Mathematics is the foundation of Pang Xuelin's life in the academic world, and it was in the field of mathematics that he became famous in one fell swoop, and then stepped into other fields, completing the research and development of high-purity carbon nanotube flying edge materials, large-size graphene materials, lithium-air batteries, etc., and then has today's academic status.

Therefore, in the inaugural issue of Discovery Magazine, a heavyweight mathematical paper is indispensable.

But the question is, which direction to choose?

In the past year, Pang Xuelin has spent most of his energy on the lithium-air battery project and the electromagnetic catapult space launch system project, and has not studied mathematical problems for some time.

At present, the editorial department of "Exploration" magazine is being formed, and according to Pang Xuelin's plan, the first issue of "Exploration" will be officially published in July.

It is already April 5th, and there are still three months before the official publication, and the actual time is only two months, taking into account the typesetting, printing and other factors.

If he wants to complete a heavyweight mathematical paper in such a short period of time, even if he has undergone transformation, the memory of the whole person and the logical thinking ability of the brain have been greatly improved, I am afraid it will be difficult to do.

As for the remaining third paper, Pang Xuelin is ready to hand it over to Shi Yi, Yang Heping, and Anderson White to solve, the relevant theory of dynamic APT technology has matured, Pang Xuelin explained the entire technical process to them, and they can write this paper before the development of dynamic APT-related equipment.

Now the only trouble is the math papers.

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

The seven difficult mathematics papers of the millennium sat in the inaugural issue of "Exploration" magazine, and Pang Xuelin was not worried that "Exploration" magazine could not fire the first shot.

The only trouble is that the paper, although it proves that the idea is feasible as a whole, has logical holes 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 about it, got up and walked to the suite next to the office, and said to Zuo Yiqiu, who was busy: "Xiao Zuo, I'll go out." ”

Zuo Yiqiu was slightly stunned, and said suspiciously: "Professor Pang, where are you going?" ”

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

"Okay."

Zuo Yiqiu nodded.

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

The Rose Home is a community specially prepared for faculty and staff of Jiangcheng University, mainly for foreign scholars working in Jiangcheng University, and some of them are sold at a discount price to faculty and staff who have settled in Jiangcheng.

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

I went upstairs through the elevator and went straight to the door of room 1302 and pressed the doorbell.

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

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

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

Perelman lives in a two-bedroom house.

The room is quite tidy, not because Perelman will take the initiative to clean it, but because Jiang University has hired a housekeeping part-time worker for him.

Every other day, I came to help him clean up the housework and tidy up.

"Would you like a cup of coffee?"

Pang Xuelin smiled and nodded.

Soon, Perelman was on the coffee table with two cups of coffee.

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

"Pang, is there anything you want me to do?"

Pang Xuelin said: "Grigory, what's the Hodge guess? ”

Perelman shook his head and said, "Still not much progress. ”

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

It seems that after all this time, Perelman still hasn't been able to find a breakthrough in Hodge's conjecture.

Pang Xuelin pondered for a moment and said, "Grigori, during this time, the Jiangcheng Institute for Advanced Study is preparing a new journal "Exploration". I'm going to add your paper to the inaugural issue, but only if you fill the gaps in the paper within two months. And this August is the International Congress of Mathematicians in St. Petersburg, as long as we complete the proof of this conjecture, we can take it to the Congress of Mathematicians and communicate with our colleagues around the world......

Perelman said: "Pang, to be honest, I feel like my research has reached a dead end during this time. It may be that he is old, and his physical and mental strength are a little unable to keep up. If you're interested in the Hodge conjecture, you can go ahead and prove it on my previous paper without asking for my opinion. For me, as long as I can prove the Hodge conjecture, it doesn't matter who proves it. ”

Pang Xuelin was slightly stunned and said, "Grigory, thank you." But I still hope you don't give up on studying the Hodge conjecture, after all, the study of mathematics is a pleasure in itself, isn't it? ”

Perelman laughed when he heard this, and said, "Professor Pang, since this is the case, let's work together." ”

……

Coming out of Perelman's house, Pang Xuelin shook his head with a smile.

He's a little bit obsessed.

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

In fact, Perelman's work has been very productive.

He's almost 95 percent done, and the remaining 5 percent is caused by a small logical loophole.

Perelman was not sure whether the spatial curvature of a T2 Klein flask, which was overlaid with a torus, was a smooth function of the Riemann manifold.

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

If you want to prove the Hodge conjecture, you can't get around it.

Mathematically, a Klein bottle is a non-orientable two-dimensional compact manifold, while a spherical or tread is a two-dimensional compact manifold that is orientable.

If you look at the picture of the Klein bottle, you can see that the neck of the bottle and the bottle body intersect, in other words, some points on the neck of the bottle and some points on the wall of the bottle occupy the same position in three-dimensional space.

But this is not the case.

The fact is that the Klein bottle is a curved surface that can only be truly expressed in the four-dimensional space, and if we must express it in the three-dimensional space in which we live, we have to settle for it and express it as if we are intersecting with ourselves.

In fact, the bottleneck of the Klein bottle passes through the fourth dimension and is connected to the bottom circle of the bottle, and does not pass through the wall of the bottle.

To use a three-dimensional kink as an analogy.

If you think of it as a curve on a plane, then it seems to intersect itself, and then it seems to break into three parts.

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

A curve can't do this on a plane, but if it has a third dimension, it can pass through it to avoid intersecting itself.

It's just that when we want to draw it on a two-dimensional plane, we have to draw it as if it intersects or breaks.

The same goes for the Klein bottle, which is a surface that is in fact in a four-dimensional space.

In our three-dimensional space, even the most skilled craftsmen have to make it intersect with themselves; It's like the most brilliant painter who has to paint knots on paper as if they intersect themselves.

The last loophole in the Riemann conjecture can be filled by proving that the spatial curvature of the T2 Klein bottle with directional double coverage as a torus is a smooth function on the Riemann manifold.

But the question is, how do you prove it?

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

This time, however, he did not go into a retreat for research.

He was confident that he would be able to complete the final piece of the puzzle in two months, even if there were other distractions.

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

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

For example, Corton Walker, Shi Yi, Wen Xiaogang, Shinichi Mochizuki, 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 had to work until one or two o'clock in the morning every night.

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

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

Pang Xuelin heard the movement, raised his head and smiled: "Xiao Xin, you 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 after eleven o'clock every night, he would change his way to boil some nourishing decoction for him.

Qi Xin pursed his lips and smiled: "I'm fine, you drink the soup first, it will be cold after a while." ”

"Okay, I'll drink it right away."

Pang Xuelin took the white fungus soup from Qi Xin's hand, and chatted with her while drinking.

"Alin, hasn't there been any progress in research during this time?"

Qi Xin said curiously.

She rarely asks about Pang Xuelin's work affairs, and has been silently helping Pang Xuelin deal with other affairs on the side.

Pang Xuelin shook his head and sighed: "I still don't have a clue, the time is too rushed, there are still fifteen days left, the first issue of "Exploration" will officially enter the printing house for printing, and now 80% of the manuscripts of the first issue have been determined, and this paper is left unfinished." ”

Qi Xin frowned and said, "It's really not good, so let's slow down." ”

Pang Xuelin said: "Do your best, if Hodge's conjecture really can't be solved, I'll go to the Ponzi Geometry Seminar to get a paper with a good level and take it to make up the numbers." ”

"Well, then I'll go take a shower first, I have to get up early for class tomorrow, and you 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 stunned, "What did you say?" ”

"I said think differently or start with the opposite way of thinking?"

Pang Xuelin's eyes flashed with an inexplicable light, and he rushed forward to hug Qi Xin and kiss him fiercely, laughing and said, "Xiao Xin, you are really my lucky star." ”

As he spoke, he no longer paid attention to Qi Xin, sat down at the desk, picked up the pen and calculated quickly.

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

……

Pang Xuelin's brain was racing.

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

Why is it necessary to prove that the spatial curvature of a T2 Klein bottle with a directional double overlay as a torus is a smooth function of the Riemannian manifold?

Wouldn't it be enough to prove that the spatial curvature of the T2 Klein flask with directional double covering non-torus is not a smooth function of the Riemannian manifold?

The painstaking thinking of the past two months is like a firewood that has been completely dried, and it only takes a spark to ignite a raging flame.

The tip of the pen in Pang Xuelin's hand was running rapidly on the paper.

[Let M be an n-dimensional smooth manifold, if a smooth second-order covariant tensor field g is given on M, (M,g) is said to be an n-dimensional Riemannian manifold, g is called the basic tensor or Riemannian metric of the Riemannian manifold, and if satisfied: is symmetrical, i.e.: g(X,Y)=g(Y,X)(X,Y∈TpM,p∈M). is positively definite, i.e., g(X,X)≥0(X∈TpM,p∈M), and the equal sign holds only when X=0......]

……

[Let D be a bicyclic directed tree, D0={e1,e2} is the set of vertices, D1={a,c} is the set of directed edges of length 1, and be the path algebra of D on the 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 derivative.] A 1-up loop refers to a linear map f:A→M that satisfies the condition f(ab)=f(a)b+af(b). If M is an ideal where the square of A is zero, and E/M ≌ A, then the algebraic expansion of A obtained by M can be studied by H2(A,M) being trivial E......]

……

Minutes and seconds passed, and before I knew it, the hour hand pointed to one o'clock in the morning.

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

"The hard work of the past two months has not been in vain, Hodge's conjecture, and finally proved it."

Pang Xuelin put the manuscript paper away neatly, 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 and has a cloakroom, living room, toilet, shower, bathtub, and much more.

Pang Xuelin first went to take a bath for half an hour, so that the muscles of his whole body relaxed, and then he dried his hair, came to the bed, quietly lifted the quilt, and got into the quilt.

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

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

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

Qi Xin shook his head and asked, "That problem, have you solved it?" ”

Pang Xuelin smiled and nodded.

Qi Xin turned over and sat on Pang Xuelin, his eyes were like water: "Alin, we haven't ...... for a long time"

"Hmm......"