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

Coming to the stage a little earlier is a necessary etiquette and respect for the audience who comes to hear the presentation in any formal presentation.

As he appeared on the podium, the crowded Alexander Auditorium instantly fell silent. Everyone stopped discussing and turned their eyes to the boy on the stage, leaving the camera alone to whisper Kacha Kacha.

Stared at by hundreds of pairs of eyes from the audience, Xu Chuan didn't have too much nervousness.

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, it was called madness.

If it weren't for the fact that there was enough security to control the crowd, I'm afraid everyone would have wanted to pounce on him in the face at that time.

Compared to the madness of the time, this scene was nothing.

.......

On the podium, Xu Chuan opened the notebook that had been prepared and clicked on the pre-compiled PPT text.

A slide projected onto a silvery white screen.

The picture above shows a golden ball on the bottom line of the grid, and the ball zigzags through various lines of blue, purple, red and black.

This image is taken from the background of Hodge's conjecture. In 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 dimensions.

】

The grid plane and the sphere, as well as the curves that can weave and weave through the sphere, can represent this idea, so it is widely used in the introduction to the Hodge conjecture.

At the top of the image, there is a large line in bold letters: "Hodge Jecture".

That's the topic for today.

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

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

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

"I'm sure you've all seen my paper, so I'm not going to repeat it here. In the following explanation, I will focus on two aspects. ”

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

The picture on the projection screen jumped.

The first official picture in the presentation PPT script jumped out.

【Algebraic Cluster and Group Mapping Tools】

[Proof of Hodge's conjecture]

Two lines of text, presented in a concise PPT text.

Xu Chuan glanced at the slides and continued, "As shown in the picture, in the following lectures, I will focus on the two aspects of 'Algebraic Cluster and Group Mapping Tools' and 'The Proof Process of Hodge's Conjecture'. ”

The former is the key to solving the Hodge conjecture, the bridge between algebraic geometry and topology, and the best part of this proof paper. The latter is the complete proof of Hodge's conjecture. ”

"I'll focus on those two, and I'll go over the rest briefly."

"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."

Highlight the theme of the presentation, which is what every academic speaker with a level of proficiency will do.

After all, everyone's time is precious, and coming to the report meeting is not to watch the speaker hold the PPT and repeat the things that are already on the paper.

It is also a common practice and a necessary etiquette in the academic community to preview the presenter's paper before the academic symposium begins.

Everyone is here to learn and understand what they don't understand.

There is no need to talk about the verification process and other things that are already clearly written in the paper.

If you have to go through all the more than 100 pages of proof papers, you may not be able to do it in a few days.

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

A few hours of reporting will be fine, but a report meeting that lasts for several days, I'm afraid most people don't have the patience.

.......

Turning a page of ppt, Xu Chuan entered the theme of this report meeting.

"The algebraic cluster and group mapping tool is the core mathematical tool for proving the Hodge conjecture, and if you want to understand the proof process of the Hodge conjecture, then you must have enough knowledge of it."

"This mathematical method originated from the mapping and torsion of Weyl groups, and its core idea was the mapping of algebraic clusters through Weyl groups, and then by the introduction of Bruhat decomposition and domain theory ......"

Following his explanation, the pictures on the PPT continued to be projected.

“..... Let gz=gl(n,c) be the general complex linear group, and b∈gz is an upper trigonometric subgroup, then gzbruhat decomposition into double culture set decomposition b\g1/b=nb is a linear isomorphism of the n*n transform matrix. ”

“..... One of the largest rings of the unitary group u(n) t:={diag(d,d2,...,dn):|dj|=1)........ The double culture set of the subgroup g?u(n) is decomposed into t\g1/t=nbwb. ”

“......”

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

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

The mathematical community has always been cautious about the receptivity of a new mathematical tool.

Therefore, 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 a report on the process of proving the Hodge conjecture for the following.

After all, if the algebraic cluster and group mapping tools don't understand it, the subsequent proof process of the Hodge conjecture will be even more incomprehensible.

For this part of the things, Xu Chuan talked very seriously, starting from the principle, to how to map, twist, expand the group and other details of all aspects.

And the audience in the auditorium also listened very carefully.

Even the math students, who had begun to understand at the beginning, stared at the stage with wide eyes.

Students who can be tutored, or who can follow their professors to participate in such large-scale math lectures, are basically interested in further mathematics.

For the study of mathematics, it is definitely much better to listen to the explanations of such top elders than to hold books and textbooks alone.

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

For students who are interested in taking the next step in mathematics, or for professors, a proof report of this major conjecture is a must-see.

......

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

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

It has been more than two years since the previous farewell at the awards party of the Morning Star Mathematics Awards.

Two and a half years is not enough for him to complete all his studies from school, and the boy who was already dazzling before is now standing at his unattainable peak.

Proof of Hodge's conjecture.

This is a problem that ordinary people spend a lifetime studying and cannot break through, but it took that person only two years to solve it.

"Professor, did you say he really solved the Hodge conjecture?" Finally, he couldn't help but whisper to the mentor Zhang Weiping on the side.

Although he had been listening hard, he had read the more than 100 pages of the paper in advance.

But sitting here today, he still can't keep up with the rhythm of the other party, and now, he can't understand the algebraic cluster and group mapping tool that is being explained.

Yes or no, math is such a realistic thing.

Hearing the inquiry, Zhang Weiping turned his head to look at his student, saw that he had a complicated expression on his face, and smiled: "What's wrong, I was hit?" ”

As for the thoughts and emotions of his disciple, he could naturally guess a few points.

After a pause, he then comforted: "You don't have to, and there is no need to compare with him, if you are a genius, then he is a real demon." ”

"Such a demonic figure, looking at the history of the entire mathematical world, can be counted on one hand."

.......

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

It was only then that he finished explaining the algebraic cluster and group mapping tools.

Of course, the real presentation could not be over in an hour, and everyone present, whether it was Xu Chuan or the audience in the auditorium, was ready to stay here until the end and then eat dinner directly.

No one cared about this long time, those who cared about it had already gotten up and left, and the people who stayed behind didn't want to explain in as much detail as possible, even if they couldn't understand it.

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

Next, there is the proof of Hodge's conjecture.

Although theoretically, the proof of the Hodge conjecture is far more important than the algebraic cluster and group mapping tools. But whether it is for Xu Chuan, or for the audience, when this tool is made and learned to use it, the rest of the things will be logical.

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

Even though the tree is unimaginably large, you can still use it to cut it down bit by bit given enough time.

Using algebraic cluster and group mapping tools to accomplish the Hodge conjecture is like chopping down a towering tree with an axe.

Perhaps one day in the future, the mathematical community will find a more efficient tool like a chainsaw, but for now, the importance and sharpness of this axe are in doubt.

It crackled through the invisible shackles of Hodge's conjecture and revealed the door to a new world before everyone's eyes.

.....

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

On either side of the old man were two other slightly younger men, one of whom was Professor Pierre Deligne of the Institute for Advanced Study in Princeton.

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

With these two world's top mathematicians by their side, it can be seen that the identity of the old man in the middle is extraordinary.

And in fact, so did he.

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

The youngest Fields Medal winner 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 a three-prize grand slam.

After the death of Pope Grothendieck in 2014, this old man is arguably the greatest scholar in mathematics today.

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

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

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

But unexpectedly, after learning the news, Serge insisted on coming over in person, even if the people around him tried to persuade him to no avail.

Staring at the young man who was giving a serious lecture on the stage, Searle's eyes were hazy, as if time had gone back to 70 years ago, when he was still a student, attending Professor Hilbert's lecture.

How similar that majestic figure is to today's teenagers.

......

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

On the podium, Xu Chuan turned over a page of PPT manuscript: "..... Based on the mapping tr, the limit map, and poincar′e, the duality theorem is all compatible with the action of gal(k/k), so the role of gal(k/k) on the upper cohomology class defined by y is also trivial. ”

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

Some of the whispered discussions that had been made up of algebraic clusters and group mapping tools had disappeared at this moment, and even scholars who had completely understood the paper presentation at the moment had a wonderful feeling in their minds.

So, all the audience couldn't help but hold their breath and stare closely at the curtain on the stage.

Above, there is the final proof step of the Hodge conjecture.

As the last step came, Xu Chuan took his eyes off the projection screen and looked at the audience.

After taking a deep breath, he calmly spoke: "When i≤n/2, the quadratic x→(?1)il?r?2i(x.x) on ai (x)n ker(l?n?2i+1) is positively definite ......"

"From this, it can be concluded that on a nonsingular complex projective algebraic cluster, any Hodge class is a rational linear combination of algebraic closed-chain classes."

"That is, the Hodge conjecture is true!"

When the last sentence fell, the Alexandria auditorium was instantly filled with thunderous applause.

After Lefschetz proved that Hodge's conjecture was correct in a low-dimensional space in 1924, it has gone through nearly a hundred years of ups and downs, regardless of the final conclusion, but at this moment, the genius boy standing on the stage ended a century of problems with his own theory.

And, conquer mathematicians from all over the world!

.......