Chapter 287 Be a dog out of context

A week passed quickly.

In Lin Xiao's room, he was still working on his work.

Based on the exchange with Peter Schultz that day, Lin Xiao’s ideas also gained many different perspectives.

It's like writing. When you get a topic, the content written by different people will be very different, whether it is the starting angle or the idea that the article wants to express, it will be very different.

The same goes for math problems.

Even if it is a question in exam-oriented education, there may be many different solutions, not to mention world-class questions like Hodge's conjecture.

Therefore, Peter Schultz's different views also broadened Lin Xiao's thinking.

But it also made him gain a lot.

Until now.

[Suppose (X, A) is a space couple, G is an arbitrary commutative group, let C (X, A) represent the singular chain complex of (X, A), and is a complete space composed of the closed p form of the differential manifold M For the subvariant space composed of appropriate p-forms, it is a Motive motivational cohomology group. 】

【HdR(M)≌R^n……】

Looking at the final definition of motivational cohomology on the draft paper, Lin Xiao also had a smile on his face.

Motive cohomology is the key to figuring out the integral Hodge conjecture, and now, he finally got it.

Of course, what he came up with is not the universal cohomology that motivation theory pursues. Universal cohomology is an encapsulation of all cohomology sets, and his current motivational cohomology is just the same as Motive motivation theory. Related.

However, it already has the prototype of a motivation theory - because this motivation cohomology has communicated with many cohomology theories such as singular cohomology, Drum cohomology, etc., it is just that it is limited to these types for the time being, not counting Truly everything, or omnipotent cohomology.

As for the real motivation theory, Lin Xiao vaguely felt that it might need more expansion.

"In that case, I should call it Lin's motivational cohomology for the time being?"

With a slight smile, Lin Xiao picked up the pen and added the word "Lin" in front of the name in the same tone as the motive.

He nodded with satisfaction and then put down his pen.

We are one step closer to having the word ‘Lin’ spread throughout the world of mathematics.

I just hope that future students won’t blame him.

But in retrospect, multidimensional field theory in physics has probably caused headaches for countless students. It is said that some university textbooks have included part of multidimensional field theory, especially for students studying particle physics. After all, the standard model under multi-dimensional field theory has replaced the standard model in the past, and the mathematical requirements of the standard model under multi-dimensional field theory have been gradually increasing, which of course is a pain for those physics students.

Of course, these have nothing to do with Lin Xiao.

Returning his attention to the present, Lin's motivation coherence is basically enough.

"Then the next step is to figure out the Hodge conjecture in integral form."

Lin Xiao's lips curled up. With Lin's motivational coherence, he could complete this step with his eyes closed.

So he picked up the pen again and started writing.

"First, express Lin's motivational cohomology in the form of differentials..."

【ω=dα+δβ+γ】

[(α,β)=∫αΛ*β]

【……】

Until ten minutes passed.

"To sum up, the Hodge conjecture in integral form should be like this: Let is the sum of the torsional and cohomological classes of algebraic rings with integral coefficients on X."

"Finally, it's done."

Seeing himself finally simplifying the Hodge conjecture after a lot of calculations with mathematical formulas, Lin Xiao's face showed a brighter smile than before.

"The Hodge conjecture in integral form turns out to be like this."

"It seems that it is indeed easier to understand than the previous form."

Lin Xiao touched his chin.

Why should Hodge's conjecture be converted into this integral form? Naturally, it is because the original form is too difficult to understand. Mathematics is the process of simplifying complex things, turning complex things into simple things, and finally solving this problem, that is This kind of process.

In a word, it is to bring those difficult problems to the same level as yourself, and then use your rich experience to solve them.

"Well... Next, it's time to prove the Hodge conjecture in integral form."

Proving the Hodge conjecture in integral form can prove the final Hodge conjecture.

One of the seven major problems of the millennium...

Even Lin Xiao couldn't help but have expectations at this time.

This is another truth in the universe. How can Lin Xiao not look forward to it?

Moreover, the role of Hodge's conjecture in promoting the unification of mathematics also makes Lin Xiao look forward to the rewards that will be brought to him after the proof.

Of course, he did not underestimate the integral Hodge conjecture at all.

The difficulty of proving the integral Hodge conjecture is at least half a millennium problem.

In other words, he has only completed half of the work from completing the Hodge Conjecture to the integral Hodge Conjecture, and now there is still half of the work waiting for him to complete.

Moreover, no one knows whether it will become more difficult towards the end.

However, just when Lin Xiao was about to take action, his phone rang again. When he saw it, it was another call from Russia.

He touched his nose. When he called last time, he said he would submit the report in a week, but now it seems like a week has passed?

After the phone was connected, the voice of the staff member from last time came from the other side: "This is the organizing committee of the International Congress of Mathematicians. Is this Professor Lin?"

"it's me."

"Hello, Professor Lin! Is your report ready? Last time you said that you would complete the report in one week, so I am here to ask you specifically."

Lin Xiao answered truthfully: "Uh... not yet, maybe it will take another week."

A helpless voice came from the other side: "Okay, then please hurry up."

"I know, don't worry."

Although he said this, Lin Xiao was not sure that he could completely solve the Hodge conjecture in the remaining seven days, unless he spent truth points. But obviously, using truth points to solve the Hodge conjecture was not in line with his concepts, and the system The rewards will also be greatly reduced, so there is no need for him to use this thing just to show off at the International Congress of Mathematicians.

Of course, he did not forget that the system rewarded him with a [Flash of Inspiration] last time, but even if he used this, it was still impossible for him to prove Hodge's conjecture in just seven days.

From this point of view, the current integral Hodge conjecture can only be submitted as a staged achievement.

But at this moment, Lin Xiao had an idea flash in his mind, and then asked: "By the way, will the Fields Medal winner also arrange a one-hour lecture?"

The staff opposite was stunned, obviously not expecting Lin Xiao to ask this question.

Is Lin Xiao scheduled for the Fields Medal?

Of course, no matter what, Lin Xiao's chances of winning the Fields Medal can be said to be the highest among all the mathematicians who are expected to win the Fields Medal at this International Congress of Mathematicians.

If Lin Xiao couldn't get it, others would probably be embarrassed to get it.

After all, this year, there are no other unsuspicious candidates.

Later, the staff member smiled and said: "Yes, of course we will arrange it. After the list of Fields Medal winners is announced, the Fields Lecture will be arranged."

Lin Xiao's heart moved, and then he smiled and said: "Okay, thank you for informing me, I will submit the report as soon as possible."

"Well, okay, Professor Lin." The staff member said, "I also wish you the Fields Medal."

"Thanks."

Lin Xiao said with a smile.

Then the phone hung up.

Lin Xiao began to plan.

Every International Congress of Mathematicians will prepare a lecture for the winners. Of course, because the winners basically have no time to prepare the content of the lecture, they will basically prepare some learning processes, share life experiences or stories, and this lecture , all in one hour.

So, if Lin Xiao can win the Fields Medal, then he will have one more lecture, and in this lecture, he can give whatever he wants, without any hateful deadline restrictions.

This is equivalent to saying that Lin Xiao can spend the next three months solving the Hodge conjecture, and then report his integral Hodge conjecture to Phil in a one-hour plenary meeting. In this lecture, the complete proof of Hodge's conjecture was completed.

Of course, the prerequisite is that Lin Xiao can win the Fields Medal, but in this regard, his confidence is still relatively high.

The Fields Medal is originally an award to reward young scholars under the age of 40, and it is not an award like the Nobel Prize that occasionally takes the age of the candidate into consideration. Therefore, even in terms of ensuring authority, Lin Xiao will not be awarded this award. It would be basically unreasonable to issue it.

So Lin Xiao has this confidence, and he has locked in a Fields Lecture.

"However, there is still a report on Lin's conjecture."

It was originally planned to be held at the International Congress of Mathematicians, but since Hodge's conjecture is ahead, there is obviously no such opportunity now.

"Doesn't this mean that I have to prepare a report on Lin's conjecture later?"

A strange expression suddenly appeared on Lin Xiao's face.

Three reports in a row?

This plot is a bit familiar.

Isn’t that what Andrew Wiles did back then?

This trouble-making prover of Fermat's Last Theorem originally gave three lectures in a row, and the titles of these three lectures were "Modular Forms, Elliptic Curves and Galois Representations."

Obviously, he did not hint in the title that he would discuss Fermat's Last Theorem, but by the end of his second lecture, people already knew his purpose, and people flocked to the third lecture. .

Even though people had guessed his purpose, Wiles still pretended not to know, and only proved the general situation of Taniyama Shimura's conjecture at the end of the third report, and then, as his colleagues thought later, pointed out that this meant Fermat's last theorem was correct - despite the fact that his proof was problematic and he didn't actually complete the proof until two years later, just like the joke: When Wiles said it was obvious, it took two years It’s really obvious later in life.

Thinking of this, Lin Xiao couldn't help but smile. This script is good and now it is his.

So then, he picked up the phone again and called their dean, Zhao Qi.

"Professor Lin, why do you have time to call me?"

Dean Zhao's laughing voice came from the opposite side.

Lin Xiao smiled and said, "Dean Zhao, I am bringing you good news."

"Oh?" Zhao Qi's tone suddenly rose, "What's the good news?"

Lin Xiao smiled and said, "I'm going to put the report on Lin's conjecture in our school."

Zhao Qi suddenly became even more excited: "This is great!"

No matter what, it was definitely a good thing for Lin Xiao to hold this report in their school.

Although the popularity of Lin's conjecture has passed and the paper has been published, if Lin Xiao decides to give a report on Lin's conjecture, he will definitely be able to attract many people.

After all, many people are still looking forward to Lin Xiao's report.

However, after being happy for a while, Zhao Qi suddenly calmed down and asked in a low voice: "Professor Lin, if you want to prove Lin's conjecture in your report at our school, then could it be that the report you gave at the International Congress of Mathematicians was Hodge’s guess?”

This time it was Lin Xiao's turn to be surprised: "How do you know?"

But when Lin Xiao said this, Zhao Qi was even more shocked, "It's actually true!"

Then, he smiled and said: "Didn't you discuss the Hodge conjecture with Schultz in class before? Then the students spread the word and said that you proved the Hodge conjecture. Now it seems that this rumor is true. That’s it.”

Lin Xiao was suddenly speechless.

That's how rumors spread.

He was obviously asking Schulz about how to prove the Hodge conjecture, but why did it become that he had already proven the Hodge conjecture?

He said helplessly: "I haven't proven Hodge's conjecture yet. This is all rumored."

"Is there no proof?" Zhao Qi was stunned, but quickly responded: "However, since you plan to give a report on Hodge's conjecture at the International Congress of Mathematicians, this should also mean that you have made great achievements, right? "

"Yes."

"Then I'm looking forward to the news then." Zhao Qi smiled. "Unfortunately, as the dean, I don't have time to go to the International Congress of Mathematicians. Otherwise, I would have to read your report."

"You can also watch it online later." Lin Xiao said with a smile.

"Well." Zhao Qi said, "Then I hope you can prove it successfully in the end."

"Haha, let me lend you some good words."

Lin Xiao smiled and thanked.

Afterwards, they discussed the time for the subsequent report on Lin's conjecture, and decided to hold this academic report at the end of June, which is only about a week away from the International Congress of Mathematicians that will start in early July.

In this way, mathematicians from all over the world can come here to listen to his report, and then go directly to Russia to participate in the International Congress of Mathematicians.

After discussing it, Lin Xiao hung up the phone, and then turned his attention again to the points Hodge conjecture in front of him.

The next step is to sort out the report as soon as possible.

Of course, in order to make the three reports cohesive, he had to process the report submitted to the International Congress of Mathematicians out of context.

Well, it’s not bad to be a dog out of context.