Chapter 290 The first scene, fragmented report

[...According to Equation 1, Equation 3, Equation 8, and Equation 12, we can get Hdg^k(X)=H^(2k)(X, Q)∩H^(k,k)(X)...]

[To sum up, we can know that when The sum of the torsion and cohomology classes of algebraic periods. 】

[So, after dividing by H^(2k)(X,Z)∩H(k,k)(X) by twisting the classes, each class is the image of the cohomology class of the integral algebraic period. 】

[Therefore, on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes]

[Certificate completed. 】

Lin Xiao wrote the last two words one stroke at a time, with a smile on his face.

Hotch guessed, it was over.

The proposer of the Hodge conjecture, William Vallance Douglas Hodge, discovered this possible conjecture during his work from 1930 to 1940. However, before 1950, this conjecture did not attract people's attention. of attention.

It was not until Hodge mentioned this conjecture in his speech at the International Congress of Mathematicians in 1950 that people were surprised to find that this was a conjecture of great significance to mathematics, especially to algebraic geometry. and algebraic topology.

With Grothendieck's subsequent great development of algebraic geometry, as well as the cohomology group that was expanded into an important mathematical tool under his single-handedness, and the subsequent proposal of the Langlands program, Hodge conjectures that this can be used to a certain extent. The conjecture on unifying the two different disciplines of algebraic geometry and topology has great significance, and naturally it has become a major mathematical problem that people are competing to solve.

It is precisely because of this that when the Clay Institute organized many mathematicians to select the Millennium Problem, the Hodge conjecture was also selected, and then it became a problem that countless mathematicians will remember.

And until today, it finally came to an end in the hands of a twenty-two-year-old mathematical genius.

From 1930 to the present, it has spanned more than 90 years of history, survived the most disastrous world war, and also spanned more than 70 years of relatively peaceful times, witnessing an increasingly prosperous mathematics community.

Its end is the end of a piece of history.

Lin Xiao put down the pen, picked up the draft paper, and blew on the pen and ink on it.

Well, although the ink has dried long ago, it is still a piece of history. Let’s blow it up and give the original Hodge conjecture a little dignity, and then clean up the dust for the upcoming Hodge-Lin theorem.

Suddenly realizing that something was missing, Lin Xiao raised his eyebrows and said softly: "Come here!"

System: "Congratulations to the host, you have completed the proof of Hodge's conjecture. The intersection of algebra and topology has enabled all qualified topologies to be expressed as polynomial solution sets, and the grand unification of mathematics has moved forward again. With this step, your understanding of mathematics has made an important leap forward. I hope that the host can continue to work hard and grasp the final truth as soon as possible! This reward is 8,000 mathematical experience, 150 truth points, and the aura of a mathematics master."

"Congratulations to the host, your mathematics level has been raised to level 6. You have officially become a true mathematics master!"

"Mathematics Master Halo: When the host solves mathematical problems, it will receive a certain inspiration bonus."

The sound of the system made Lin Xiao's face reveal surprise and surprise.

Actually, 8,000 math experience points were directly awarded!

The last time he proved Lin's conjecture, he only got 4,500 points of mathematical experience. Unexpectedly, this time it almost tripled.

To a certain extent, this is also due to the fame of Hodge's conjecture. Of course, the most important thing is that Hodge's conjecture is more important to the unification of mathematics. After all, it directly connects the relationship between topology and algebraic geometry.

Because the proof of Hodge's conjecture will allow mathematicians to know what conditions are required for a shape to be described by a polynomial after topological transformation.

Then, we can easily find how to algebraically describe a shape and its topologically transformed homeomorphism.

This is of great significance not only to mathematical research, but also to many other fields.

No matter what, as one of the millennium problems, its significance to the world goes without saying.

Of course, having more experience is one thing, but Lin Xiao soon paid attention to another thing: "So, since mathematics has reached level 6, is it time for me to develop my brain a little bit?"

The next moment, the familiar heaviness came to his mind again.

However, due to his now more powerful brain development, this feeling disappeared after a moment, replaced by the same familiar feeling of "I am getting stronger".

Of course, his hair was still black and shiny, and now even thicker because he hadn't gotten a haircut for another ten days.

"Well, compared to many mathematicians, it's obvious that my hair volume is pretty good."

Lin Xiao nodded with satisfaction.

He even had the luxury of getting a haircut.

"It's time to take care of it. In five days, I have to give a report."

He let down his hair and asked how much the system had improved his brain development. This time it increased by 0.7%, and his brain development reached 8.85%.

There is only a 1.15% gap from 10%.

With a slight sigh, Lin Xiao stopped thinking about it and went out to get her hair cut.

…

Lin Xiao naturally had to wait until the International Congress of Mathematicians to officially reveal the proof of Hodge's conjecture. Before that, the mathematics community would be itching for more than ten days because of his fragmented conference report, and Let’s hope he gets the Fields Medal.

Otherwise, if you want to witness the proof of Hodge's conjecture, you can only go to Shangjing University to listen to his report.

I don’t know if the dean of their School of Mathematical Sciences wants a non-permanent professor who is a Fields Medal winner, or if the academic report that proves the Hodge conjecture is to be conducted here?

Just like that, the time came to June 30th.

On this day, Lin Xiao will give a report on the proof of Lin's conjecture in the auditorium.

Because the time of this report was very close to that of the International Congress of Mathematicians, and it was more convenient to fly from Beijing to St. Petersburg, many mathematicians came to listen to this report.

In addition, this is also Lin Xiao's first report on Lin's conjecture, so many mathematicians are eager to communicate with Lin Xiao about the issues involved.

Although they could communicate with Lin Xiao via email before, obviously face-to-face communication is more convenient.

Until nine o'clock in the morning, the report was held as scheduled.

After some grooming and a specially styled hairstyle, Lin Xiao appeared in great spirits in front of many mathematicians present.

Now twenty-two years old, he has become more mature than before, both in appearance and in other aspects.

Facing these mathematicians, he said with a smile: "Dear mathematicians who have come from afar, welcome to Beijing University. I hope that my few days here can leave a good impression on you."

"Lin, if you can improve the last part of your report at the International Congress of Mathematicians, I will definitely leave a good impression."

At this time, a mathematician sitting in the first row said.

Lin Xiao looked and saw that this mathematician was none other than Pierre Deligne who flew from the United States.

Deligne's words quickly resonated with other mathematicians.

"Yeah! Lin, you either don't write that part, or you write more! You're making us uncomfortable!"

"Lin, you're driving me crazy like this!"

…

Seeing that at least six or seven mathematicians, all of whom were familiar with him, expressed their opinions, Lin Xiao couldn't laugh or cry, and then smiled and said: "You will get a satisfactory answer by then."

"This is what you said, and we all remember it!"

"certainly."

Lin Xiao made an OK gesture.

So these mathematicians left him alone for the time being and listened to his report.

Regarding the proof of Lin's conjecture, since it has been approved by the "Annals of Mathematics", of course there is basically no problem. Lin Xiao's academic report can also be regarded as a process.

Mainly for future communication.

So Lin Xiao spoke very quickly. He finished the paper in about twenty minutes.

Next comes the question and answer session.

Lin Xiao left enough time to communicate with the mathematicians present about the proof process of Lin's theorem and some methods of application.

The question-and-answer session lasted for more than forty minutes. The students who came to join in the fun, or some less capable teachers, were already confused, and were extremely confused by the questions asked by these big guys and Lin Xiao's answers. at a loss.

They could only lament in their hearts: It’s too high-end!

Of course, after forty minutes, when no one asked any more questions, Lin Xiao finally began his conclusion.

He smiled and said: "Actually, there is a big connection between Lin's theorem and Hodge's theorem... conjecture. I believe everyone is aware of it. And after my subsequent research on Hodge's theorem... conjecture, I also discovered a This magical method can help us prove the Hodge conjecture and bring great help."

"Of course, if you want to know what happened, please listen to the breakdown of my next report. Today's report ends here. Thank you all, and I look forward to seeing you at the International Congress of Mathematicians."

Then he bowed deeply, and then, as if fleeing, he left the podium.

The mathematicians present were stunned for a while before suddenly coming back to their senses.

Lin Xiao said "Hodge's theorem" twice. Is it a coincidence or intentional?

But according to common sense, when they mention the Hodge conjecture, they must always say "Hodge-conjecture", so how can they say "Hodge-theorem"?

Could it be?

Deligne reacted first, then quickly got up and ran to the top, shouting at the same time: "Lin! Don't run! Stop!"

Other mathematicians also reacted one after another. This damn Lin Xiao actually broke the chapter again?

Suddenly, for the first time since its completion, a strange scene appeared in the Great Hall of Beijing University: a group of world-famous mathematicians ran onto the podium to chase a young mathematical genius.

The audience behind them, who still didn't know what was going on, looked confused.

What happened?