Chapter 190 Conquering Mathematicians All Over the World

The report meeting was held at 2 o'clock, and Xu Chuan could not have gone on stage at 2 o'clock.

Going on stage a little earlier is a 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 instantly became quiet. Everyone stopped discussing and turned their eyes to the young man on the stage, leaving only the camera clicking quietly.

Being stared at by hundreds of eyes in the audience, Xu Chuan did not feel too nervous.

After all, he had experienced all this before.

Let alone a speech in front of hundreds of people, when he discovered dark matter and dark energy in his previous life, it was called crazy.

If there were no sufficient security to control the crowd, I am afraid that everyone would want to throw themselves on his face at that time.

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

On the podium, Xu Chuan opened the notebook he had prepared long ago and clicked on the PPT copy that had been compiled in advance.

A slide was projected onto the silver-white screen.

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

This picture comes from the background of the Hodge conjecture. In the 20th century, mathematicians found a powerful way 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 with increasing dimensions.

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

Above the picture, there is a line of bold words: "Hodge Conjecture".

This is today's topic.

After clicking on the homepage of the PPT, Xu Chuan turned around and looked at the crowd in the Alexander Auditorium, and said calmly:

"Thank you very much for coming here from all over the world. I would like to express my most sincere gratitude to you all."

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

"I believe everyone has read my paper, so here I will not repeat the full picture of the paper. In the following explanation, I will focus on two aspects."

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

The picture on the projection screen suddenly jumped.

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

[Algebraic clusters and group mapping tools]

[Proof process of Hodge conjecture]

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

Xu Chuan glanced at the slides and continued, "As shown in the figure, in the following explanation, I will focus on the two aspects of 'algebraic clusters and group mapping tools' and 'the proof process of the Hodge conjecture'."

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

"I will focus on these two aspects, and I will briefly mention other things."

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

Highlighting the theme of the report is what every academic speaker with a certain level will do.

After all, everyone's time is precious, and coming to the report meeting is not to watch the speaker holding a PPT and repeating what has been written in the paper.

And previewing the speaker's paper before the academic report meeting is also a common practice and a necessary etiquette in the academic community.

Everyone comes here to learn and understand the knowledge that they don't understand.

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

If you want to go through all the proof papers of more than 100 pages in detail, you may not be able to do it without several days.

And for most people who attend the lecture, such as students who follow the professor to broaden their horizons, or professors who take the initiative to participate in the lecture, they come to witness history.

A lecture of a few hours is fine, but most people may not have the patience for a lecture that lasts for several days.

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

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

"This mathematical method originated from the mapping and twisting of the Weyl group. Its core idea is to map the algebraic cluster through the Weyl group, and then introduce Bruhat decomposition and field theory."

As he explained, the pictures on the PPT kept showing.

".Let Gz=GL(n,C) be a general complex linear group, and B∈Gz be an upper triangular subgroup. Then, the Bruhat decomposition of Gz is a bi-training decomposition B\G1/B=∏BωB. The Weyl group W is a linear isomorphism of N*N transformation matrices."

".A maximal ring T:={diag(d,d2,…,dn):|dj|=1) of the unitary group U(n) is a bi-training decomposition of the subgroup GU(n) is T\G1/T=∏BωB."

"."

In the entire paper proving the Hodge conjecture, there is no doubt that this algebraic cluster and group mapping tool is the most important and essential thing.

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

For a brand-new mathematical tool, the acceptance of the mathematical community has always been 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 it.

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

After all, if the algebraic cluster and group mapping tool are not understood, the subsequent proof process of the Hodge conjecture will be even more confusing.

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

And the audience in the auditorium also listened very seriously.

Even those math students who had begun to not understand were staring at the stage with their eyes wide open.

Students who can be mentored or follow professors to attend such large-scale mathematics conferences are basically determined to make further progress in mathematics.

For those who study mathematics, it is definitely better to listen to the explanations of these top masters than to study alone with textbooks.

Even if you don’t understand the process, there are always some concepts and ideas that can be recorded, and these things combined with the knowledge in your mind can often bring them inspiration.

For students or professors who are determined to make further progress in mathematics, this kind of proof report of major conjectures is something that cannot be missed.

On the stage, Xu Chuan explained the algebraic cluster and group mapping tools in an orderly manner.

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

It has been more than two years since they parted at the Morning Star Mathematics Award Ceremony.

Two and a half years are not enough for him to complete all his studies in school, and the boy who was originally dazzling before has now stood at the peak that he can’t reach.

The proof of Hodge's conjecture.

This is a difficult problem that ordinary people spend their whole lives studying but cannot make any breakthroughs, but that person solved it in just two years.

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

Although he has been trying hard to listen to the lectures and has read the more than 100 pages of papers in advance.

But sitting here today, he still can't keep up with the other party's pace, and now, he can't understand the algebraic cluster and group mapping tools that are being explained.

Whether it works or not, mathematics is such a realistic thing.

Hearing the question, Zhang Weiping turned his head and looked at his student. Seeing his complicated expression, he smiled and said, "What's wrong, are you hit?"

He could naturally guess a little about the thoughts and emotions of his disciple.

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

"Such monsters can be counted on one hand throughout the history of the development of mathematics."

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

At this time, he had just finished explaining the algebraic cluster and group mapping tools.

Of course, a real report meeting could not end in an hour. Everyone present, whether Xu Chuan or the audience in the auditorium, was ready to stay here until the end and then have dinner directly.

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

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

Next, it was the proof of the Hodge conjecture.

Although theoretically, the proof of the Hodge conjecture is far more important than the algebraic cluster and group mapping tools. But for Xu Chuan and the audience, once this tool is made and learned to use, the rest will follow naturally.

It's like using an axe to chop down a big tree.

Although the tree is so huge that it is unimaginable, as long as there is enough time, you can still chop it down bit by bit.

Using algebraic clusters and group mapping tools to complete the Hodge conjecture is like using an axe to chop down a towering tree.

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

It successfully split the invisible shackles of the Hodge conjecture and opened the door to the new world to everyone.

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

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

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

With these two world-class mathematicians accompanying him on the left and right, it can be seen that the old man in the middle has an extraordinary status.

And in fact, he is also like this.

Just because this old man is called Jean-Pierre Searle.

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 the three awards.

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

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

It's just that Searle is now 91 years old and has retired to enjoy his old age.

In fact, the Institute for Advanced Study at 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 learning the news, Searle insisted on coming in person, no matter how much the people around him persuaded him.

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

That tall figure was so similar to the boy today.

At the same time, with Xu Chuan's explanation, the proof process of the Hodge conjecture 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 the action of Gal(k/k) on the cohomology class defined by Y is also ordinary."

When the final moment came, the whole auditorium was silent, and you could hear a pin drop.

Some of the whispered discussions that originally emerged due to the algebraic cluster and group mapping tools disappeared at this moment. Even the scholars who could no longer understand the paper report at this moment had a wonderful feeling in their hearts.

So, all the audience couldn't help holding their breath and staring at the curtain on the stage.

On it, there were the final proof steps of the Hodge conjecture.

As the last step came, Xu Chuan moved his eyes away from the projection screen and looked at the audience below.

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"

"From this, 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."

"That is, the Hodge conjecture holds!"

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

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

And, it has conquered mathematicians from all over the world!