"Congratulations to the user, you have been selected for a high-level academic paper ("On non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes")."

"What?" Xu Shengzhou was shocked and screamed.

"On non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes." It sounds rather mysterious. If I change it to a more understandable name, many people may have an impression: Hodge (

Hodge) conjecture.

——Well, for ordinary people, not many people know about Hodge’s conjecture. After all, its reputation is still a bit weak, far less famous than Goldbach’s conjecture, Fermat’s last theorem, and not as famous as Zhang Yitang.

The noisy twin prime conjecture may not even be as good as the Sitapan conjecture solved by an undergraduate at a key university in central China. But in the field of algebraic geometry, it is the highest throne. In the field of mathematics, there are as many important problems as stars, and it is still the highest on the crown.

One of the shining diamonds!

In early 2000, the Clay Mathematics Institute's Scientific Advisory Board selected seven "Millennium Prize Problems", and the Clay Mathematics Institute's board of directors decided to establish a $7 million grand prize fund. Each "Millennium Prize Problem" can be solved.

Received a reward of US$1 million. These seven problems include the Poincaré Hypothesis, which is the only one that has been solved so far, and the Riemann Hypothesis, which Hilbert would most like to know whether it has been solved if he can return to the world in 500 years.

As well as NP-complete problems, BSD conjectures, etc., including Hodge's conjecture.

The mathematical idea of ​​Hodge's conjecture is very profound and difficult to understand, but we might as well use figurative language to get a rough understanding, that is, it wants to prove that "any complex and magnificent palace can be built from a pile of wood."

, or "any geometric figure of a shape, no matter how complex it is, can be made up of a bunch of simple geometric figures." The reason why Hodge's conjecture is so important to the development of mathematics is that it will combine algebraic geometry with

The connection of topology, two extremely important but seemingly unrelated fields in mathematics, allows mathematicians to try to unify the two fields to a certain extent.

Why is the Hodge Conjecture so important yet so little attention?

First of all, the foundation of algebraic geometry in China is naturally weak. But the more important point is that this problem is so complex that it is very difficult for ordinary science and engineering students to understand, including "non-singular complex projective algebraic varieties" and "Hodge

"Classes", "algebraic closed chain classes", "rational linear combinations", what are they talking about? Civilian science is even more daunting. It cannot be hyped, and naturally it has no popularity.

Xu Shengzhou has studied algebraic geometry for so long, so he naturally knows the importance of Hodge's conjecture. It can be said that since the conjecture was proposed in 1950, the progress has not been smooth. Especially in 1982, Freedman (Fields 1986)

winner) discovered the Freedman E8 manifold. No matter what kind of topological transformation the manifold in this four-dimensional space undergoes, it cannot be described by a polynomial. As a result, the content of the Hodge conjecture has undergone some changes, that is, it has become "finding a method that ensures

"The condition under which a shape can be described by a polynomial after topological transformation", or "under what conditions a geometry can be transformed into a figure determined by an equation". As long as this condition can be found, the Hodge conjecture can be put to an end.

At the same time, it received US$1 million from the Clay Mathematics Institute.

But to find this condition, we must first consider all imaginable shapes and equations, which puts the conjecture into trouble. Many times the work done is equivalent to climbing a dangerous peak with an unknown height. The only thing we know is that it

The height is beyond imagination. You just laid a climbing nail on the journey. No one can know whether it is located halfway up the mountain or 3,000 meters, 5,000 meters or 10,000 meters away from the top of the mountain!

As one of the researchers of algebraic geometry, Xu Shengzhou once had some wild thoughts about the Hodge conjecture, but he was soon poured cold water on his head by the cold reality, and he started to study other problems honestly. Nothing happened.

I thought that the system uncle actually offered this gift package!!!

Xu Shengzhou didn't care that he was at the defense scene, and opened the system with trembling hands, eager to read the content of the paper. There was a progress bar behind the reward, and it was still 0% motionless. He couldn't help but want to roll his eyes: What the hell are you doing?

Which network disk company did you learn the system from? Do you need to top up the value and open a super membership to increase the download speed?

Fortunately, there was a question mark icon behind the progress bar, as if he expected Xu Shengzhou to have questions. After he clicked on it, a paragraph of text appeared: "Senior academic papers refer to papers that are groundbreaking and pioneering in a certain discipline or a certain research direction.

, can effectively promote the scientific and technological development of the planet where it is located. However, in order to avoid the huge impact caused by the sudden appearance of papers, the system will display part of the paper content in a timely manner based on the user's reading of documents, in-depth thinking, discussion with others, writing papers and other activities."

Hi! I thought I could get a free ticket again, but it turned out that the dog chewed the urine and soaked it for joy, and I had to work hard to redeem the points.

As soon as he relaxed his mind, he felt like someone was poking him. When he looked back, he saw Hengping with a flustered face: "Hey, if you have something to say, why are you poking me?"

Heng Ping lowered his voice: "Why do you want me to poke you? You have been here in a daze for several minutes, shaking all over and rolling your eyes. Teacher Cheng called you several times and you didn't respond. Why do you think I poke you?"

How many minutes have I been in a daze?

Xu Shengzhou felt a little unbelievable, and looked around again, and found that Mr. Cheng, Dean Zhang Anping, including Tian Ziliang and others were looking at him with concern. He introduced with some embarrassment: "I suddenly had an idea just now. I thought of a certain person."

A clever solution to a problem, but I didn’t expect it to cause trouble to all the teachers. I’m sorry!”

Due to Heng Ping's informant, Mr. Cheng and Dean Zhang Anping knew that Xu Shengzhou often had fantastic ideas - in fact, it was the effect of "wisdom time (permanent solidification)" and "junior's extraordinary academic vision", which worked smoothly

He accepted Xu Shengzhou's explanation. But Tian Ziliang was not sure and asked curiously: "What are you thinking about?"

Discussing with others can speed up downloading!! Xu Shengzhou will not give up this opportunity: "About Hodge's conjecture."

"Hodge's conjecture?!"

Everyone present was shocked to death. Because they were basically majoring in algebraic geometry, and even if they were not majoring in algebraic geometry, they were professionals in the field of mathematics. Naturally, they knew the lofty status and importance of Hodge's conjecture.

.

A master's student has an ingenious proof of Hodge's conjecture? Is this Fermat-style fishing or Gauss-style genius? They felt shocked and incredible, but also felt somewhat credible. After all, the young man in front of them

However, publishing two "Big Four" articles on algebraic geometry is a miracle in itself. What if there is an even bigger miracle?

Mr. Cheng quickly asked: "Which aspect is it about Hodge's conjecture? What specific progress is it?"

Xu Shengzhou said: "It's about the conditions for the establishment of the Hodge conjecture. But currently I just have a vague idea, but if I want to straighten it out and write it down, I estimate that it will be possible in as little as one or two years, and as many as three to five years?"

Zhang Anping suddenly felt numb all over, and blood rushed to his head: What about the conditions for the establishment of the Hodge conjecture? That is to completely solve the Hodge conjecture? And the tone of Xu Shengzhou's words sounded very familiar! It seems that he was right last time.

I have some ideas for the group mass number one conjecture. It took me a few months to sort it out and write it down. What was the result? It was published on the "Big Four" a few months later! If it took a year or two, the Hodge conjecture would have been completely solved.

, isn’t that a student from the School of Mathematics at Beijing Normal University who is going to win the country’s first Fields Medal?!

So pretty!!

He was trembling with excitement. He quickly stepped forward and grabbed Xu Shengzhou and said, "Then you write! Hurry up and write! What are the conditions? The college will prepare a dedicated study room for you? I will also find you some related directions.

Teacher? What are you missing? We will solve it for you immediately, immediately, no, now. If I can’t solve it, don’t we still have teachers and schools?”

Mr. Cheng was also so excited that he couldn't control himself: "Xiao Xu, I'll give you my office. You can use it as long as you want. Oh, you are about to graduate from graduate school, and your dormitory——"

Xu Shengzhou said quickly: "I think it would be good to live with my current roommates."

"Okay, I'll say hello to the school and your dormitory won't move. I'll make a room next to your dormitory for you to use specifically as a study room. It's convenient for you to go straight to work when you're inspired.

Disturb your roommate." Mr. Cheng considered the issue very carefully, "Do you have any other requirements?"

Xu Shengzhou thought of ways to speed up the download speed, as well as reading literature: "Can you trouble the college to help collect papers, books and other information about the Hodge conjecture in recent years? I can refer to it."

Zhang Anping patted his chest and said confidently: "No problem! I will arrange for the data clerk in the department to be responsible for this work later. I will try my best to collect as much information about the Hodge conjecture at home and abroad as possible, and I will send a batch of information tomorrow morning.

Go to the table in your office. I will arrange for a graduate student to be responsible for this work on a daily basis and to connect with you specifically."

Heng Ping suggested: "Teacher Cheng, Dean Zhang, I personally suggest that we select one or two students from each of the graduate students and junior students recruited this year, and let Xiao Xu and I guide them. In the future, we can both

It can help with work, conduct daily discussions, promote research, and strengthen the effective force of the algebraic geometry direction in the college. What do you think?"

Mr. Cheng immediately made a decision: "Xiao Heng's suggestion is good. I think it's okay. What do you think, Dean Zhang?"

"It's indeed a good idea! I'll be responsible for implementing it."

Tian Ziliang saw a group of people from the School of Mathematical Sciences of Beijing Normal University, from Academician Cheng Deru to Dean Zhang Anping, to ordinary Qingjiao, all seriously discussing helping a graduate student to study the Hodge conjecture, and he vaguely regretted not having done so just now

He agreed to allow Xu Shengzhou to teach at Yanda, but he thought it was ridiculous. Huo Qi guessed that you could have researched it if you wanted to? Did a bunch of people from Beijing Normal University also fool around with him?

So he asked: "What are you doing?"

Mr. Cheng smiled sheepishly: "We are not like your School of Mathematics at Yan'an University, where we are all elites selected from thousands of people across the country. Although our students also have some talents, they are a little weaker after all. But as long as we find out

If there is a one in 10,000 chance, we will give 100% support. As for success or failure, it all depends on his personal efforts. But this matter has not been revealed yet, so I would like to ask Academician Tian to keep it a secret to avoid any changes in the future.

You have to make people laugh and be generous."
Chapter completed!