Scholar's Advanced Technological System

930 Manifold Problem

“Achoo!”

Lu Zhou was in his Zhongshan International mansion, sitting at his study room desk. He had a pen in his hand.

“Do I have a cold...”

He sniffed and looked at the draft paper in front of him. He gently tapped his pen on the table.

“This is kind of hard.”

[... When n is greater than 2, the two n-dimensional complex dimensions intersect X ^ n (d), X ^ n (d ‘), there exists a differential homeomorphism if and only if their Euler number, total degree, and Pontrjagin class are equal.]

Strictly speaking, this was neither a complex analysis problem nor a partial differential equation problem. It was a unique classification of smooth manifolds.

In fact, this was a fairly popular research direction in differential topology. It was mainly about studying the invariant properties of differential manifolds under homoeomorphic mapping.

However, the problem was that even though Lu Zhou had researched differential manifolds and topology before, he had had little contact with the mathematical branch of differential topology.

One could make the case that he was completely new to this field.

However, this was expected to happen. After all, Chen Yang was researching the Hodge conjecture, which was completely different from his research on Riemann’s hypothesis.

He was introduced to this problem only because the hyperelliptic curve analysis method could be used to research the Cauchy-Riemann equations...

“This is not my cup of tea... Should I just give up?”

Lu Zhou pondered for a while, then shook his head. He didn’t want to give up just yet.

He was the one who proposed a competition against Chen Yang on who could solve this first. Plus, he wasn’t a person who would give up.

Even though differential topology wasn’t his area of research, with his knowledge of differential manifolds and topology, he was able to pick up some of the core concepts of the field.

Not to mention that his research on Riemann’s hypothesis had reached a bottleneck. Instead of trying to move an immovable object, he decided to venture on other paths.

If he could bridge the gap between topology and complex analysis, he might be able to apply the hyperelliptic curve analysis method to the Riemann zeta function...

“Xiao Ai, bring me a cup of coffee, no sugar!”

Xiao Ai: [But Master, instant coffee already has sugar in it? (°ー°〃)]

“... Shut up, just bring me the coffee.”

[Yes, Master! (・∀・ *)]

The drone left the bookshelf and flew out of the study room.

Lu Zhou placed all of his attention on the draft paper in front of him.

He opened the pen cap with his thumb and began writing.

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

[Let X be a complex complete intersection. If the complex dimension of X is n, then when X is a smooth manifold, a smooth complex complete intersection exists. Set X as a 2n-dimensional smooth closed manifold...]

The first step is done.

There was a twinkle of excitement in Lu Zhou’s eyes.

He had a slight feeling that he was walking in the right direction.

“Set the smooth closed manifold with dimension 2n, establish an association on the complex projective space...”

Once he was in the zone, ideas began to flow like a river.

Lines and lines of calculations appeared on the draft paper, and soon, the entire draft paper was filled with numbers and symbols.

Time quickly passed by, and draft papers began to pile up on the table.

Xiao Ai was lying on the bookshelf, quietly observing. Ever since Lu Zhou banned Xiao Ai from using his laptop camera, Xiao Ai could only look through the drone camera.

Even though Xiao Ai didn’t like seeing its master working so hard, when it came to creative work like this, there was nothing Xiao Ai could do.

The sun gradually began to set.

Lu Zhou turned on his desk light.

However, he didn’t seem to be moving in the right direction.

As the sun in the sky gradually disappeared, his writing began to slow down. Beads of sweat began to drip down Lu Zhou’s forehead.

This problem wasn’t as easy as he had thought.

The deeper he dived into the problem, the more he realized how complex this maze was.

“Ah... why is it so difficult!”

After being stuck again, Lu Zhou turned into a grumpy old man. He scrunched up his draft paper and threw it in the bin.

“This can’t be! If the proposition is true, there’s no way I can’t find a proof...”

He spun the pen in his hand and stared at the draft paper. He began to mutter to himself, “The strange thing is that I can’t prove the proposition is wrong either... No matter how many parameters I set, the output is correct.”

Lu Zhou sighed and opened his laptop. He logged onto the literature database and typed in a few words.

Rows of research papers appeared in front of him.

Lu Zhou sorted them by the number of citations and selected some of the top-ranking theses.

“There’s no way no one has ever researched this problem before...

“I should be able to find similar examples.”

Lu Zhou’s eyes suddenly lit up when an interesting thesis title caught his eye.

The author of the thesis was Professor Kreck, an expert in the field of partial differential equations and topology. The thesis was published in Annual Mathematics and had quite a few references.

According to its abstract, the main intent of this paper was to improve surgery theory and to transform the problem of manifold classification into an edge group problem.

Lu Zhou was intrigued. He quickly downloaded the paper and began reading it.

After ten minutes went by, he finally finished reading the thesis. He suddenly spoke with excitement.

“This is it!

“I knew it, there has to be research done on this problem before... At least for a similar problem!

“If I can transform a manifold classification problem into an edge group problem... Jesus, this might be the last missing piece of the puzzle!”

Lu Zhou excitedly stood up from his chair and walked to the paper bin. He quickly pulled out the scrunched up draft paper.

“It’s just a bit wrinkled, should be alright!”

As Lu Zhou flattened the draft paper on his table, his eyes were filled with determination.

“Let’s do this!”

He started to write again.

And this time, he didn’t stop until the sun came up...

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