From all-round academic master to chief scientist

At 8:30 am on July 10, 2022.

St. Petersburg National Theater, main venue.

Countless mathematicians poured into the venue from the door, and then began to take their seats. The top mathematicians sat at the front, and then basically moved to the back according to their level of strength in the mathematical world.

Here, regardless of age or seniority, whoever is most likely to understand the report will sit in the front.

In the front row of seats, Pierre Deligne chose his seat, and suddenly found that the old man next to him was very familiar. He couldn't help but be surprised: "Professor Serge, you are here too? You seem to be in good health. .”

This old man is none other than Professor Jean-Pierre Serre, Lin Xiao’s doctoral supervisor.

Serge smiled and said: "It's just a few hours' flight. I can still bear this distance."

Then, he turned his attention to the stage and said with emotion: "What's more, Lin was once my student. He has made such important achievements. Of course, I, the teacher, have to come and take a look."

Hearing this, Deligne's old face twitched.

He still couldn't forget this matter. Wouldn't it be better if Lin Xiao studied for his doctorate?

And look, Lin Xiao's recent mathematical achievements are all in algebraic geometry and topology, which are all areas in which he is good at.

But no matter what, Lin Xiao was already someone else's doctor. He could only sigh and express his helplessness, and then said: "You are so lucky."

"Hahaha, I think so too." Serge laughed, and then said: "Look, I won the Fields Medal at the age of 28, and I have always been the youngest winner. Now my students are successful I have accomplished this breakthrough for me, and it will be difficult for others to break through it in the future. Think about it, in the future, the mathematics community will know that the two youngest winners of the Fields Medal are a pair of teachers and students. How interesting is that?"

"It's really interesting."

Deligne was even more envious.

"However, it is a pity that I was not there on the day when he presented the award. Otherwise, I would have had a few drinks with him."

"Tonight is fine too."

"Haha, no need for tonight." Serge smiled, and then took out a bottle of champagne from a bag on the ground.

Deligne was stunned for a moment, and then he couldn't help but laugh: "You are always so well prepared."

"Of course."

Serge smiled slightly, and after Lin Xiao completed the proof, he would personally go up and deliver the bottle of champagne.

At the same time, on the other side.

The team from China was also sitting together and talking.

"There are so many people."

A mathematics professor from Beijing University looked at the number of people in the venue and couldn't help but said.

Even in the previous plenary meetings, there were never so many people, not even the opening ceremony.

Now, the entire main venue is full of people, including many people in the aisles. It can be said that it is not as good as the original situation when more than 10,000 people listened to the report in the Great Hall of the People.

Next to him, Xu Ji, who had just arrived yesterday, smiled and said: "After all, what Lin Xiao wants to prove is the Hodge conjecture. There are so many people, so normal. When he proved Goldbach's conjecture, there were even more people than this." Woolen cloth."

"Oh, sometimes I really envy Lin Xiao's talent. I thought I could treat him calmly before, but now, I still can't bear it."

On the other side, another professor couldn't help but say.

Everyone looked at it and said with a smile: "Professor E, let's not compete with Lin Xiao. You have received an invitation to give a one-hour report this year. You have already made history for our country."

This Professor Enan is an academician of the Chinese Academy of Sciences and is well-known in the domestic mathematics community. At the same time, he also received an invitation to give a report from the International Congress of Mathematicians, and it was only one hour. This is indeed considered a success for China. History has been made.

After all, there are not many mathematicians who hold Chinese nationality and have received one-hour reports.

Professor Enan smiled. It was indeed a happy thing for him to receive the invitation to give a one-hour report. After all, this is considered a high honor in the international mathematics community.

Of course, compared to Lin Xiao’s Fields Medal and the Hodge conjecture that is about to be proved in his hands, the one-hour report suddenly becomes nothing worth mentioning.

However, while they were chatting, there was a sudden commotion from behind, and many people turned their attention to the door.

When they saw this scene, they couldn't help but wonder, what happened?

So they all turned their heads and looked towards the door at the back, and then they all showed surprised expressions.

Because, a bearded and rather slovenly looking man suddenly appeared at the door.

However, it was precisely because of this man's beard that everyone present quickly recognized him.

"Perelman?!"

The mathematicians present all showed surprised expressions.

The prover of the Poincaré conjecture, who had not been seen for more than ten years, actually came to the scene?

Of course, this is probably because Perelman originally lived in St. Petersburg, so it was very convenient, but no matter what, Perelman's appearance here still surprised everyone present.

It seemed that even he, who had almost no interaction with the mathematical community, couldn't help but come out and listen to the proof of Hodge's conjecture.

Of course, thinking of Perelman's character, the people present quickly looked away, lest they get bored and run away again. After all, he had said at the beginning that he did not want to be displayed like an animal in a zoo.

Soon, Perelman found a seat in a corner and sat down, and people stopped paying attention.

At this time, it was already 8:55, and there were only 5 minutes left before the report officially started.

Backstage, Lin Xiao was sitting in his seat, closing his eyes and relaxing.

In his mind, he recalled all his proof processes.

There were many staff members standing next to him, all of whom from time to time cast their eyes on this person who is known as the smartest person in the world today, not even one of them.

He has proved Goldbach’s conjecture, proposed multi-dimensional field theory, and is now going to prove Hodge’s conjecture. Of course, he is definitely the smartest person in the world. He can even be called the smartest human being in history. They are all I want to see how it is different from myself.

But obviously, everyone has one nose and two eyes, there is no difference.

These staff members had no choice but to give up their thoughts, and then their eyes turned to admiration. Russia has always paid attention to basic mathematics education, and now facing Lin Xiao, they could only feel admiration.

Soon, nine o'clock arrived.

First, the host went up to control the stage, and then a staff member said to Lin Xiao: "Professor Lin, it's nine o'clock now, you can go on stage."

Lin Xiao nodded, "Okay."

Then he stood up, smoothed down his well-tailored clothes, then took steps and walked out of the backstage.

There was a wave of applause, welcoming the appearance of this mathematical genius.

And until the applause died down, Lin Xiao smiled and said: "Hello, friends."

"Thank you all for traveling thousands of miles to come here and listen to my report. Now, I have felt your enthusiasm. Next, please allow me to give you a satisfactory report."

Then, on the big screen behind them, the page turned and a PPT appeared in front of them.

On the first page of the PPT, everyone who participated in Lin Xiao’s last report was familiar with the colorful spring toy called Slinky.

And then, Lin Xiao took out the toy from his arms again.

This time, it was probably because he had practiced a few times in the past few days, so he was much better at playing with this toy than before.

However, people in the audience couldn't help but smile when they saw Lin Xiao playing with toys during such a serious academic report.

But soon, Lin Xiao played with the spring a few times, and then he said: "Okay everyone, I also showed you this toy in the last report. As I said last time, this toy It is a very typical one-dimensional manifold topological homeomorphism."

"According to Lefschetz, the Hodge conjecture holds for the (1,1) class, then we can obviously get a result, that is, in a certain low-dimensional space, our Hodge conjecture holds."

"Now, we let H^2(X,C) you can decompose into..."

Lin Xiao's narration began to get on the right track, and the mathematicians present also followed Lin Xiao's narration and became fascinated.

As for those who came to join in the fun, they just thought they were here to join in the fun. After all, it would be bad to miss such a historic scene. In addition, there are students who come from all over the world and basically look at the expressions of gods/gods. Look at Lin Xiao above.

Everyone is in their twenties, why are you different?

At the very back of the hall, there was still a pile of media, even more than those at the opening ceremony.

After all, this is a world-class difficult conjecture, with at least two Fields Medals awarded every year. However, in the history of mathematics, there is only one Hodge conjecture. Of course, these media want to join in the fun.

Among them, the reporters from CCTV were the same ones from the past few days. They looked at Lin Xiao on the stage, chatted with the Russian reporter next to them with a smile, and said very proudly: "Lin Xiao is the genius of our country! "

The Russian reporter was not sour at all. He pouted in Perelman's direction and said, "We also have Perelman."

Reporters from the United States, the United Kingdom, France and other countries nearby were all wondering, why don’t they have a mathematician who has proved the Millennium Problem?

…

“…In the special case where D)) is the Poincaré dual of the homology class given by D. This therefore establishes the usual formulation of the Hodge conjecture for divisors of projective varieties."

"At this point, we have fully understood the Hodge conjectures in low dimensions. Obviously, they are established, that is, when K=1."

"But how do we extend the Hodge conjecture to other dimensions? That is when K=n."

Speaking of this, Lin Xiao showed a smile.

The mathematicians below immediately sat up straight.

Next, is the point.

How does Lin Xiao expand from low dimensions to high dimensions?

Hodge conjectures that whether the proof can be completed depends entirely on this key point.

For algebraic topology, it is very common to study multiple dimensions. They do not have the three dimensions that humans live in, but study them together from one dimension to n dimensions.

Therefore, if you want to prove the Hodge conjecture, you also need to extend it to n dimensions.

In fact, in the past few days, the mathematicians who participated in the last report have also studied how to solve the problem along the lines of ideas given by Lin Xiao that day. However, the most they have studied is what Lin Xiao is talking about now. In this part, I still have no idea what to do next.

Seeing so many pairs of eyes eager for knowledge, Lin Xiao raised the corner of his mouth and said, "Everyone, do you still know about multidimensional field theory?"

The mathematicians present were all stunned. Multidimensional field theory?

Isn’t that a theory of quantum physics?

What does it have to do with the purely mathematical theory of Hodge’s conjecture?

But suddenly, many mathematicians who had studied multidimensional field theory were shocked.

Has Lin Xiao ever done the same work in multidimensional field theory?

From low dimensions to high dimensions!

Although it is a physical theory and a mathematical problem, aren't the mathematical methods used in physical theory also universal for mathematical problems?

All the mathematicians couldn't help showing their astonishment.

Lin Xiao could actually think of using methods from physical theory to solve this problem!

What a wonderful flash of inspiration!

"This is the true light of genius!"

Below, Deligne couldn't help but murmured, and Serge next to him nodded in the same way.

As for other mathematicians, many also showed surprise.

Among them was Perelman, who was sitting in the corner, with the same "I see" expression on his face.

Although multidimensional field theory is a physical theory, the mathematical methods used in it are very good. There are many mathematicians who have studied those mathematical methods. At this moment, they all understand clearly. As for those who do not understand clearly, Then go back and study it yourself.

Seeing the reactions of those mathematicians, Lin Xiao smiled lightly and continued: "Of course, the methods in multidimensional field theory are not completely suitable for our solution to the Hodge conjecture. Therefore, I need to study the methods in multidimensional field theory. method and make some changes.”

Then he turned to the next page of PPT again and introduced his method after changes to the people present.

From then on, the steps to solve Hodge's conjecture became gradually clearer.

The eyes of the mathematicians on the field became brighter and brighter, and Lin Xiao's PPT gradually bottomed out.

finally--

"So, we can now prove that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes."

"In 1950, Mr. Hodge announced his conjecture to the mathematical community, and today, I can confidently say that this conjecture will become a theorem from today."

"So, please let me congratulate the mathematics community here, because we have once again added a foundation to the completion of the Langlands Program Building!"

After Lin Xiao finished speaking, he bowed slightly to the audience.

The next moment, thunderous applause resounded throughout the entire venue.

The mathematicians in the front row stood up one after another and sent their celebrations for the birth of truth.

The audience in the back row also stood up one after another. Even if they didn't understand, it didn't stop them from shouting awesome 666.

At this point, in the mathematics column of mankind's problem book, another bar can be added.