February 16, 2022.

I am sixteenth on the first month.

The Lantern Festival just passed.

Quiet for a month, Jiang University campus became noisy again.

Early in the morning, Pang Xuelin had just arrived in the office, and there was a sudden noise outside.

Immediately afterwards, Pang Xuelin's office door was slammed open.

雷 Perelman, who had not seen him for a long time, hurried in.

He Zuo Yiqiu followed up from behind and said, "Professor Pang is sorry, I did not stop this gentleman ..."

Pang Xuelin smiled slightly, and said, "It's all right, Xiao Zuo, you go out first."

Then he turned his attention to Perlman, "Grigory, is there anything wrong with me?"

Perelman looked unkempt, with a beard, and curled hair covered his head. He looked greasy and didn't know how long it hadn't been washed.

穿 He wore a brown jacket with dark cuffs.

Pang Xuelin hasn't seen Perelman for nearly four or five months. The last time the two met was at the inauguration ceremony of Jiangcheng University Pang Xuelin Mathematical Science Research Center. Perelman came over to show his face, and then hurriedly Left.

For more than a year, he devoted all his energy to the research of Hodge conjecture.

"Professor Pang, I prove Hodge's conjecture!"

Perelman waved the manuscript paper in his hand, his expression exhilarated.

证明 "Prove Hodge's conjecture?"

Pang Xuelin took a moment's notice, and the difficulty of Hodge's conjecture was clear.

穿越 In the interstellar world, when he was trapped by Shu Lao on the fifth planet, he spent more than half a year tackling this conjecture, but he never succeeded.

He never expected that, in the real world, Perelman had solved this conjecture.

"Let me see."

Perelman handed the manuscript paper to Pang Xuelin.

He Pang Xuelin drafted the paper and started to page through it.

雷 Perelman was not in a hurry, and Ma Dadao sat down on the small sofa beside him.

After a while, Zuo Yiqiu came in with a steaming cup of coffee and put it in front of Perelman's life.

Subsequently, Zuo Yiqiu quietly closed the office door.

Looking at it for nearly an hour, Pang Xuelin put down the manuscript and groaned for a moment, and said, "You have a little proof of this proof method. Have you read the manuscript?"

Pang Xuelin just browsed ~ lightnovelpub.net ~ this manuscript, and probably clarified Perelman's proof of proof. ,

However, the specific proof process requires careful study.

"not yet."

Perelman shook his head.

He Pang Xuelin said, "I brought Professor Mochizuki Shinichi, and let him follow along."

He said, Pang Xuelin picked up the phone on the desk and dialed Wang Yuexin.

Twenty-five hours later, Mochizuki hurried to Pang Xuelin's office.

Seeing Perelman's presence, Mochizuki's face looked surprised: "Say, Grigori, how are you here?"

Immediately afterwards, Mochizuki Shinichi seemed to think something, with an incredible look in his eyes, saying, "Should you solve Hodge's conjecture?"

Perelman has been in retreat for this time, he knows.

Today, he suddenly came to find Pang Xuelin, and in addition to Pang Xuelin, called himself, Mochizuki guessed Perelman's intention all at once.

Perelman nodded and said nothing.

Pang Xuelin laughed and said, "New one, this is Perlman's proof manuscript about Hodge's conjecture. You can also take a look. Is there any problem?"

He said, Pang Xuelin had just copied it, and handed a warm copy of the manuscript to Mochizuki Shinichi.

Just as Mochizuki came over, Pang Xuelin copied the manuscript aside.

"it is good!"

Despising Wang Yuexin was also polite, took the manuscript, found a chair and sat down opposite Pang Xuelin.

Xi Pang Xuelin also took out a manuscript paper and wrote on it.

The office was quiet.

Pang Xuelin and Mochizuki Shinichi are carefully studying the manuscript of Perelman.

Perelman himself drank coffee with pleasure.

He is a very patient man, even if no one talks to him, he can sit all day alone.

Every minute and a second passed, and at noon, Pang Xuelin called Zuo Yiqiu and asked her to order three takeaways.

After eating, Pang Xuelin and Mochizuki continued to study Perelman's manuscript.

According to Perelman's ideas, Pang Xuelin tried to deduct the entire Hodge conjecture from beginning to end.

Unconsciously, it was more than three in the afternoon.

Wang Yueyue finally raised her head and said, "I feel that there is nothing wrong with the overall idea, but the detailed inference needs further study."

Perelman could not help but breathe a sigh of relief, with a smile on his face, and turned his eyes to Pang Xuelin: "Professor Pang, what do you think?"

Pang Xuelin didn't speak, he groaned for a moment, and said, "Grigory, come here. In the fifth page of the manuscript, Lemma 3.3.4: ?? is defined on the region Ω in the Riemannian manifold ?? 4. Smooth function without critical point. The fastest descending line in the region Ω is the orthogonal curve of the level set. In other words, the fastest descending line without the critical point function is the tangent vector field in the region. Integral curve. How are you going to solve the level set and steepest descent curve curvature here? "

Perelman pondered for a moment, picked up his pen, and wrote on the manuscript paper:

[Let {??? 1, ??? 2} be unit orthogonal tangent frames. If ??? 1 is the unit tangent vector of the curve, then the geodesic curvature of the smooth curve is ?? =, where ?? is the curve The parameter of arc length. {??? 1, ??? 2} is a unit orthogonal tangent frame, and the geodesic curvature can also be expressed as ?? =? =? **** (??? 2), which is equivalent to saying that the geodesic curvature of a smooth curve is a derivative of the unit normal vector of the curve. 】

Pang Xuelin smiled a little, could not agree with Perelman's explanation, and turned to page ten, pointing to the above proof: "Here, in the space form ????, ?? is defined in a strictly convex ring The harmonic function on ?? 2 ??? 1, ?? continues to ?? 2 ??? 1. If ?? satisfies ?? | ???? 1 = 1, ?? | ???? 2 = 0, Then, there are | ??? | (??)> 0, ??? ∈ ?? 2 ??? 1, and the level set of ?? is strictly convex. How do you give the principle of extreme value in the last part? "

Perelman went on to explain: [Ω is a bounded connected region in ????, ?? ∈ 2 (Ω) ???? (Ω), consider the operator on Ω ?????? = ?????? (??) ???????? + ???? (??) ?????? + ?? (??) ?? ……】

"What about here? ?? is a smooth function on a Riemannian manifold with constant cross-section curvature. ????????? and ???? are the Riemannian curvature tensor and Curvature, then ?????? = ???????? + ?????????????? and ???????? = ??????? ???? 2 ???????????????? + ???????????? + R ?????????? ...... how about this prove?"

[Take 1≤ ??, ??, ??, ??, ?? ≤ ??, 1≤ ?? ≤ ?? + 1. Take the orthogonal frame field {??? 1, ??? 2, ..., ?????, ????? + 1} in ??????, where ????? + 1 is Outer normal direction, then {??? 1, ??? 2, ..., ??? i} is the tangent frame field, and ??? = ????? + 1, the equation of motion is ...]

Uh ...

I watched the new moon watching new moon a little strangely, how Pang Xuelin always turned around the Riemannian manifold question, and asked some relatively simple questions, some lemmas or definitions, it is very obvious to deduce it.

I was perplexed, but Perelman did not show much impatience, basically Pang Xuelin asked what he explained.

The time elapsed minute by minute, and unknowingly, another hour passed.

Pang Xuelin finally figured it out: "You have here a homogeneous group Hn (M, Z) = 0 of a compact and boundless n-dimensional manifold M. It is inferred that M is non-directional. Then we can see from Theorem 4.6.7 The even-dimensional projective spaces are non-orientable, and their directional double-covered spaces are spherical with the same dimension. Then I want to ask, the directional double-covered Klein bottle with torus T ^ 2, its space Is curvature a smooth function on a Riemannian manifold? "

As soon as Pang Xuelin said this, not only was Perelman stunned, but even Wang Yuexin was still stunned.

This is a very subtle logic loophole. From the initial setting all the way to the orientation of the four-dimensional Klein bottle, it is equivalent to Hodge's conjecture to prove the basis of the whole process.

问题 If something goes wrong in this paragraph, it basically means that the whole certification process has major flaws.

But Mochizuki was not shocked by this.

But Pang Xuelin was able to detect such a small logical loophole in such a short period of time.

I want to know that Perelman's manuscript has more than thirty pages, and he has omitted many links. If this part of the manuscript is converted into a thesis, at least half of the content must be added.

It took me nearly five hours before Mochizuki Shinichi to finish reading this essay.

If you want to understand, Mochizuki Shinichi can only say that he understands Perelman's overall proof of thought. He will spend a few days to study some of the details.

Pang Xuelin, after reading this thesis, even understood Perelman's proof of proof in such a short period of time, and even discovered very subtle holes in it.

的 The amazing thinking ability and mathematical intuition shown here are a bit beyond the imagination of the new moon.

In general, the difference in thinking ability between top mathematicians like Perelman and Mochizuki Shinichi is not large.

What really reflects the gap between mathematicians is to see if the other party has creative thinking and can open up a whole new battlefield in areas that others don't expect.

For this, it takes a long time to accumulate and flash of occasional aura.

Wang Yuexin thought that even if there was a gap between him and Pang Xuelin, at least in terms of logical thinking ability, there was no qualitative difference.

今天 But today, Pang Xuelin's performance is completely beyond his imagination.

哪 Where is this monster?

Perelman also realized this, but he didn't think so much at this time.

He took the manuscript from Pang Xuelin and deduced it from beginning to end.

The final result proves that Pang Xuelin is correct.

雷 Perelman's face is difficult to hide the loss of color, after all, after so much effort, but in the end because of a small loophole, and lost all his achievements, it is really difficult to accept.

But he adjusted his mentality quickly.

In the mathematics world, it is normal for a research loophole to be picked up after a research result comes out.

I was like Andrew Wiles of the year, and when he proved Fermat's Theorem, he had been picked out by the academic community.

He only later proved that Fermat's theorem was proved after he spent another year filling up the loophole.

Look at the new moon is a good player.

证明 In order to prove the ABC conjecture, he invented a set of cosmic Tessie Miller theory. As a result, no one in the academic world could understand it.

If it wasn't for Pang Xuelin's later birth, which proves this conjecture, maybe Mochizuki is still talking with people in the mathematical world.

"Pang, if there is nothing else, I will go back first, I have to think about it, there is no way to remedy this loophole."

The three of them talked for a while, and Perelman left to take the initiative.

Watching Perelman's back disappear behind the door, Mochizuki curiously said, "Pang, do you think Perelman can prove Hodge's conjecture?"

Pang Xuelin shook his head and said, "I don't know, see if Perelman can fill the loophole himself. At least in terms of the overall direction of thinking, I think there is nothing wrong. By the way, how is your research during this time? Anymore? "

被 Since the ABC conjecture was proved, Mochizuki shifted his research direction to the field of continuous power.

The so-called continuum potential is very simple to express. It refers to how many real numbers are contained in the set of real numbers? Or, how big is the potential of the set of real numbers?

Continuum power determination is the oldest, most basic and most natural problem in set theory.

For (infinite) sets, a sufficient and necessary condition for the equipotentiality of two sets is that there is a one-to-one correspondence or bijection between them.

As we all know, natural numbers can be used as a measure of the number of elements contained in a finite set: a sufficient and necessary condition for the equipotentiality of two finite sets is that they contain the same number of elements ~ lightnovelpub.net ~ Therefore, each The potential of a finite set is uniquely determined by a natural number.

Similarly, the potential of an infinite set is uniquely determined by a cardinality? Α.

The smallest infinite cardinality is? 0, which represents the potential of a set of all natural numbers.

? The first cardinality after 0 is? 1, then the first cardinality after that is? 2, then? 3, etc ...

In general, the cardinality immediately after the cardinality? Α is? Α + 1: the comparison of the sizes of the two cardinality? Α and? Β is uniquely determined by the length of their subscripts (ordination numbers α and β).

Each natural number n is a cardinality smaller than? 0. For infinite cardinality,? 0 <? 1 <? 2 <? 3 <...

Tor proved in December 1873 that the set consisting of all real numbers (ie, continuum) has a potential of at least? 1.

Now the question arises: which cardinality? Α is the potential of continuum?

Is? 1? Or 22, 33, or something else? Α?

Tor once conjectured that the potential of the continuum was the first uncountable cardinality? 1.

This is the tor continuum conjecture and the first of the 23 questions raised by Hilbert in 1900.

Xuan Wangyue shook her head and smiled bitterly: "I just have a clue now. It will take a long time to really understand the problem.

Next, Wang Yuexin talked with Pang Xuelin about the problems of the recent Ponzi geometry seminar, and then he left.