Chapter 285 Integral Hodge Conjecture

Lin Xiao probably didn't expect that such an unintentional video would make Asmail think so much.

He didn't pay much attention to the video at first. Anyway, there was no key information leaked in it. At most, it was just a servo motor that could be seen. If others knew about it, they would know it. It would also allow the outside world to guess their current progress. Lest the outside world think that they can't survive without Shan Tianbo.

As a result, he originally wanted to let people outside know that they had made progress, but unexpectedly, Asmail, under the reasoning of a wise man, completely went in the opposite direction.

If Lin Xiao knew about it, he would probably be speechless about it.

Of course, no matter what, whether they can understand or not will have no impact on the normal progress of the After a technology breakthrough is made, it will be directly promoted.

For Lin Xiao, any other trick is just a trick. Directly building their lithography machine is the big road and a conspiracy.

So, just like that, time passed quickly.

Regarding Lin Xiao's drawings, the popularity quickly passed. For everyone, this matter is just an anecdote, and it is no big deal if it is a hot search.

However, Lin Xiao's two papers have become more influential, especially his electronic topological bonding theory, which has increasingly attracted research from relevant scholars, including scientists who previously studied Lin's bonding mechanism. Many of us have devoted their attention to the study of electronic topological bonding theory. After all, Lin's bonding mechanism is just a manifestation of electronic topological bonding theory, and the latter is the essence. Naturally, there are more Many people are willing to invest time into researching this stuff.

Of course, there are many people who continue to study Lin's bonding mechanism, and many people have even joined the study of Lin's bonding mechanism. As for the purpose, it is to use the electronic topological bonding theory to combine Lin's bonding of other elements. Once the mechanism is worked out, it will be a one-area paper, so no one will be excited when faced with such a treasure.

Of course, the theory of electronic topological bonding is very meaningful to the academic community, and now the proof of Lin's conjecture is also very meaningful to the mathematical community. Of course, since Lin Xiaofa posted it on arxiv, he has basically obtained the mathematics The significance of this paper being published now is that people can cite it.

It couldn't be published before because this key paper has not been published. They can't cite it. If they can't cite it, then they can't use the conclusion that "all functions are layered".

As for why Lin Xiao can use it in the electronic topological bonding theory, it is because he simplified the entire proof process to a dozen pages and put it into this paper, so it was used.

Although there is suspicion that he plagiarized himself, with Lin Xiao's reputation and the importance of his two papers, the academic community will only think that Lin Xiao is eager to publish these two papers and will not care too much. Many, they even have to thank Lin Xiao for publishing the two papers as early as possible, so that the academic community can invest time in research as soon as possible.

Just like those scholars who are eager to quote, such as Zheng Ronger’s parents.

"Professor Lin, thank you very much. Our paper has been sent to Science and has entered the peer review process. It should basically be ready for publication."

Listening to Zheng Cheng's voice coming from the other side, Lin Xiao smiled and said, "That's good. Congratulations first."

"Hey, what's the point of congratulations? I still have to congratulate you, Professor Lin. Two papers in the same period were published in the world's top journals. We are so envious."

Lin Xiao laughed, "It's nothing, we just happened to be together."

"Well, next time we go back to Beijing, we will definitely treat you to a meal. Professor Lin, don't refuse."

"No problem at all."

After saying a few more polite words, Lin Xiao hung up the phone and returned his attention to the draft paper on the desk.

Now after about half a month, his research on Hodge's conjecture has become more in-depth. Of course, it is not said that he can completely solve the problem, but at least he has a clearer idea of ​​what problem he wants to overcome.

“If Equations 1 and 2 of §4 are algebraic, their motivation on C will form a semi-simple Abelian category with tensor products, and be the representation category of a certain reservation group scheme..."

"In this case, the relationship between Hodge's conjecture and motivation theory is very close..."

Lin Xiao pinched her chin and fell into thinking.

Motive theory is an idea proposed by Grothendieck at the age of 32. Its purpose is to find a unified theory that can serve as a path between algebraic geometry and various cohomology theories, and to make it a universal theory for everything. The motivation behind homology tools.

Cohomology is an extremely difficult knowledge point in the world of mathematics. As a theory involving a lot of difficult fields in mathematics such as algebraic topology, algebraic geometry, and representation theory, it plays many roles. However, there are many categories of cohomology. , such as singular cohomology, Drum cohomology, etc., and Grothendieck's motivation theory is a universal cohomology theory that attempts to unify these different cohomology theories.

This is in line with Grothendieck's attempt to use algebraic geometry to unify many mathematical theories and thereby realize the Langlands Program's vision.

Because of this, finding such a theory is also called the dream of this mathematical emperor.

Thinking of this great figure in mathematics who died only eight years ago, Lin Xiao couldn't help but sigh. It was a pity that he missed communicating with this great man in mathematics.

If you had to rank all mathematicians, Grothendieck would definitely be ranked first.

Just looking at his title of Mathematics Emperor, you can tell that his achievements in mathematics are even half as good as those of Lin Xiao today.

Of course, after all, Lin Xiao has only been cheating for a few years now, and he has not devoted all his time to the study of mathematics. Now he is only half a point behind, and he is already considered extremely good.

Then, he no longer thought about it. Grothendieck had passed away, but he still had a lot of energy to continue his research.

He lowered his head again and looked at the thing in front of him. If Hodge's conjecture could be related to the Motive theory, then maybe he could start from this aspect.

His eyes moved slightly, and then he continued writing on the scratch paper.

[H2p(X, Q)∩ Hp, p(X)= H2p(X, Q)∩ F p H2p(X, C)]

【……】

Time passed quietly, from the beginning of March to the end of March.

During this period of time, Lin Xiao basically lived in seclusion. Except for occasionally caring about the progress of the X-ray project, he basically devoted the rest of his time to the research of Hodge's conjecture.

But obviously, Hodge's conjecture can be classified as a millennium problem, and even Lin Xiao has no solution for the time being.

On March 28, in Lin Xiao’s house.

"Well, it's such a troublesome thing."

Putting down the pen, Lin Xiao scratched his hair, showing a helpless expression.

"Hotch, Hotch, why did you come up with this kind of problem? Can't you make it simpler?"

However, it is obvious that the difficulty of this kind of problem will not be affected by human will. After all, this kind of thing involves the truth of the universe.

Of course, although it is helpless, it is not a pity. After all, Lin Xiao is studying the Hodge conjecture during this period just to pass the time. If the millennium problems are really so easy to solve, then they are not millennium problems. It's a difficult problem. Lin Xiao spent several months trying to solve Goldbach's conjecture.

At this moment, Lin Xiao's cell phone suddenly rang.

He glanced at it and saw that it was the phone number from Russia.

Russia?

Lin Xiao was stunned for a moment, and then suddenly realized that the call was most likely from the International Congress of Mathematicians.

Like the International Congress of Mathematicians held in Rio de Janeiro, Brazil, the host country formed a conference committee and was responsible for various matters.

Therefore, this International Congress of Mathematicians is also organized by the Institute of Mathematics of the Russian Academy of Sciences.

After connecting the phone, a voice came from the other side: "This is the International Congress of Mathematicians Committee. Is this Professor Lin Xiao?"

"It's me." Lin Xiao responded.

"Hello, Professor Lin Xiao!" The voice on the other side became more respectful: "It's like this. It's less than four months until this International Congress of Mathematicians. According to previous regulations, you are the speaker. People need to submit the report three months in advance. There are still fifteen days left before the deadline for report submission. When do you plan to submit it?"

"Well, it probably won't take more than a week. I'm basically ready."

"Okay, that's good, we all look forward to seeing your report."

"Um."

"Then I won't disturb you. Goodbye."

After hanging up the phone, Lin Xiao shook his head. The deadline was still coming, his deadline.

Of course, he had already made preparations. Since he still had no idea about the Hodge conjecture, there was no need to dwell on the Hodge conjecture. He could just prepare and then put up the proof of Lin's conjecture.

He had almost prepared a report on Lin's conjecture before, and he could send it if he wanted to make any additions.

Of course, people are more or less prone to procrastination, so let’s wait until the seven days he just mentioned before talking.

I believe that the deadline will give him more productivity.

Then, he put down his phone and turned his attention to the formula on the scratch paper again.

"However, the direction of the Motive theory is definitely possible. The only question is, how to come up with this universal cohomology theory?"

He touched his chin.

Cohomology theory is an important tool he uses to study the Hodge conjecture. If the Motive theory can be confirmed, it will be very helpful to solve the Hodge conjecture.

"Hmm, but I don't have to come up with this theory of universal cohomology. If I can express the current Hodge conjecture in the form of integrals, then using cohomology I can explain this characteristic in differential form, then the distance Solving the Hodge conjecture is very close.”

"But how to do this?"

Lin Xiao fell into thinking again.

The integral Hodge conjecture has long been proposed by predecessors, and two forms were proposed based on it, but without exception, both forms were proved to be wrong.

Until now, people are still looking for ways to express Hodge's conjecture in the form of integrals.

Once we can express the Hodge conjecture in terms of integrals...

Lin Xiao's mind flashed through the five-element mathematical formulas he had obtained from the bug in the system. Two of them, which he thought were strange at first, could be completely substituted into the Hodge conjecture in the form of integrals. And this, too, It will be of great help to him in proving Hodge's conjecture.

However, after thinking for a long time, Lin Xiao finally sighed helplessly.

Alas, I can't think of anything.

We all understand the principle, but if we want to implement it, it is not that simple.

He shook his head and looked at the time. He still had a class this afternoon. Now he could prepare to go to Beijing University.

So he packed up various things and headed to Beijing University.

…

In the afternoon, Lin Xiao's class started on time.

"Dear students, do you want me to draw a picture today?"

Arriving at class, Lin Xiao showed a smile.

"don't want!"

In the classroom, all the students in his class were wailing.

At the beginning of this month, Lin Xiao's random drawing and then the question that combined algebraic geometry and number theory tortured them terribly.

In the end, there were so few that were actually made that I could count them on one hand.

So the few students who could count on one hand won two ordinary points, and for the remaining students who failed to perform, Lin Xiao ruthlessly deducted two ordinary points from them.

As a result, these students almost have PTSD from drawing pictures.

Seeing their appearance, Lin Xiao smiled: "Are you scared?"

"Afraid!"

"No, you guys."

Lin Xiao shook his head with a smile, and then said: "Okay, I won't tease you anymore. In this class, we will talk about o-adic theory."

"p-adic is the content of algebraic number theory. We have discussed it before. There is a very close relationship between algebraic number theory and algebraic geometry..."