Chapter 289: Coming to Hodge’s Conjecture

For Oleyevich, he knew very well that Lin Xiao's report was of great importance to the study of Hodge's conjecture.

In the past, both forms of the integral Hodge conjecture had certain errors, which meant that mathematicians had found counterexamples. Therefore, mathematicians are still pursuing a brand-new integral Hodge conjecture that can perfectly combine Hodge's conjecture. Guess it in a nutshell.

Now in Lin Xiao's report, he used a very rigorous discussion process to propose this new motivated cohomology person, and then came up with this new integral Hodge conjecture. From this point of view, it is enough for Ole Yevich trusted it very much, and later Lin Xiao also used a proof by contradiction to prove that his integral Hodge conjecture was completely correct and carried all the information of the original Hodge conjecture.

"Loss of information" is a common situation that hinders the development of mathematics, and it usually occurs in changes in form.

For example, Lin Xiao's Lin's theorem solves the change from function to layer. In the past, not all functions could be converted into layers because part of the information would be lost during the conversion process. In other words, some of the information would be lost. Information will be lost after the transformation.

The form of incomplete information will naturally have an impact on the process of solving mathematical problems. Therefore, how to solve this problem requires the development of new tools.

For Lin's theorem, it is a tool to achieve lossless transformation between functions and layers.

In the process of transforming the Hodge conjecture into the integral Hodge conjecture, it is necessary to take advantage of the fact that cohomology can be converted into differential form. However, due to the limitations of tools, this conversion process will lead to the loss of some information, so it is This led to the errors of the two integral Hodge conjectures in the past.

But now, Lin Xiao's new motivation cohomology has once again solved such a fatal problem, allowing the original "information" to be transformed in two forms without loss.

Therefore, Orejevich was very amazed by Lin Xiao's report. Undoubtedly, this was a very remarkable achievement.

The importance of this report cannot be overstated.

"I'm really looking forward to Professor Lin's report."

Oleyevich shook his head, then looked at the end of the report, and suddenly he was stunned.

At the end, there are several lines of mathematical formulas.

"Huh? Why is there something at the end... Professor Lin hasn't finished it yet?"

With some doubts, he looked at it a few more times.

After a while, his eyes suddenly showed a solemn color.

"Is this...Professor Lin actually starting to prove this conjecture?"

The few steps at the end clearly prove the beginning of the integral Hodge conjecture.

But even though it was just a few simple steps, it was enough to arouse the imagination of a mathematician like Oleyevich.

According to the steps given by Lin Xiao, if we continue, maybe we can achieve some important phased results?

At least for now, Lin Xiao's last few lines of expressions already have such potential.

Not to mention the actual proof, but the staged results are also very important in mathematics. For example, Perelman was able to prove the Poincaré conjecture because of a mathematical tool called Ricci flow, which played a very important role. Important, this also led Perelman to publicly say that the mathematician who created the Ricci flow, named Richard Hamilton, made a greater contribution than him.

But even so, but——

"Why are there only so many steps?"

For the first time, Olejevich felt the abomination of the broken dog, and even felt like sending a blade to Lin Xiao.

This can obviously be written in the future. At least with Lin Xiao's ability, I think it is definitely possible to write in the future.

But he got stuck at such a crucial step and stopped writing.

Olejevich scratched his hair, feeling itchy.

I believe that no mathematician can resist the temptation to prove the Hodge conjecture, or in other words, any of the seven millennium problems. The $1 million reward offered by the Clay Institute alone is enough to convince many mathematicians. I'm so greedy.

Mathematicians are often not that wealthy, so they were surprised when they learned that Perelman did not even accept the $1 million.

And beyond that, not to mention the fame it brings and the need for self-actualization, which is, after all, a millennium problem.

So Oleyevich really wanted to know how to write after this crucial step.

"What is this Professor Lin doing?"

But no matter how doubtful he was at this moment, no matter how hard he thought about it, he probably never imagined that Lin Xiao wanted to recreate the process of Andrew Wiles proving Fermat's last theorem.

If you can’t figure it out, then don’t think about it.

So Orejevich stopped taking care of it and put the report in the archives. After it is reviewed later and it is completely determined that there are no problems with the content of the report, it can be uploaded to their 2022ICM official website.

Orejevich could even imagine that when other mathematicians downloaded Lin Xiao's report and read it, they would, like him, hate Lin Xiao's step that was stuck at a critical point.

Maybe, after Lin Xiao finishes his report in his speech, the audience will make a fuss.

Olejevich is already looking forward to that time.

Shaking his head and putting the matter behind him, he was about to start other work, but at this moment, someone suddenly called him.

When he saw it, he saw that it was his friend Zorelovsky, who was a professor of mathematics at St. Petersburg State University.

St. Petersburg State University is second only to Moscow State University in Russia, and its status is equivalent to Qingbei in China.

Although in the so-called international university rankings, St. Petersburg State University's ranking can only compete with the 985 schools outside Qingbei, China, its own strength cannot be objectively expressed by those international rankings, because these international rankings are all Paying attention to internationalization, in layman's terms, is the requirement for English. Therefore, in the QS university rankings, universities in English-speaking areas are basically at the forefront.

Because of this, St. Petersburg State University is actually very strong.

At the same time, Russia’s basic mathematics education is also very strong, and St. Petersburg State University has naturally inherited this strength. Even Perelman graduated from St. Petersburg State University.

It is precisely because of this that Zorelovsky is also quite famous in the mathematics community.

After answering the phone, he smiled and said, "Brother, why are you in the mood to call me?"

Zorelovsky on the opposite side did not greet him, but said directly: "Did you know? Lin Xiao from China has decided to hold a report on the proof of Lin's conjecture at their Beijing University. Do you plan to go?"

Oleyevich was stunned, "Proof of Lin's conjecture?"

Of course, he is also very interested in this matter. As a person who studies algebraic geometry, he is certainly not unfamiliar with Lin's theorem. In some of his recent research, he has used research on Lin's theorem.

But then he said helplessly: "I am currently in charge of the International Congress of Mathematicians, so I probably won't have time."

"That's right, that's true." Zorelovsky said: "This report will be held on June 30th. I guess you really don't have time."

"Um...wait, you mean June 30th?"

Suddenly, Oleyevich sat up directly. When he heard Zorelovsky say that Lin Xiao planned to give a report on June 30, he suddenly felt something was wrong.

A week after June 30th, it will be the International Congress of Mathematicians, and there will be a one-hour report by Lin Xiao.

So what is Lin Xiao planning to do?

Two reports in ten days?

No, there’s also the Fields Lecture.

Although he, the chairman of the organizing committee, cannot evaluate the Fields Medal, which is decided by the International Mathematical Union, he can still get some information. Lin Xiao has a 10-10 chance of winning the Fields Medal. big.

This made him fantasize even more.

Lin's guess? Integral Hodge conjecture?

There are also several steps that Lin Xiao tried to prove Hodge's conjecture at the end of the report.

Oleyevich always felt that Lin Xiao wanted to cause something big.

"It's June 30th, what's wrong?"

On the other end of the phone, there was a question from a friend.

Olejevich came to his senses, and then said with a smile: "It's nothing, I just think that this year's mathematics world may be unusual."

Zorelovsky said with a smile: "Professor Lin proved his own Lin's conjecture in January. This alone is enough to make this year's mathematics world extraordinary."

Olijevich shrugged: "Well, what I want to say is that this year's conference should not be too ordinary."

"Really? Then I'm looking forward to it."

"Well, look forward to it." Oleyevich said: "I'm looking forward to it too."

…

Time flies.

April and May were spent in peace, and June arrived on time.

In Lin Xiao's room.

After being in seclusion for almost three months, Lin Xiao's hair now reaches her eyebrows, collar and ears, and even her beard has grown.

The young man who was originally twenty-two years old seemed to have turned into a sloppy uncle.

Of course, this is also because his classes at school have been completed before the ninth week of the semester, so he has not been able to go out.

However, despite being in seclusion for so long, he still has not completed the final proof of Hodge's conjecture.

"Only the last bit left..."

He rested his chin on the table with one hand, and many thoughts ran through his mind.

[Flash of Inspiration] has not been used yet because he has not yet grasped the last and most critical idea.

"Convert a shape into small pieces in integral form, and then subject them to topological transformation. Then how to describe its homeomorphism in a polynomial way?"

Lin Xiao scratched his head, and the thick feeling told him that his hair was still there, and it was still thick, but it was a bit long.

He took a long breath, then stood up and twisted his body to relax his somewhat stiff body.

Then look at the time.

"It's already June 15th."

There is still half a month left before the report on June 30. Similarly, there is only less than a month left before the speech at the International Congress of Mathematicians.

Shaking his head, this won't work.

Now it seems that this car called Hodge's conjecture cannot be built behind closed doors.

"Let's go to school for a walk. It's been a long time since we've been there."

With this thought in his mind, he no longer hesitated. After staying in the room for so long, people were about to get moldy, so it was time to go out for a walk.

Afterwards, he left the house and called Shi Hao at the opposite door. Then Shi Hao drove him to Beijing University.

"Just find a parking lot and walk around the school."

Entering the Beijing University campus, Lin Xiao spoke.

"Okay, Professor Lin." Shi Hao responded.

Quickly finding a parking lot, Shi Hao parked the car, and the two of them got out of the car and started walking around the school.

Because it happened to rain in the afternoon, and it was already evening, the temperature was no longer so hot, and it was a good time to go out for a walk.

Lin Xiao, who looked like a slovenly uncle, was walking around the campus. Fortunately, he didn't get to the first floor before being called out by the students, allowing him to relax a little.

In this way, they walked all the way to the Weiming Lake. Looking at the slightly wavy water, Lin Xiao was not thinking about any mathematical problems at this time, but was just sighing at the beauty of the natural scenery.

Then, he curiously asked Shi Hao: "If you become my bodyguard, how long will you stay?"

Shi Hao replied: "Until I can no longer protect you."

"What happens when you can't protect me?"

"When my physical fitness declines, or I become disabled and lose my ability to move normally." Shi Hao's tone seemed to always be so confident.

Lin Xiao was helpless and said: "There is no deadline? Three or four years?"

"It depends on the organizational decision."

Lin Xiao shook his head and was about to speak when suddenly there was a burst of exclamation from behind.

"Holy shit! Awesome!"

"6666!"

Lin Xiao was stunned, and turned around to look, only to see a bunch of students watching a boy playing with a toy that everyone was familiar with. It looked like a colorful plastic spring, and its official name was Slinky.

And this boy played this thing very well, like an acrobat, which made many students around him look at him in surprise.

But compared to the surprised looks of these students, a bunch of mathematical formulas appeared in Lin Xiao's eyes.

Isn't this thing just a classic one-dimensional manifold topological body?

Seeing the colorful spring beating like a smart ghost in the student's hand, Lin Xiao's eyes gradually lit up, because he grasped the solution to Hodge's conjecture from this one-dimensional manifold. The essential.

Hodge conjecture studied the algebraic representation of topological bodies in three dimensions, and now he has found a similar answer in this one-dimensional manifold.

This is probably the magic of mathematics.

Looking at the boy, Lin Xiao wanted to thank him for the inspiration he brought to him, but at this time, he discovered that the boy was still a student he had taught before.

He couldn't help but walked up and shouted: "Wenmin, have you finished writing your thesis?"

The boy was startled. He thought he had met his mentor, but when he turned around, he saw a slovenly man walking towards him. Then he breathed a sigh of relief. Fortunately, he was not his mentor.

However, he stared at Lin Xiao for a while, and finally recognized this former teacher, and then was extremely surprised: "Damn, Professor Lin, what's wrong with you?"

At this time, the people around him also stared at Lin Xiao for a long time before they recognized him as Lin Xiao, and they were also surprised.

How did Professor Lin become like this?

Lin Xiao waved his hand and said, "Don't worry about it. Where did you buy this toy? Is it sold near the school?"

When Wen Min saw Lin Xiao talking about the toy in her hand, she said, "Oh, this, I bought it online. I don't know about it near the school."

Then he quickly handed the thing to Lin Xiao and said with a smile: "If you want it, I will give it to you. If I get drawn to you during the defense, you have to let me go."

Lin Xiao couldn't help but laugh, took the Slinky, played with it a few times, and then said: "You are bribing the teacher in public. This is not okay. I will keep this stolen goods for now. You can write the paper well for me. No." If you know how, you can ask me. As for letting you go, don't even think about it."

The students around laughed, and Wenmin said in a longer voice: "Ah~"

Lin Xiao smiled and patted him on the shoulder: "Goodbye, I told you if you don't understand, you can ask me, do you understand the value of Professor Lin's personal guidance?"

Wen Min was taken aback for a moment and suddenly smiled: "Yes! This is what you said! I remember it."

The students around him suddenly showed envious expressions.

If you give a toy, you can get personal guidance from Professor Lin. They also want such a good thing!

Lin Xiao smiled and then said: "Okay, that's it. Thank you for the toy. You can play with it slowly."

Then, he left the crowd of students and said to Shi Hao: "Let's go back!"

"yes!"

Although Lin Xiao didn't understand that Lin Xiao had only been out for less than an hour, and then took a toy from a young student's hand and planned to go back, Shi Hao always obeyed orders.

Afterwards, the two returned to the parking lot and then back to Yanbei Garden.

Returning directly to his room, Lin Xiao put Slinky on the table and played with it for a while. Mimi next to him was also curious about this moving thing, and finally took it over and started playing with it by herself.

Lin Xiao couldn't help but smile, and then let Mimi play by herself, while he picked up the pen again and wrote an equation describing Slinky on the scratch paper.

Recalling the process of Slinky's movement and the model of its rules, he smiled slightly.

"Hodge guesses it's officially over."

Then, "Use [Flash of Inspiration]."

System: "[Flash of Inspiration] has been used."

The next moment, Lin Xiao grasped the idea and found the inspiration he needed.

Then, his pen wrote smart mathematical formulas on the paper.

One of the seven major mathematical problems of the millennium, Hodge guessed that one of the most enchanting girls in mathematics began to take off her veil and clothes, waiting for the genius to arrive.