February 16, 2022.

The sixteenth day of the first month.

The Lantern Festival has just passed.

The campus of Jiang University, which had been silent for a month, became noisy again.

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

Immediately afterwards, Pang Xuelin's office door was pushed open with a bang.

Perelman, who he hadn't seen for a long time, hurriedly barged in.

Zuo Yiqiu also followed closely and chased after him, saying, "Professor Pang is sorry, I didn't stop this gentleman......"

Pang Xuelin was slightly stunned, laughed, and said, "It's okay, Xiao Zuo, you go out first." ”

Then he turned his gaze to Perlman and said, "Grigory, is there anything you are looking for me?" ”

Perelman looked unkempt, with a big beard, curled hair draped behind his head, looking greasy, and he didn't know how long he hadn't washed it.

He wore a brown jacket with pitch-black cuffs.

Pang Xuelin hadn't seen Perelman for nearly four or five months, and the last time the two met was at the inauguration ceremony of the Pang Xuelin Mathematical Science Research Center of Jiangcheng University.

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

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

Perelman waved the manuscript paper in his hand and looked excited.

"Proof Hodge's conjecture?"

Pang Xuelin was slightly stunned, he knew very well the difficulty of Hodge's conjecture.

In the interstellar world, when he was trapped on Planet 5 by the tree elder, he spent more than half a year tackling this conjecture, but he never succeeded.

He didn't expect that in the real world, Perelman would solve this conjecture.

"I'll see."

Perelman handed the manuscript paper in his hand to Pang Xuelin.

Pang Xuelin's manuscript paper began to flip through the pages page by page.

Perelman was not in a hurry, and sat down on the small sofa next to him.

After a while, Zuo Yiqiu came in with a cup of steaming coffee and placed it in front of Perelman.

Then, Zuo Yiqiu quietly closed the office door.

After reading for nearly an hour, Pang Xuelin put down the manuscript, pondered for a moment, and said, "Your proof method is a bit interesting, have you shown your manuscript to Shinichi?" ”

Just now, Pang Xuelin has gone through this manuscript and probably clarified Perelman's proof idea. ，

However, the specific proof process still needs to be carefully studied.

"Not yet."

Perelman shook his head.

Pang Xuelin said: "I'll also find Professor Shinichi Mochizuki and let him take a look." ”

As he spoke, Pang Xuelin picked up the phone on the table and dialed Mochizuki Shinichi.

Half an hour later, Shinichi Mochizuki hurriedly came to Pang Xuelin's office.

Seeing that Perelman was also there, Shinichi Mochizuki showed a look of surprise on his face: "Say, Grigory, why are you here?" ”

Immediately after, Shinichi Mochizuki seemed to think of something, with an incredible look in his eyes, and said, "You shouldn't have solved the Hodge conjecture, right?" ”

Perelman has been in retreat all this time, and he knows it.

Today he suddenly came to find Pang Xuelin, and after Pang Xuelin called him, Mochizuki Shinichi guessed Perelman's intention at once.

Perelman nodded and did not speak.

Pang Xuelin laughed and said, "Shinichi, this is Perlman's proof manuscript of Hodge's conjecture, you can also take a look, is there any problem?" ”

As he spoke, Pang Xuelin handed the copy of the manuscript that had just been copied and had a hint of warmth to Shinichi Mochizuki.

Just in the process of Mochizuki Shinichi coming over, Pang Xuelin copied the manuscript aside.

"Good!"

Shinichi Mochizuki was not polite, took the manuscript, found a chair and sat down opposite Pang Xuelin.

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

There was silence in the office.

Both Pang Xuelin and Shinichi Mochizuki were poring over Perelman's manuscript.

Perelman, for his part, sipped, coffee.

He was a very patient man, and even if no one spoke to him, he could sit alone and stay all day.

Time passed minute by minute, and when it was close to noon, Pang Xuelin found Zuo Yiqiu and asked her to help the three of them order three takeaways.

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

According to Perelman's ideas, Pang Xuelin tried to deduce the entire proof process of Hodge's conjecture from beginning to end.

Before I knew it, it was more than three o'clock in the afternoon.

Shinichi Mochizuki finally raised his head and said, "I don't think there's anything wrong with the overall idea, but the details need to be further studied." ”

Perelman couldn't help but breathe a sigh of relief, a smile appeared on his face, and he turned his gaze to Pang Xuelin and said, "Professor Pang, what do you think?" ”

Pang Xuelin didn't speak, he pondered for a moment, and said loudly: "Grigory, come over for a moment." On the fifth page of the manuscript, in Lemma 3.3.4: ?? Is it defined in the Riemannian manifold?? A smooth function with no critical point on the region Ω in 4. In the regional Ω?? The fastest descent line is the orthogonal curve of the horizontal set. In other words, there is no critical point function?? The fastest descent line is the tangential vector field within the region???? The integration curve. Here how are you going to solve the level set and the curvature of the fastest descent line? ”

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

【Set {???? 1，???? 2} is the unit orthogonal cut frame, if ???? 1 is the unit tangent vector of the curve, so the geodesic curvature of a smooth curve is ?? =, where?? is the arc length parameter of the curve. by {???? 1，???? 2} is the unit orthogonal tangent scale, and the geodesic curvature can also be expressed as?? =?? =?? div(???? 2), which is equivalent to saying that the geodesic curvature of a smooth curve is a differentiation of the unit normal vector of the curve. 】

Pang Xuelin smiled faintly, and his explanation of Perelman was undeniable, and he turned to the tenth page, pointed to the proof above: "Then here, in the form of space???? Middle?? Is it defined in strict convex ring?? 2???? Harmonic function on 1,?? Consecutive to?? 2???? 1。 If?? Satisfied??|???? 1= 1，??|???? 2=0, then, there is |????|(??) >0，???? ∈?? 2???? 1, and?? The horizontal set is strictly convex. How do you give the principle of extremums in the last section? ”

Perelman went on to explain: [Ω is???? Bounded Connectivity Area,?? ∈?? 2(Ω)?????? (Ω) to consider operators on Ω?????? =?????? (??)???????? +???? (??)?????? +?? (??)?? ……】

"And what about here??? is a Riemann manifold with constant cross-sectional curvature???? smooth function on ,???????? And???? They are???? on the Riemannian curvature tensor and the Ricci curvature, then?????? =???????? +?????????????? And???????? =???????????? 2???????????????? +???????????? +R?????????? …… How can this be proven? ”

【Take 1 ≤??,??,??,??,?? ≤??， 1 ≤?? ≤?? + 1。 Take???? in the orthogonal frame field {???? 1，???? 2，……，??????，?????? +1}, where ?????? +1 is the outer normal, then {???? 1，???? 2，……，???? i} is the cutting frame field, and ???? =?????? +1, the equation of motion is ......]

……

Shinichi Mochizuki, who was watching from the sidelines, was a little strange, why did Pang Xuelin always spin around on the Riemann manifold problem, and asked some relatively simple questions, some lemmas or definitions, and the deduction was very obvious.

However, Perelman did not show much impatience, basically Pang Xuelin asked what he explained.

Time passed minute by minute, and before I knew it, more than an hour had passed.

Pang Xuelin finally figured out the dagger: "You here from a compact and boundless n-dimensional manifold M of the homology group Hn(M,Z)=0, deduced that M is not orientable, and then we can see from theorem 4.6.7 that all even-dimensional projective spaces are non-orientable, and their directional double coverage space is a sphere of the same dimension, so I want to ask, is the spatial curvature of the Klein bottle with a directional double coverage of the torus T^2 a smooth function on the Riemannian manifold?" ”

As soon as Pang Xuelin said this, not only Perelman was stunned, but even Mochizuki Shinichi was stunned.

This is an extremely subtle logical loophole, from the initial setting all the way to the orientation problem of the four-dimensional Klein bottle, which is equivalent to the basis of the whole process of proving the Hodge conjecture.

If there is a problem with this paragraph, it basically means that the entire proof process is seriously flawed.

But that's not what shocked Shinichi Mochizuki.

It was Pang Xuelin who was able to detect such a subtle logical loophole in such a short period of time.

You must know that Perelman's manuscript is more than 30 pages in total, and he also omits many links, and if this part of the manuscript is converted into a thesis, at least more than half of the content must be added.

It had taken Shinichi Mochizuki nearly five hours to read the paper carefully.

If you want to understand, Shinichi Mochizuki can only say that he understands Perelman's overall proof idea, and he will spend a few days studying some of the details in it.

While reading this paper, Pang Xuelin completely understood Perelman's proof idea in such a short period of time, and even found very subtle loopholes in it.

The amazing thinking ability and mathematical intuition displayed in this are somewhat beyond the imagination of Mochizuki Shinichi.

In general, there is not much difference between top mathematicians like Perelman and Shinichi Mochizuki in terms of thinking ability alone.

What really shows the gap between mathematicians is whether they have creative thinking and can open up new battlefields in areas that others can't think of.

And this point requires a long period of accumulation and an occasional flash of inspiration.

Mochizuki Shinichi originally thought that even if there was a gap between himself and Pang Xuelin, at least in terms of logical thinking ability, there was no qualitative difference.

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

Where the hell did this monster come from?

Perelman was aware of this, but he didn't think much of it at this point.

He took the manuscript of the paper from Pang Xuelin's hand and deduced it from beginning to end.

The final result proved that Pang Xuelin was right.

Perelman's face could not hide the look of loss, after all, it took so much effort, but in the end, because of a small loophole, the previous efforts were lost, which is really a little unacceptable.

However, he quickly adjusted his mentality.

In the world of mathematics, it is normal for a research result to be found with loopholes.

Just like Andrew Wiles, when he proved Fermat's theorem, he was also singled out by the academic community for loopholes.

It only took him another year to close the loophole before he proved Fermat's theorem.

Shinichi Mochizuki is even better at this.

At the beginning, in order to prove the ABC conjecture, he invented a set of Tessimüller theory of the universe, but no one in the academic community could understand it, and he argued for more than ten years.

If it weren't for Pang Xuelin's later appearance to prove this conjecture, maybe Mochizuki Shinichi would still be arguing with people in the mathematical community.

"Pang, if there is nothing else, I'll go back first, I have to think about whether there is a way to remedy this loophole."

The three chatted for a while, and Perelman took the initiative to leave.

Seeing Perelman's back disappear behind the door, Shinichi Mochizuki was curious: "Pang, do you think Perelman can prove Hodge's conjecture?" ”

Pang Xuelin shook his head and said: "I don't know, it depends on whether Perelman can fill that loophole, at least in the overall direction of thinking, I don't think there is any problem." By the way, how has your research been during this time? ”

Since the ABC conjecture was proven, Shinichi Mochizuki has shifted his research direction to the field of continuum.

The so-called continuous potential, which is very simple to formulate, refers to how many real numbers are in the set of real numbers? In other words, what is the potential of a set of real numbers?

The problem of continuum potential determination is one of the oldest, most basic and most natural problems in set theory.

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

It is well known that 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 equipotential of two finite sets is that they contain the same number of elements.

Thus, the potential of each finite set is uniquely determined by a natural number.

Similarly, the potentials of infinite sets are uniquely composed of a cardinality?? α to be sure.

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

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

In general, immediately followed by the cardinal ?? What is the base after α?? α+1: Two bases?? α and?? The comparison of the magnitudes of β is uniquely determined by the length of their subscripts (ordinal α and β).

Every natural number n is a ratio?? 0 small cardinality. For infinite cardinality,?? 0＜?? 1＜?? 2＜?? 3＜……

In December 1873, Cantor proved that the potential of a set of all real numbers (i.e., the continuum) is at least ?? 1。

Now the question arises: which cardinal base exactly? α is the potential of the continuum?

Be?? 1?or?? 2，?? 3, or something else?? α？

Cantor once conjectured that the potential of the continuum was the first uncountable cardinality?? 1。

This is the Cantor continuum conjecture, the first of the 23 questions Hilbert asked in 1900.

Shinichi Mochizuki shook his head and said with a wry smile: "I just have a clue now, and it will take a long time to really understand this problem." ”

Then, Shinichi Mochizuki chatted with Pang Xuelin about the recent Ponzi geometry seminar before he said goodbye and left.

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