Great Country Academician

Hodge's conjecture, one of the seven millennium puzzles.

is a great unsolved problem in algebraic geometry.

Proposed by William Valens Douglas Hodge, it is a conjectural problem concerning the association of the algebraic topology of nonsingular complex algebraic varieties and its geometry expressed by polynomial equations defining subvarieties.

In short, Hodge's conjecture is that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraically closed classes.

Together with Fermat's last theorem and Riemann's conjecture, it constitutes the m-theory structural geometric topology carrier and tool that integrates general relativity and quantum mechanics, and its importance is self-evident.

If he could solve the Hodge conjecture, it would take a giant step forward in the validity of general relativity and M-theory.

For Xu Chuan, the temptation of this matter is undoubtedly quite great.

After all, he studied physics in his previous life, and he studied under Edward Witten. He is very familiar with both general relativity and m-theory.

Suddenly, the cell phone next to the desk vibrated again, and the loud ringing interrupted Xu Chuan's thoughts. He picked up the cell phone, and it was Professor Wei Teng calling.

"Hey, teacher, what can I do for you?"

"Where are you? Is it convenient now?" Wei Teng's voice came from the computer.

"I'm in the dormitory, what's the matter, teacher, what's the matter?" Xu Chuan replied.

"Then come to Professor Deligne's office now."

"Okay, I'll go right away."

After hanging up the phone, Xu Chuan glanced at the screen of the mobile phone that was turned on automatically, and the date on it surprised him.

August 27th.

He actually stayed in the dormitory unknowingly for more than a month, which was far beyond the time he had asked Professor Deligne for leave.

More importantly, Professor Deligne didn't even ask about it in the past month.

It's outrageous, the student asked for leave for seven days, and then didn't go to class for more than a month, and the instructor didn't even ask.

Shaking his head, Xu Chuan went into the bathroom to wash his face, and tidied up his somewhat messy hair. He had devoted himself to studying mathematics for more than a month, and his hair was long enough to cover his ears. He had to find some time to trim it.

As soon as he stepped out of the dormitory and was about to close the door, Xu Chuan paused, turned around and entered the room again, found the manuscript paper that he had sorted out from his previous research on the problem of 'incompressible decomposition of differential algebraic varieties', copied it in his hand, and prepared to go together take it.

Although the mathematical tools made for Hodge's conjecture are more important than this, they also need to be brought over for the two tutors to help check. But those things are still lying in the entire dormitory in a mess, on the table, on the floor, on the bed, everywhere, and there is no time to tidy them up.

On the contrary, the mathematical tools that are suitable for the problem of "irreducible decomposition of differential algebraic varieties" have been sorted out before, and now they can be taken away directly.

Mathematics tutor Professor Deligne has a lot of experience in differential equations, so you can show him to see if there is anything that needs to be revised before submitting the manuscript.

After all, he is alone, and what he considers may not be comprehensive, and sometimes he can see something different from other perspectives.

Carrying the manuscript paper to solve the problem of "incompressible decomposition of differential algebraic varieties", Xu Chuan walked across the Princeton campus and rushed to the Princeton Institute for Advanced Study.

He knocked on the door of the instructor's office and walked in. The two instructors, Witten and Deligne, were there.

Seeing his sloppy look, Deligne couldn't help frowning, and asked, "How long have you been out?"

Xu Chuan scratched his head and said with a smile, "Maybe two months?"

"Are you studying the manuscript that Professor Mirzakhani left for you? What is it about?" Edward Witten asked curiously, but he didn't care about Xu Chuan's image.

It's actually quite normal for scientific researchers to make this appearance, and purely theoretical calculations may be slightly better. Except for that weirdo Perleman, there are still very few mathematicians who will make themselves this appearance.

But in many other disciplines, various experiments are often done. When he was at CERN, he dealt with many staff members.

Sometimes when some equipment is being repaired, those staff often make themselves unkempt, which is normal.

It was Deligne who said that Xu Chuan was studying the manuscript left to him by Professor Mirzakhani, which made him a little curious.

Is my student related to Professor Mirzakhani?

"Um."

Xu Chuan nodded, and continued: "Some thoughts on algebraic varieties are related to the problem of 'incompressible decomposition of differential algebraic varieties'."

Hearing this, Professor Deligne raised his eyelids, leaned forward slightly, and asked with interest: "Can I read the manuscript?"

"The manuscript is still in my dormitory, but I brought some of my own research today, and I asked the two teachers to help me see if there are any flaws in it."

As he spoke, Xu Chuan raised the manuscript paper in his hand, then found the printer in the office, made a copy of the manuscript, and handed it to Deligne and Witten respectively.

Needless to say, Professor Deligne is the only two Grand Slam players in mathematics, not to mention differential algebra and algebraic geometry are still his professional fields.

Although Witten is a physicist, he is also very capable in mathematics. After all, he has won the Fields Medal. From his perspective, he may be able to find some loopholes.

The two tutors took the manuscript from Xu Chuan with some curiosity, and began to read it.

The student in front of him has a strong mathematical ability, and they all know that more than 99.99% of the Fields Medals will be awarded to him one year later.

Although the age is a little immature, the subject of mathematics does not mean that the older the better.

Between the ages of twenty-five and forty-five is the golden career of studying mathematics. No matter how young you are, the basic knowledge in your mind is insufficient to lay a good foundation. No matter how old you are, your thinking will start to solidify and age, and it will be difficult to do anything. Such results.

Of course, this age group does not apply to everyone, especially the gifted with excellent mathematical talents.

For example, genius mathematicians favored by God, such as Schultz and Terence Tao, all made great contributions in the mathematics field in their early twenties.

There is no doubt that Xu Chuan is also such a genius, and even more so than Schultz and Tao Terence. After all, the first two did not have the achievement of solving world-class math problems at the age of eighteen or nineteen.

Therefore, both Deligne and Witten are very interested in Xu Chuan's research.

"Irreducible Differential Algebraic Variety Decomposition of 'Irreducible Decomposition of Differential Algebraic Varieties' - Field Theoretic Algebraic Variety Association Method."

On the first manuscript paper, the eye-catching title occupying the top layer caught the eyes of Deligne and Professor Witten, which shocked the two of them. They raised their heads and looked at each other at the same time, then looked down at the proof process.

The irreducible decomposition of differential algebraic varieties is another world-class mathematical problem after the Weyl-Berry conjecture.

After more than a year at Princeton, has the student finally refocused his attention on mathematics?

Compared with the Weyl-Berry conjecture, the irreducible decomposition problem of differential algebraic varieties is not much less difficult, because it is a bridge between algebraic geometry and differential equations.

If this problem can be solved, the mathematical community can extend algebraic geometry to algebraic differential equations and differential polynomials.

However, although the difficulty is not bad, compared with the completeness of the Weyl-Berry conjecture, the completeness of the irreducible decomposition of differential algebraic varieties is still much worse.

The Weyl-Berry conjecture is a complete conjecture, from the weak Weyl-Berry conjecture to the complete Weyl-Berry conjecture proof, no one has ever broken through.

However, the irreducible decomposition problem of differential algebraic varieties has been defined long ago, and the irreducible decomposition of differential algebraic varieties exists.

It's just that mathematicians haven't been able to find a way to the final definition so far.

On the other hand, this problem has another "half brother": "irreducible decomposition of differential algebraic varieties".

The irreducible decomposition of differential algebraic varieties and the irreducible decomposition of differential algebraic varieties actually come from the Ritt-Wu zero-point decomposition theorem, and are partly solved by the Ritt-Wu zero-point decomposition theorem.

However, the Ritt-Wu zero-point decomposition theorem still has certain limitations on these two problems.

One is that the irreducible decomposition needs to be further obtained, and the other is that an algorithm cannot be given to decompose the solution set of the difference algebraic equation into an irreducible difference algebraic variety.

If these two problems can be solved at the same time, the difficulty of the system can surpass the Weyl-Berry conjecture, but the difficulty of the irreducible decomposition of a single differential algebraic variety is indeed not as difficult as the Weyl-Berry conjecture.

However, it is not easy to solve these two problems.

In particular, the irreducible decomposition of differential algebraic varieties is not much less difficult than the Weyl-Berry conjecture.

Although as early as the 1930s it was proved by Ritt et al.: "Arbitrary differential algebraic varieties can be decomposed into unions of irreducible differential algebraic varieties."

But today, nearly a century has passed, and no one has been able to give an algorithm to decompose the solution set of differential algebraic equations into irreducible differential algebraic clusters.

In the past seven or eighty years, it is not that no one has tried to solve this problem.

Ritt et al. who proved that "any differential algebraic variety can be decomposed into a union of irreducible differential algebraic varieties" also tried to extend the Ritt-Wu zero-point decomposition theorem to algebraic difference equations.

But the obtained result can decompose the differential algebraic variety into the form of Zero(S)=∪/kZero(SAT(ASk)), and the rest cannot be advanced.

If no one can solve this problem in more than ten years, it will become a typical century-old problem.

In the office, Deligne and Witten were immersed in the manuscript in their hands.

And Xu Chuan skillfully took out a copy of the latest issue of "Annual of Mathematics" from the tutor's office and read it.

In the Institute for Advanced Study in Princeton, there are many such top-level journals. Almost any professor, whether it is mathematics, physics, or other natural subjects, basically has a lot of various journals in his office.

Some are subscribed by professors themselves, while others are sent by journals. Deligne and Witten are naturally the latter.

This has something to do with the fact that these two top leaders are also academic editors of various top journals.

After all, in academia, under normal circumstances, peer review is a kind of voluntary labor without any monetary reward.

In this case, in order to find suitable reviewers, the journal will naturally pay some other things. For example, the submission of previous reviewers is free of page fees, and journal papers are given away.

Of course, in addition to these, there are other invisible benefits, such as improving personal reputation, always updating one's grasp of current scientific research hotspots, and so on.

After all, peer reviewers review all the latest academic papers, and can obtain different ideas, techniques and perspectives from the reviewed manuscripts, broaden their horizons, and learn from the mistakes made by other researchers as a warning, Help improve your own research and more.

Two old and one young, the three of them were immersed in their respective manuscripts and papers, and it didn't know how long it took before the office became active again.

"It's really wonderful. I didn't expect that Bruhat decomposition and Weyl group can be introduced into domain theory in this way." In the office, after reading the manuscript in his hand, Deligne let out a sigh of emotion.

The irreducible decomposition of differential algebraic varieties is a difficult problem in differential equations and algebraic geometry, but it is not oriented to the most advanced mathematics, on the contrary, it was born on the basis of the two.

This is like opening a channel on the ground floor of two mathematics buildings to connect the two.

Although everyone knows that this can be done as long as it does not affect the load-bearing walls.

But the difficulty lies in the fact that the materials used to construct the walls of these two buildings are too hard. Whether it is a hammer, a hammer or a chisel, these commonly used mathematical tools in the past cannot dig a hole out of them.

But now, Xu Chuan constructed a new tool, dug a hole in the originally indestructible wall, successfully connected the two buildings, and further decomposed the differential algebraic variety into an irreducible differential algebraic variety, thus giving The process of irreducible decomposition of differential algebraic varieties.

In this tool, Deligne saw some tricks and shadows of the Weyl-Berry conjecture, but also something about algebraic groups, subgroups, and torus.

I just don't know how many of these things belonged to Professor Mirzakhani, and how many belonged to his student.

After all, he had never read Professor Mirzakhani's manuscript, so he didn't know how many things there were in that manuscript.

But no matter what, a difficult problem in the palace of mathematics can be removed with a high probability.

He didn't say for sure, but at least he was 80 to 90% sure.

Of course, the draft paper in hand is not 100% perfect, and there are still some places that can be slightly adjusted, but these are just minor details.

As for whether there are other major defects, it is impossible to judge now. After all, this is not a simple problem. The difficulty lies there. Simply going through it is not enough for him to guarantee that there must be no problems.