Great Country Academician

The report meeting was held at two o'clock, and it was impossible for Xu Chuan to be on stage until two o'clock.

Going to the stage a little earlier is the necessary etiquette and respect for the audience who come to listen to the report in any formal report meeting.

As he appeared on the podium, the crowded Alexander Auditorium fell silent in an instant. Everyone stopped discussing, and turned their attention to the young man on the stage, leaving only the camera quietly clicking.

Being stared at by hundreds of pairs of eyes from the audience, Xu Chuan didn't feel too nervous.

After all, he had already experienced all this.

Not to mention a speech in front of hundreds of people, when he discovered dark matter and dark energy in his previous life, that was called madness.

If there were not sufficient security to control the crowd, I am afraid that everyone would have wanted to jump on his face at that time.

Compared to the madness at that time, the scene at this meeting is nothing.

On the podium, Xu Chuan opened the notebook he had prepared, and clicked on the pre-compiled PPT copy.

A slide was projected onto a silver-white screen.

In the picture above, there is a golden ball on the bottom line of the grid, and various lines of blue, purple, red and black meander through the ball.

This picture comes from the background of the Hodge conjecture. During the twentieth century, mathematicians discovered powerful ways to study the shape of complex objects. The idea is to what extent the shape of a given object can be formed by gluing together simple geometric building blocks of increasing dimensionality.

The grid plane and sphere, as well as the curves that can shuttle and interweave in the sphere, can express this idea, so it is widely used in the introduction of Hodge's conjecture.

Above the picture, there is a line of bold characters: "Hodge Conjecture (Hodge Conjecture)".

This is today's topic.

Clicking on the homepage of the PPT, Xu Chuan turned to look at the crowd in the Alexander Great Hall, and said calmly:

"Thank you very much for coming here from all over the world, and here I would like to express my sincerest gratitude to all of you."

"The topic of today's report is the proof paper of Hodge's conjecture."

"I believe that everyone has read my thesis, so here, I will not repeat the whole picture of the thesis. In the next explanation, I will focus on two aspects."

After a pause, Xu Chuan lightly tapped the pen in his hand.

The picture on the projection screen suddenly jumped.

The first official picture in the speech PPT manuscript jumped out.

【Algebraic Variety and Group Mapping Tool】

[The proof process of Hodge's conjecture]

Two lines of text are presented in a concise PPT copy.

Xu Chuan glanced at the slide, and then said: "As shown in the picture, in the next explanation, I will focus on the two aspects of 'algebraic varieties and group mapping tools' and 'the proof process of Hodge's conjecture' .”

"The former is the key to solving Hodge's conjecture, it is a bridge connecting algebraic geometry and topology, and it is also the most essential part of this proof paper. The latter is the complete proof idea of ​​Hodge's conjecture."

"I will focus on these two aspects. As for other things, I will briefly mention them."

"Of course, if you have any questions about this proof paper, you can ask them in the follow-up question session, and I will do my best to answer them."

Highlighting the key points of the report meeting is what every competent academic speaker will do.

After all, everyone's time is precious, and attending the report meeting is not just to watch the speaker repeat what is already in the paper with a PPT.

It is also a practice and a necessary etiquette in the academic world to preview the speaker's thesis before the academic report meeting begins.

You come here to learn and understand what you don't know.

There is no need to talk about the verification process and other things that have been clearly written in the paper at the report meeting.

More than a hundred pages of proof papers, if you want to go through everything without detail, it may not be possible to do it in a few days.

And for most of the people who participated in the report meeting, such as the students who followed the professor to increase their knowledge, or the professors who took the initiative to participate in the report meeting, they came to witness history.

A report meeting for a few hours is fine, but for a report meeting lasting several days, I am afraid that most people do not have the patience.

Turning over a page of PPT, Xu Chuan entered the theme of this report.

"Algebraic varieties and group mapping tools are the core mathematical tools for proving Hodge's conjecture. If you want to understand the proof process of Hodge's conjecture, you must have a good understanding of it."

"This mathematical method originated from the mapping and torsion of Weyl groups. Its core idea is the mapping of algebraic varieties through Weyl groups, and then through the introduction of Bruhat decomposition and field theory."

Following his explanation, the pictures on the PPT were continuously shown.

".Assume that Gz=GL(n, C) is a general complex linear group, and B∈Gz is an upper triangular subgroup, then GzBruhat decomposes into a double-set decomposition B\\G1/B=∏BωB. The Weyl group W is Linear isomorphism of N*N transformation matrices."

".A maximum ring T of the unitary group U(n):={diag(d, d2,...,dn): |dj|=1), then the double-set decomposition of the subgroup GU(n) is T\\G1/ T=∏BωB."

"."

In the whole paper proving Hodge's conjecture, there is no doubt that this algebraic variety and group mapping tool is the most important and essential thing.

It is based on the algebraic group, subgroup and torus structure method proposed by Professor Mirzakhani, but it has been completely reborn. It can be said that it has completely broken away from the original foundation and structure and has become a brand new mathematical method.

For a brand new mathematical tool, the acceptance of the mathematical community has always been relatively cautious.

So at today's report meeting, Xu Chuan focused on explaining this tool.

On the one hand, it is to let more mathematicians understand.

On the other hand, it is for the next report on the proof process of Hodge's conjecture.

After all, if the algebraic variety and group mapping tools do not understand, the subsequent proof process of Hodge's conjecture will be even more unclear.

For this part, Xu Chuan spoke very seriously, starting from the principle, and then covering all the details of how to map, reverse, and expand the group domain.

And the audience in the auditorium also listened very carefully.

Even those math students who had begun to fail to understand were staring closely at the stage with wide-eyed eyes.

Students who can be tutored, or can follow the professor to participate in this kind of large-scale mathematics report meeting, are basically interested in furthering mathematics.

For researching mathematics, it is definitely better to listen to the explanations of such top-level experts on problems than to gnaw on books and textbooks alone.

Even if they don't understand the process, there are always some concepts and ideas that can be recorded, and combining these things with the knowledge in their minds can often bring them inspiration.

For students or professors who are interested in going further in mathematics, the proof report of this major conjecture will be something not to be missed.

On the stage, Xu Chuan methodically explained the tools of algebraic varieties and group mapping.

In the corner of the auditorium, Hu Xingjian, who followed his tutor Zhang Weiping to participate in the mathematics exchange meeting, looked at the talking peer on the stage with complicated eyes.

It has been more than two years since I said goodbye at the Morning Star Mathematics Awards Gala.

Two and a half years was not enough for him to complete all his studies in school, and the boy who was originally extremely dazzling before had already stood at the peak that he could not reach.

A proof of Hodge's conjecture.

This is a difficult problem that ordinary people can't break through after spending a lifetime of research, but it was solved by that person in only two years.

"Professor, do you think he really solved Hodge's conjecture?" Finally, he couldn't help but whispered to his mentor Zhang Weiping.

Although he has been working hard to listen to the lecture, he has also read the more than one hundred pages of papers in advance.

But sitting here today, he still couldn't keep up with the opponent's rhythm, and now, he couldn't understand the algebraic variety and group mapping tool that was being explained.

Yes or no, mathematics is such a realistic thing.

Hearing the inquiry, Zhang Weiping turned his head to look at his student, saw his complex expression, smiled and said, "What's wrong, you were hit?"

Naturally, he could guess a little bit about his own disciple's thoughts and emotions.

After a pause, he continued to comfort him: "You don't need to, and there is no need to compare with him. If you are a genius, then he is a real monster."

"Such monstrous characters can be counted on one hand when looking at the entire history of the mathematics world."

The report time passed quickly. During Xu Chuan's explanation, half of the scheduled one-hour report meeting passed in the blink of an eye.

Only at this time did he complete the explanation of algebraic varieties and group mapping tools.

Of course, it is impossible for the real report to end in an hour. Everyone present, whether it is Xu Chuan or the audience in the auditorium, is ready to stay here until the end and then eat dinner directly.

No one cared about this long time. Those who cared about this had already got up and left, and everyone who stayed behind hoped that the explanation would be as detailed as possible, even if they couldn't understand it.

On the stage, Xu Chuan finished explaining the tools of algebraic clusters and group mapping, and looked at the audience.

Next, is the proof of Hodge's conjecture.

Although theoretically speaking, the proof of Hodge's conjecture is far more important than algebraic varieties and group mapping tools. But whether it is for Xu Chuan or for the audience in the audience, when this tool is manufactured and learned to use, the rest will be a matter of course.

It's like chopping down a big tree with an axe.

Even though this tree is unimaginably huge, as long as you have enough time, you can still use it to chop it down bit by bit.

Using algebraic varieties and group mapping tools to complete the Hodge conjecture is like cutting down a towering tree with an axe.

Perhaps one day in the future, the mathematics community will be able to find a more efficient tool like a 'chainsaw', but right now, the importance and sharpness of this ax are beyond doubt.

It successfully split the invisible shackle that Hodge conjectured, and revealed the gate of the new world to everyone.

On the other side, in the front row of the lecture hall, among the rows of seats that had been arranged in advance, an old man looked at the youth on the stage with cloudy but profound eyes.

On both sides of this old man, there are two other slightly younger old men, one is Professor Pierre Deligne of the Institute for Advanced Study in Princeton.

The other is Professor Gerd Faltings of the Max Planck Institute for Mathematics.

Accompanied by these two top mathematicians in the world, one on the left and one on the right, it can be seen that the old man in the middle has an extraordinary identity.

And in fact, so did he.

Just because the old man's name is Jean-Pierre Serre.

The youngest winner of the Fields Medal in history, the first winner of the Abel Prize, the Wolf Prize in Mathematics, and the first genius mathematician in the history of mathematics to win the Grand Slam of three awards.

After the death of Pope Grothendieck in 2014, the old man can be said to be the greatest scholar in mathematics today.

He has deep research in pure mathematics such as topology, algebraic geometry, and number theory. Even Faltins, who is now vaguely known as the first person, is like a student in front of him.

It's just that Searle's age is as high as ninety-one years old now, and he has already retired to enjoy his old age.

In fact, the Institute for Advanced Study in Princeton did not send an invitation letter to Searle. After all, you have to consider whether his age and physical condition can withstand the toss.

But unexpectedly, after hearing the news, Searle was determined to come here in person, no matter how much the people around him tried to persuade him, it was useless.

Staring at the young man who was seriously explaining on the stage, Searle's eyes were hazy, as if time had returned to seventy years ago, when he was still a student attending Professor Hilbert's lecture.

How similar is that stalwart figure to the young man of today.

At the same time, with Xu Chuan's explanation, the proof process of Hodge's conjecture has entered the most core final stage.

On the podium, Xu Chuan turned over a page of the PPT manuscript: ".Based on the mapping Tr, restriction mapping and Poincare, the duality theorem is compatible with the action of Gal(k/k), so Gal(k/k) is defined by Y Functions on the homology class are also trivial."

When the final moment came, the entire auditorium fell silent, and a needle could be heard.

Some of the whispered discussions that had sprung up due to the tools of algebraic varieties and group mappings disappeared at this moment, and even scholars who could not understand the paper report at all at this moment felt a strange feeling in their hearts.

As a result, all the audience couldn't help holding their breath, and stared closely at the curtain on the stage.

On it, there is the final proof step of Hodge's conjecture.

With the arrival of the last step, Xu Chuan turned his eyes away from the projection screen and looked at the audience in the audience.

After taking a deep breath, he said calmly: "When i≤n/2, the quadratic form x→(1)iLr2i(x.x) on Ai (X)∩ker(Ln2i+1) is positive definite"

"Therefore, it can be obtained that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes."

"In other words, the Hodge conjecture is established!"

When the last sentence fell, the Alexander Auditorium was instantly filled with thunderous applause.

After Lefschetz proved that the Hodge conjecture is correct in low-dimensional space in 1924, it has gone through more than a hundred years of ups and downs. No matter what the final conclusion is, at this moment, the genius boy standing on the stage , ended a century-old problem with his own theory.

And, conquered mathematicians from all over the world!