Mathematics is the foundation of Pang Xuelin in the academic world. At first, he became famous in the field of mathematics and then entered other fields. He completed high-purity carbon nanotube flying blade materials, large-scale graphene materials, and lithium-air batteries. R & D work has gained academic status today.

Therefore, the first issue of Discovery Magazine is an indispensable mathematical paper.

The question is, which direction should I choose?

In the past year, most of Pang Xuelin's energy was involved in the lithium air battery project and the electromagnetic ejection space launch system project. He has not studied mathematical problems for some time.

At present, the editorial department of "Exploration" magazine is being established. According to Pang Xuelin's plan, the first issue of "Exploration" will be formally published in July.

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

If he wants to complete a heavy-weight mathematical dissertation in such a short time, even after his transformation, the entire person's memory and brain logical thinking ability have been greatly improved, I am afraid it is difficult to do so.

As for the remaining third paper, Pang Xuelin is going to hand it over to Shi Yi, Yang Heping, Anderson White. The relevant theory of dynamic APT technology has matured. Pang Xuelin explained the entire technical process to them. Before the development of related equipment, write this paper.

The only trouble now is the mathematical thesis.

Pang Xuelin pondered for a moment, and could not help thinking of the Hodge conjecture that Perelman had seen himself a few months ago.

Mathematics papers on the seven major levels of the Millennium were at the inaugural issue of "Exploration" magazine. Pang Xuelin was worried that "Exploration" magazine could not start the first shot.

The only trouble is that although the paper proved the idea as a whole, there were logical loopholes in the details.

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

I don't know how well Perelman's Hodge guessed.

Pang Xuelin thought about it, got up and walked to the suite next to the office, and said to the busy Zuo Yiqiu, "Xiao Zuo, I'll go out."

Zuo Yiqiu froze slightly, wondering, "Professor Pang, where are you going?"

Pang Xuelin laughed: "I'll go find Perelman, maybe I will be in retreat for a while, you are here, and if there is something important, call me directly."

"it is good."

Zuo Yiqiu nodded.

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

The Rose Home is a community specially prepared by Jiangcheng University for faculty and staff, mainly for foreign scholars who work at Jiangda University, and some of it is sold to faculty and staff who settle in Jiangcheng at a discount.

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

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

About ten minutes later, his beard was unshaven, and Perelman, with a tired face, opened the door.

Seeing Pang Xuelin standing in the doorway, Perelman was a little surprised: "Pang, why are you here?"

Pang Xuelin walked into the room and laughed: "I'll come and see you."

Perelman lives in a two-bedroom house.

The room was fairly tidy. This was not because Perelman would take the initiative to clean it, but that Jiangda had hired a housekeeper for him.

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

"A cup of coffee?"

Pang Xuelin smiled and nodded.

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

Pang Xuelin took a sip of a sip, and a bitter aroma with a bitter taste quickly transmitted to the central nervous along the taste buds.

"Pang, is there anything wrong with me?"

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

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

Before Pang Xuelin entered the biochemical crisis world, Perelman once found Pang Xuelin with a Hodge conjecture. At that time, Pang Xuelin pointed out that there was a slight logical loophole in this paper.

It looks like this time has passed, and Perelman still can't find the breakthrough of Hodge's conjecture.

Pang Xuelin groaned for a moment and said, "Grigory, during this time, the Jiangcheng Institute of Advanced Studies is preparing a brand new journal" Exploration ". I plan to include your paper in the inaugural issue, but the prerequisite is Fill in the loopholes of the paper within two months. And this August is the St. Petersburg International Mathematics Conference. As long as the proof of this conjecture is completed, we can take it to the Mathematics Conference and communicate with colleagues around the world ... "

Perelman said: "Pang, to be honest, this time I feel that my research has entered a dead end. It may be old, physically and mentally a bit behind. If you are interested in Hodge conjecture , You can continue to do the next proof on my previous paper, without asking for my opinion. As far as I can prove the Hodge conjecture, it doesn't matter who proves. "

Pang Xuelin said for a moment, "Gregory, thank you. But I still hope you don't give up the research on Hodge's conjecture, after all, mathematical research is a pleasure in itself, isn't it?"

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

...

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

He's a little distracted.

With Perelman's temperament, he can really help him complete the proof of Hodge's conjecture, he will only be happy, and there will be no extra thoughts at all.

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

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

Perelman was unable to determine whether the spatial curvature of the T2 gram Rhein bottle with a directional double cover as a torus was a smooth function on the Riemannian manifold.

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

To prove Hodge's conjecture, it is impossible to bypass it.

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

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

But this is not the case.

The fact is: the Klein bottle is a curved surface that can only be truly expressed in a four-dimensional space. If we must express it in the three-dimensional space in which we live, we will have to click on it and have to behave as if we are ourselves Intersect with yourself.

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 bottle wall.

For example, use a three-dimensional kink.

If it is regarded as a curve on a plane, then it seems to intersect itself, and it seems to be broken into three sections again.

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

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

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

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

In our three-dimensional space, even the most brilliant craftsman has to make it look like it intersects; just like the most brilliant painter, when painting kink on paper, he has to draw them as intersecting himself. appearance.

As long as the space curvature of the T2 Klein bottle with directional double cover as a torus is proved to be a smooth function on the Riemannian manifold, the last loophole of the Riemann conjecture proof can be filled.

The question is, how do you prove it?

The next time, Pang Xuelin entered the research state again.

However, this time, he did not conduct a special retreat.

He is confident that he can complete the last piece of Hodge's conjecture in two months even if there are other business disturbances.

In the past two months, Pang Xuelin ~ lightnovelpub.net ~ has begun to set up the editorial department of the "Exploration" journal.

He himself is the editor-in-chief, and the editorial board is also a gangster-level figure 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 brought in by Pang Xuelin.

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

Basically every night until 1 or 2 in the morning.

Time passed day by day, before unknowingly, it entered June.

That night, Pang Xuelin was meditating in the study, and Qi Xin carried in a bowl of Tremella lotus seeds and red dates.

Pang Xuelin heard the movement, and looked up and smiled: "Xiao Xin, go to bed early, you don't have to stay so late every night."

During this time, Pang Xuelin went to bed at one or two in the morning every morning. Qi Xin was distressed by Pang Xuelin. After eleven o'clock every night, he changed his way to give him some tonic soup.

Qi Xin grinned and said, "I'm fine. You can drink soup first, and then it will be cold."

"Okay, I'll drink right away."

Pang Xuelin took the Tremella fuciformis from Qi Xin and talked with her while drinking.

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

Qi Xin curiously.

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

Pang Xuelin shook his head and sighed: "There is still no clue, time is too fast, and fifteen days, the first issue of" Exploration "will officially enter the printing house to print, and now 80% of the manuscripts of the first issue have been identified. This paper is left unfinished. "

Qi Xin frowned. "It's not working, just take it easy."

Pang Xuelin said: "Try your best. If Hodge's conjecture is really unsettled, I will go to the Ponzi Geometry Workshop to get a essay that can pass the level."

"Well, then I'll take a shower first. I have to get up early for class tomorrow, and you should go to bed earlier. If you can't find a breakthrough, change your mind or start with reverse thinking. Maybe it can be solved."

Pang Xuelin froze slightly, "What did you say?"

"I said to change my mind or start with reverse thinking?"

Pang Xuelin's eyes flashed inexplicably, and he rushed forward and hugged Qi Xin and kissed him fiercely, laughing: "Xiao Xin, you are my lucky star."

Then, he stopped paying attention to Qi Xin and sat down at the desk, picking up his pen and calculating quickly.

Qi Xin looked at Pang Xuelin, who was immersed in the research, and 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 is moving fast.

Qi Xin's reverse thinking, like a light, illuminated the vast night sky ahead.

Why do we have to prove that the directional double-covered T2 gram rhine bottle is toroidal, and its spatial curvature is a smooth function on the Riemannian manifold?

It is only necessary to prove that the T2 Klein bottle with a directional double-covering non-annular surface, and its spatial curvature is not a smooth function of the Riemannian manifold?

The last two months of contemplative thinking, as if the firewood has dried up, just a spark can ignite the blazing flames.

The pen tip in Pang Xuelin's hand moved quickly on the paper.

[Let M be an n-dimensional smooth manifold. If a smooth second-order covariant tensor field g is given on M, call (M, g) an n-dimensional Riemannian manifold, and g is called the basic tensor of the Riemannian manifold. Or Riemannian metric, if: 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 only holds when X = 0 ...]

...

[Let D be a two-cycle directed tree graph, D0 = ｛e1, e2｝ is the set of vertices, D1 = ｛a, c｝ is the set of directed edges of length 1, is the path algebra of D on the domain K, then H0 (C) ≌K⊕K, H1 (C) = 1 ……】

...

[If e: K → M satisfies ~ lightnovelpub.net ~ 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 the ideal with zero squared in A, and E / M≌A, then we can study the algebraic extension E obtained by M through M by H2 (A, M) being trivial ...]

...

The time passed minute by minute. Unconsciously, the hour hand pointed at one o'clock in the morning.

Looking at the mathematical symbols on the manuscript paper, Pang Xuelin's smile was relieved.

"The efforts of the past two months have not been wasted, Hodge guessed, and finally proved it."

Pang Xuelin put the manuscript paper neatly aside and got up and walked out of the study.

The living room was dark, and 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 first took a bath for half an hour to relax the muscles in the whole body. Then, he dried his hair, came to the bed, quietly lifted the quilt, and got into the bed.

Pang Xuelin was preparing to sleep, and a white armband came over.

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

Pang Xuelin froze slightly, looking at Qi Xinmei's face and said, "Xiao Xin, did I just wake you up?"

Qi Xin shook her head and asked, "You got the problem?"

Pang Xuelin smiled and nodded.

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

"Well……"