Chapter 190 Conquering Mathematicians Around the World
The report meeting was held at two o'clock sharp, and it was impossible for Xu Chuan to come on stage at two o'clock.
Going on stage a little early 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 Alexandria Auditorium fell silent instantly. Everyone stopped their discussion and turned their attention to the young man on the stage, leaving the camera to click quietly.
Being stared at by hundreds of pairs of eyes in the audience, Xu Chuan didn't feel too nervous.
After all, he has experienced all this before.
Not to mention giving 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 there hadn't been adequate security to control the crowd, everyone would have wanted to jump in his face.
Compared to the craziness of that time, the scene here was nothing.
On the podium, Xu Chuan opened the notebook he had prepared and clicked on the pre-written PPT copy.
A slide was projected onto a silver-white curtain.
The picture above shows a golden ball on the bottom line of the grid, with various blue, purple, red and black lines zigzagging through the ball.
This picture comes from the background of the Hodge conjecture. During the 20th century, mathematicians discovered powerful ways to study the shapes 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 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".
That's today's topic.
Clicking on the homepage of the PPT, Xu Chuan turned to look 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."
"The theme of today's report is the proof paper of Hodge's conjecture."
"I believe everyone has already read my paper, so I will not repeat the entire paper here. I will focus on two aspects in the following explanation."
After a pause, Xu Chuan gently clicked the control pen in his hand.
The picture on the projection screen suddenly jumped.
The first official picture in the speech PPT document jumped out.
[Algebraic Varieties and Group Mapping Tools]
[The proof process of Hodge’s conjecture]
Two lines of text, presented in concise PPT copy.
Xu Chuan glanced at the slide and continued: "As shown in the picture, in the following 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 the Hodge conjecture, a 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 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 subsequent question session, and I will do my best to answer them."
Highlighting the theme of the report meeting is something that every competent academic speaker will do.
After all, everyone’s time is precious, and attending a report meeting is not just to watch the speaker repeat what is already in the paper with a PPT.
It is also a common practice and a necessary etiquette in the academic community to preview the speaker's paper before the start of the academic seminar.
Everyone comes here to learn and understand the knowledge that they do not understand.
There is no need to talk about the verification process and other things that have been clearly written in the paper again at the report meeting.
If you want to go through the proof paper of more than 100 pages in detail, you probably won't be able to do it if you don't have a few days.
And for most people who attend the lectures, such as students who follow the professors to gain knowledge, or professors who take the initiative to participate in the lectures, they are here to witness history.
A report meeting of a few hours is fine, but for a report meeting that lasts for several days, I am afraid that most people will not have the patience.
Turning over the PPT page, Xu Chuan entered the theme of this report meeting.
"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 sufficient understanding of it."
"This mathematical method originated from the mapping and torsion of the Weyl group. Its core idea is the mapping of algebraic varieties through the Weyl group, and then through the introduction of Bruhat decomposition and field theory."
Following his explanation, the pictures on the PPT were constantly shown.
".Suppose Gz=GL(n,C) is a general complex linear group, and B∈Gz is an upper triangular subgroup, then GzBruhat is decomposed into a double-pertained set decomposition B\G1/B=∏BωB. Weyl group W is Linear isomorphism of N*N transformation matrix. ”
".A maximum ring T of the unitary group U(n):={diag(d, d2,...,dn): |dj|=1), then the double-performed set of the subgroup GU(n) is decomposed into T\G1/ T=∏BωB.”
"."
In the entire paper that proves 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 methods proposed by Professor Mirzakhani, but it is completely reborn. It can be said that it is completely separated from the original foundation and structure and has become a brand new mathematical method.
As for a brand-new mathematical tool, the mathematical community has always been cautious in accepting it.
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 a report on the subsequent proof process of Hodge’s conjecture.
After all, if the algebraic varieties and group mapping tools cannot be understood, the subsequent proof process of Hodge's conjecture will be even more unclear.
Regarding this part, Xu Chuan was very serious about it, starting from the principles to details on how to map, reverse, and expand the group domain.
The audience in the auditorium also listened very carefully.
Even the math students who had begun to fail to understand were staring at the stage with their eyes wide open.
Students who can be tutored, or can follow professors to participate in such large-scale mathematics seminars, are basically interested in making further progress in mathematics.
For studying mathematics, it is definitely much better to listen to the explanations of problems by such top experts than to read textbooks alone.
Even if they don’t understand the process, there are always some concepts and ideas that can be recorded, and these things can often bring inspiration to them when combined with the knowledge in their own minds. For students or professors who are interested in making further progress in mathematics, the proof report of this major conjecture will be something not to be missed.
On the stage, Xu Chuan explained algebraic varieties and group mapping tools in an orderly manner.
In the corner of the auditorium, Hu Xingjian, who came to attend the mathematics exchange meeting with his tutor Zhang Weiping, looked at the talkative peer on the stage with complicated eyes.
More than two years have passed since we said goodbye at the Morning Star Mathematics Prize awards night.
Two and a half years were not enough for him to complete all his studies in school, but the young man who was already so dazzling before was now at the pinnacle that was beyond his reach.
Proof of Hodge's conjecture.
This was a problem that ordinary people could spend a lifetime studying and could not break through, but it was solved by that person in only two years.
"Professor, do you think he really solved the Hodge conjecture?" Finally, he couldn't help but whispered to his instructor Zhang Weiping.
Although he had been trying to listen to the lectures and had read the more than 100 pages of the paper in advance.
But sitting here today, he still can't keep up with the opponent's rhythm, and now, he can no longer understand the algebraic varieties and group mapping tools that are being explained.
Whether it works or not, mathematics is such a realistic thing.
Upon hearing the inquiry, Zhang Weiping turned to look at his student. Seeing the complicated expression on his face, he smiled and said, "What's wrong? Were you hit?"
He can naturally guess a certain amount of his disciple's thoughts and emotions.
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 monster."
“If you look at the entire history of the development of mathematics, you can count such evil figures on one hand.”
The reporting time passed very quickly. During Xu Chuan's explanation, half of the scheduled one-hour reporting session passed in the blink of an eye.
At this time, he had just completed the explanation of algebraic varieties and group mapping tools.
Of course, the real report meeting cannot end in an hour. Everyone present, whether it is Xu Chuan or the audience in the auditorium, are 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. Everyone who stayed wanted the explanation to be as detailed as possible, even if they couldn't understand it.
On the stage, Xu Chuan finished explaining algebraic varieties and group mapping tools 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 for both Xu Chuan and the audience, once this tool is made and learned to be used, the rest becomes a matter of course.
It's like using an ax to chop down a tree.
Although this tree is unimaginably huge, you can still use it to chop it down bit by bit as long as you have enough time.
Using algebraic variety and group mapping tools to complete the Hodge conjecture is like using an ax to chop down a towering tree.
Perhaps one day in the future, the mathematical community will be able to find a more efficient tool like a 'chainsaw', but for now, the importance and sharpness of this ax are beyond doubt.
It successfully split open the invisible shackles that Hodge guessed, revealing the door to the new world in front of everyone's eyes.
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 young man on the stage with cloudy but profound eyes.
On both sides of the old man were two other slightly younger men. One was Professor Pierre Deligne from the Institute for Advanced Study in Princeton.
The other one is Professor Gerd Faltings of the Max Planck Institute of Mathematics.
With two of the world's top math masters by his side, one can see that the old man in the middle has an extraordinary status.
In fact, so is he.
Just because this 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 to win three grand slam prizes in the history of mathematics.
After the death of Pope Grothendieck in 2014, this old man can be said to be the greatest scholar in the world of mathematics today.
He has profound research in pure mathematics such as topology, algebraic geometry, and number theory. Even Faltings, who is now vaguely known as the number one person, is like a student in front of him.
It's just that Serge is now ninety-one years old and has already retired to enjoy his old age.
In fact, the Institute for Advanced Study in Princeton did not send an invitation letter to Serge. After all, you have to consider whether his age and physical condition can withstand the torment.
But unexpectedly, after learning the news, Serge was determined to come in person, no matter how much the people around him tried to persuade him, it was of no use.
Staring at the young man who was explaining seriously on the stage, Serge's eyes were hazy, as if time had returned to seventy years ago, when he was still a student attending Professor Hilbert's lecture.
That majestic figure was so similar to the young man today.
At the same time, with Xu Chuan's explanation, the proof process of Hodge's conjecture entered the core final stage.
On the podium, Xu Chuan turned over a page of 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 The effect on the homology class is also ordinary. ”
When the final moment came, the entire auditorium fell silent and you could hear a pin drop.
Some of the whispered discussions that had arisen over algebraic varieties and group mapping tools disappeared at this moment. Even the scholars who could not understand the paper report at this moment felt a strange feeling in their hearts.
As a result, all the audience could not help but hold their breath and stared closely at the curtain on the stage.
On it, there are the final steps to prove the Hodge conjecture.
As the last step came, Xu Chuan moved 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"
"From this, it can be concluded 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 is established!"
When the last words fell, the Alexandria Auditorium was instantly filled with thunderous applause.
After Lefschetz proved that Hodge’s conjecture was correct in low-dimensional space in 1924, it has gone through nearly a hundred years of ups and downs. Regardless of the final conclusion, at this moment, the genius boy standing on the stage , used his own theory to end a century-old problem.
Moreover, he conquered mathematicians from all over the world!
(End of chapter)