The Science Fiction World of Xueba

Mathematics is the foundation of Pang Xuelin's career in academia. It was in the field of mathematics that he became famous at the beginning, and then he stepped into other fields and completed the research of high-purity carbon nanotube flying blade materials, large-size graphene materials, lithium-air batteries, etc. Research and development work, and then has the academic status of today.

Therefore, a heavyweight mathematics paper is indispensable for the first issue of Discovery Magazine.

But the question is, which direction to choose?

In the past year, most of Pang Xuelin's energy has been involved in the lithium-air battery project and the electromagnetic catapult space launch system project. It has been a while since he has studied mathematical problems.

Currently, the editorial department of "Exploration" magazine is being established. According to Pang Xuelin's plan, the inaugural issue of "Exploration" will be officially published in July.

It is now April 5th, and there are still three months before the official publication. Considering factors such as typesetting and printing, the actual time is only two months.

If he wants to complete a heavyweight mathematics paper in such a short period of time, even if his memory and logical thinking ability have been greatly improved after the transformation, it may be difficult to do so.

As for the remaining third paper, Pang Xuelin is going to hand it over to Shi Yi, Yang Heping, and Anderson White. The relevant theories of dynamic APT technology have matured. Pang Xuelin explained the entire technical process to them. Before the relevant equipment is developed, this paper will be written.

The only trouble now is the math paper.

Pang Xuelin pondered for a moment, and couldn't help thinking of the paper on Hodge's conjecture that Perelman had shown him a few months ago.

Mathematics papers at the level of the seven major problems of the millennium are in the first issue of "Exploration" magazine, and Pang Xuelin is not worried that "Exploration" magazine will not be able to make the first shot.

The only trouble is that although the paper proves that the idea is feasible as a whole, there are logical loopholes in the details.

How to fill this loophole is a problem that Pang Xuelin needs to face.

I don't know how Perelman's Hodge conjecture is being studied now.

Pang Xuelin thought for a while, got up and walked to the suite next to the office, and said to the busy Zuo Yiqiu, "Little Zuo, I'll go out for a while."

Zuo Yiqiu was slightly taken aback, and asked in confusion, "Professor Pang, where are you going?"

Pang Xuelin said with a smile: "I'll go find Perelman, maybe I'll have to retreat for a while, you stay here, if you have something important, call me directly."

"good."

Zuo Yiqiu nodded.

More than an hour later, the car parked in the underground parking lot of Rose Home next door to Jiangcheng University.

Rose Home is a residential area specially prepared by Jiangcheng University for faculty and staff. It is mainly for foreign scholars working in Jiangcheng University to live in, and some of them are sold at discounted prices to faculty and staff who make their homes in Jiangcheng.

Pang Xuelin asked Zhou Chen to wait downstairs and got out of the car.

Go upstairs through the elevator, go straight to the door of Room 1302, and press the doorbell.

After about 10 minutes, Perelman, with a beard and a tired face, opened the door.

Seeing Pang Xuelin standing at the door, Perelman was slightly taken aback: "Pang, why are you here?"

Pang Xuelin walked into the room and said with a smile, "I'll come and see you."

Perelman lived in a two-bedroom house.

The room was fairly tidy, not because Perelman would take the initiative to clean it, but because Jiang University hired a housekeeper for him.

Come over every other day to help him clean up the housework and tidy up.

"Would you like a cup of coffee?"

Pang Xuelin smiled and nodded.

Soon, Perelman placed two cups of coffee on the coffee table.

Pang Xuelin took a sip, and a bitter taste in the strong fragrance was quickly transmitted to the central nervous system along the taste buds.

"Pang, what can I do for you?"

Pang Xuelin said: "Grigory, what happened to Hodge's guess?"

Perelman shook his head and said, "There is still no progress."

Before Pang Xuelin entered the world of Resident Evil, Perelman once approached Pang Xuelin with a paper on Hodge's conjecture. At that time, Pang Xuelin pointed out that there was a slight logical loophole in this paper.

It seems that after such a long time, Perelman still failed to find a breakthrough for Hodge's conjecture.

Pang Xuelin pondered for a moment, and said: "Grigory, Jiangcheng Institute for Advanced Study is preparing a new journal "Exploration" during this time. I plan to add your paper to the first issue, but the prerequisite is that it must be in Fill in the loopholes in the paper within two months. And this August is the St. Petersburg International Conference of Mathematicians. As long as the proof of this conjecture is completed, we can bring it to the Conference of Mathematicians to communicate with colleagues around the world... "

Perelman said: "Pang, to be honest, I feel that my research has reached a dead end during this period of time. It may be that I am getting older, and my physical and mental strength can't keep up. If you are interested in Hodge's conjecture , you can continue to do the next proof on my previous paper without asking for my opinion. For me, as long as I can prove Hodge’s conjecture, it doesn’t matter who proved it.”

Pang Xuelin was slightly taken aback, and said: "Grigory, thank you. But I still hope that you will not give up the research on Hodge's conjecture. After all, mathematical research itself is a kind of fun, isn't it?"

Perelman laughed and said, "Professor Pang, since that's the case, let's work together."

...

Coming out of Perelman's house, Pang Xuelin smiled and shook his head.

He's getting a little out of shape.

With Perelman's temperament, he would only be happy if he could really help him complete the proof of Hodge's conjecture, and would not have any extra thoughts at all.

In fact, Perelman's work is very effective.

He has almost completed 95% of the work, and the remaining 5% is caused by a small logical loophole.

Perelman has no way to determine whether the space curvature of the T2 Klein bottle whose directional double covering is a torus is a smooth function on the Riemannian manifold.

This question is very basic, but it is the basic link of the whole Hodge conjecture theory building.

To prove the Hodge conjecture, it is impossible to get around it.

Mathematically, a Klein bottle is a non-orientable two-dimensional compact manifold, while a sphere or tire surface is an orientable two-dimensional compact manifold.

If you look at the pictures of the Klein bottle, you can find that the neck and the body of the Klein bottle intersect. a location.

But this is not the case.

The fact is: the Klein bottle is a curved surface that can only be shown in the four-dimensional space. If we must show it in the three-dimensional space we live in, we have no choice but to make it appear as if it is our own surface. Same as intersecting with oneself.

In fact, the bottleneck of the Klein bottle passes through the fourth-dimensional space and connects with the bottom circle of the bottle, and does not pass through the wall of the bottle.

Use the analogy of a three-dimensional kink.

If you look at it as a curve on a plane, it seems to intersect with itself, but when you look at it again, it seems to be broken into three pieces.

But this figure is actually a curve in three-dimensional space, it does not intersect with itself, and it is a continuous curve.

Naturally, a curve on a plane cannot do this, but if there is a third dimension, it can pass through the third dimension to avoid intersecting itself.

It's just because when we want to draw it on a two-dimensional plane, we have to make do with it and draw it as intersected or broken.

The same goes for the Klein bottle, which is actually a curved surface in four dimensions.

In our three-dimensional space, even the best craftsman has to make it intersect itself; just as the best painter has to draw kinks on paper to make them intersect themselves. appearance.

As long as it is proved that the spatial curvature of the T2 Klein bottle whose directional double cover is a torus is a smooth function on the Riemannian manifold, the last loophole in the proof of the Riemann conjecture can be filled.

But the question is, how to prove it?

In the following time, Pang Xuelin entered the research state again.

But this time, he didn't specialize in retreat for research.

He is confident that even if other affairs interfere, he will be able to complete the last piece of the Hodge conjecture puzzle within two months.

In the past two months, Pang Xuelin has started to set up the editorial department of the "Exploration" journal.

He himself serves as the editor-in-chief, and the editorial board members are also bigwigs in various fields.

For example, Kirton Walker, Shi Yi, Wen Xiaogang, Mochizuki Shinichi, Anderson White, Yang Heping, Perelman, Xu Xingguo, Li Changqing and others were all pulled in by Pang Xuelin.

On the other hand, Pang Xuelin did not relax his research on Hodge's conjecture.

Basically, I have to work until one or two in the morning every night.

Time passed day by day, and before I knew it, it entered June.

That night, Pang Xuelin was meditating in the study when Qi Xin walked in with a bowl of white fungus, lotus seed and red date soup.

Hearing the movement, Pang Xuelin raised his head and said with a smile, "Xiao Xin, you should go to bed early, you don't have to stay up so late every night."

During this time, Pang Xuelin went to bed at one or two o'clock in the morning every day, Qi Xin felt sorry for Pang Xuelin, and every night after eleven o'clock, he would cook some nourishing soup for him in different ways.

Qi Xin pursed her lips and said with a smile: "I'm fine, you can drink the soup first, it will be cold after a while."

"Okay, I'll drink right away."

Pang Xuelin took the white fungus soup from Qi Xin, drank it and chatted with her at the same time.

"Alin, hasn't the research progressed during this time?"

Qi Xin asked curiously.

She seldom asked about Pang Xuelin's work affairs, and she always silently helped Pang Xuelin handle other affairs.

Pang Xuelin shook his head and sighed: "I still don't have a clue, the time is too fast, and there are still 15 days before the first issue of "Exploration" will officially enter the printing factory for printing, and now 80% of the manuscripts for the first issue have been confirmed. This paper is left unfinished."

Qi Xin frowned and said, "It's really not possible, so let's take it easy."

Pang Xuelin said: "Do your best. If Hodge's conjecture is really uncertain, I will go to the Ponzi Geometry Seminar to get a paper with a decent level and use it to make up the numbers."

"Well, then I'm going to take a shower first. I have to get up early for class tomorrow, and you should go to bed early. If you really can't find a breakthrough, change your thinking or start with reverse thinking, maybe you can solve it."

Pang Xuelin was slightly taken aback, "What did you say?"

"I said change your mind or start with reverse thinking?"

There was an inexplicable light in Pang Xuelin's eyes, he rushed forward to hug Qi Xin and kissed him hard, he laughed and said, "Xiao Xin, you are really my lucky star."

As he said that, he ignored Qi Xin as well, sat down at the desk, picked up a pen and quickly calculated.

Qi Xin looked at Pang Xuelin who was immersed in research, with a gentle smile on her face, she picked up the empty bowl on the table, and quietly exited the study.

...

Pang Xuelin's brain was running fast.

Qi Xin's reverse thinking sentence was like a ray of light illuminating the vast night sky ahead.

Why it is necessary to prove that the space curvature of the T2 Klein bottle whose directional double cover is a torus is a smooth function on the Riemannian manifold?

It is only necessary to prove that the space curvature of the T2 Klein bottle with directional double-covered acyclic surface is not a smooth function of the Riemannian manifold?

The concentrating thinking of the past two months is like dry firewood, only one spark is needed to ignite a raging flame.

The nib in Pang Xuelin's hand was spinning rapidly on the paper.

[Assuming that M is an n-dimensional smooth manifold, if a smooth second-order covariant tensor field g is given on M, (M, g) is called an n-dimensional Riemannian manifold, and g is called the basic tensor of the Riemannian manifold. Quantity or Riemannian metric, if it satisfies: 1.g is symmetric, that is: g(X, Y)=g(Y, X)(X, Y∈TpM, p∈M). 2. g is positive definite, that is: g(X, X)≥0 (X∈TpM, p∈M), and the equal sign is only established when X=0...]

...

[Assuming that D is a two-cycle directed tree graph, D0={e1, e2} is a vertex set, D1={a, c} is a set of directed edges with length 1, which is the path algebra of D on domain K, then H0(C)≌K⊕K, H1(C)=1...]

...

[If e:K→M satisfies e(1)=M, then δ(e)(a)=am-ma is a linear derivation. A 1-up cycle refers to a linear mapping f:A→M that satisfies the condition f(ab)=f(a)b+af(b). If M is an ideal in A where the square is zero, and E/M≌A, then the algebraic expansion E obtained by A through M can be studied by H2(A, M) being trivial...]

...

The time passed by one minute and one second, and before I knew it, the hour hand pointed to one o'clock in the morning.

Looking at the manuscript paper full of mathematical symbols, Pang Xuelin showed a relieved smile on his face.

"The efforts of the past two months have not been in vain, Hodge conjectures, and finally proved."

Pang Xuelin tidied up the manuscript papers and put them aside, got up and walked out of the study.

The living room was dark, Pang Xuelin stretched his waist, and went straight to the master bedroom on the third floor.

The master bedroom is large, with a cloakroom, living room, toilet, shower, bathtub and more.

Pang Xuelin went to take a bath for half an hour to relax all the muscles in his body. After that, he dried his hair, came to the bed, quietly lifted the quilt, and got into the bed.

Pang Xuelin was about to go to bed when a pair of snow-white lotus root arms came over.

Qi Xin's black and white eyes stared at Pang Xuelin without blinking.

Pang Xuelin was slightly taken aback, looked at Qi Xin's delicate face and said, "Xiao Xin, did I wake you up just now?"

Qi Xin shook his head and asked, "Have you solved that problem?"

Pang Xuelin smiled and nodded.

Qi Xin got on his horse and sat on Pang Xuelin's body. His eyes waved like water and he said, "Alin, it's been a long time since we..."

"Well……"