting classification problem for smooth manifolds.
This might sound a bit awkward, but in fact, this was a popular research area in differential topology. It mainly studied the invariant properties of differential manifolds under diffeomorphisms.
However, the problem was that even though Lu Zhou had done a lot of research on differential manifolds and topology, he hadn't done much research on differential topology.
It could even be said that this was a completely new field for him.
However, this was understandable. After all, Professor Chen Yang was researching the Hodge conjecture, which was completely different from the Riemann conjecture.
It was only because the hyperelliptic curve analysis method could be used to study the Cauchy- Riemann equation and the generalization of the Riemann surface …
"This is really making things difficult for me, Fat Tiger … Should I just leave it be?"
Lu Zhou thought for a bit and shook his head. In the end, he got rid of this disappointing idea.
Not to mention that they had agreed to compete to see who would be the first to come up with this result. Even if he didn't act cool, giving up in the face of a difficult problem wasn't his style.
Even though differential topology was not his research direction, with his understanding of differential manifolds and topology theory, he could quickly familiarize himself with the key research points in this field.
Not to mention that the research on Riemann's conjecture had reached a bottleneck. Instead of fighting on a dead end, it was better to look around and see if there were any clues worth paying attention to.
If he could build a bridge between topology and complex analysis, his hyperelliptic curve analysis method might be able to work wonders on the Riemann zeta function …
"Xiao Ai, make me a cup of coffee. No sugar!"
Xiao Ai: [Master, instant coffee doesn't need sugar. (° ー ° 〃)]
"… So long-winded. Anyway, help me make a cup of coffee."
[Yes, yes, Master! (???*)】
The drone floated up from the bookshelf and flew out of the study.
Lu Zhou threw away the distracting thoughts in his mind and focused all his attention on the draft paper in front of him.
His thumb pushed open the cap of the pen, and he wrote the first line on the paper.
[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) …
[X is called a complex complete intersection. If the complex dimension of X is n, when X is a smooth manifold, then it is called a smooth complex complete intersection. Then X is a 2n-dimensional smooth closed manifold …]
The first step was done.
Lu Zhou's eyes flashed with excitement.
Although this feeling was very weak, he had a very clear premonition that he had found the correct path to the end of the labyrinth.
"Using a 2n-dimensional smooth closed manifold as an entry point, establishing a correlation in the complex projective space …"
Once a breakthrough was made in one's thinking, inspiration would surge out like a flood, and it couldn't be stopped.
Lines of calculations flowed down from his pen like a flowing river. Soon, the entire piece of draft paper was filled with densely packed calculations and numbers.
Time quickly passed by, and more and more draft papers piled up on the table.
Xiao Ai, who was lying on the bookshelf, quietly peeked at Lu Zhou. Ever since Lu Zhou forbade it from turning on the camera, it could only use the camera on the drone.
Even though Xiao Ai felt bad seeing its owner working so hard, it couldn't help much with this kind of creative work.
The sky outside the window began to darken.
Other than reaching out and turning on the lamp, Lu Zhou didn't do anything else other than calculate.
However, the situation didn't seem to be developing in a more optimistic direction.
As the sky outside the window gradually darkened, not only did the originally smooth strokes of his pen gradually slow down, but beads of sweat gradually seeped from Lu Zhou's forehead.
The situation didn't seem to be as optimistic as he thought.
The more he delved into this problem, the more he realized how complicated the maze behind this problem was.
"Ahhh … This is so f * cking difficult!"
Lu Zhou didn't know how many times he walked into a dead end. Finally, he couldn't hold it in anymore. He crumpled the draft paper into a ball and threw it into the wastebasket next to him.
"No way! If the proposition is true, I should be able to figure it out after thinking for so long … "
The ballpoint pen in Lu Zhou's hand spun faster and faster. He frowned and stared at the first question. His lips gently opened and closed.
"But the strange thing is, I can't prove that the proposition is wrong … No matter how many sets of parameters I put in, I can get a result that matches the proposition."
Lu Zhou took a deep breath and grabbed the spinning pen. He turned on the computer on the corner of the table, logged into the literature database, and entered a few keywords.
Rows of theses quickly appeared in front of him.
After selecting a few tags, Lu Zhou quickly picked out a few top-ranked theses based on the number of citations.
"… There's no way no one has researched such an interesting problem.
"I should be able to find something similar."
Suddenly, Lu Zhou's eyes lit up. He captured an interesting title from the vast database.
The author of the thesis was Professor Kreck, a famous expert in the field of partial differential equations and topology. This thesis was published in Annual Mathematics in 1999, and it was highly cited.
According to the description in the abstract, the main content of this thesis was to improve the surgery theory and transform the problem of manifold classification into a calculation problem of cobordism groups.
Lu Zhou had a look of interest in his eyes. He quickly downloaded the thesis and began to read the thesis line by line.
About ten minutes later, when he finished reading the thesis, the look of interest in his eyes suddenly turned into a look of excitement.
"That's it!
"I knew it, there's no way no one has researched such an interesting problem before … At least someone has researched similar topics!
"If I can transform the problem of manifold classification into a calculation of cobordism groups … Jesus Christ, this is the last piece of the puzzle I'm missing!"
Lu Zhou excitedly stood up from his chair and walked to the wastepaper basket. He quickly found the draft paper that he had crumpled into a ball.
"Good thing it's just a little wrinkled, it's not torn!"
Lu Zhou held the draft paper and sat back down on his chair. He placed the draft paper on the table and his eyes lit up with fighting spirit.
"It's about to begin!"
As soon as he finished speaking, the sound of brush strokes rang out in the study once more.
This time, it didn't stop until dawn …
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