Hodge's conjecture, one of the seven millennium puzzles.
It is a major unsolved problem in algebraic geometry.
It was proposed by William Valens Douglas Hodge as a conjecture problem about the connection between the algebraic topology of non-singular complex algebraic varieties and their geometry expressed by the polynomial equations that define sub-varieties.
In short, the Hodge conjecture is that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes.
Together with Fermat's Last Theorem and Riemann's Hypothesis, it constitutes the geometric topological carrier and tool of m-theory structure that integrates general relativity and quantum mechanics. Its importance is self-evident.
If he can solve the Hodge conjecture, the correctness of general relativity and m-theory will take a big step forward.
For Xu Chuan, the temptation of this matter is undoubtedly quite great.
After all, he studied physics in his previous life and studied under Edward Witten. He was familiar with both general relativity and m-theory.
Suddenly, the phone next to the desk vibrated again. The loud ringing interrupted Xu Chuan's thoughts. He touched the phone and saw that it was Professor Witten calling.
"Hey, mentor, what do you want from me?"
"Where are you? Is it convenient now?" Witten's voice came from the other end of the computer.
"I'm in the dormitory, what's wrong, teacher, is there something wrong?" Xu Chuan replied.
"Then come to Professor Deligne's office now."
"Okay, I'll be there right away."
...
After hanging up the phone, Xu Chuan glanced at the screen of his mobile phone that automatically lit up. The date on it shocked him.
August 27th.
He actually stayed in the dormitory for more than a month without knowing it, which was far longer than the time he asked for leave from Professor Deligne.
What's more important is that Professor Deligne didn't even ask about this matter in the past month.
It's outrageous. The student took seven days off and then didn't go to class for more than a month. The instructor didn't even ask.
Shaking his head, Xu Chuan went into the bathroom to wash his face and tidy up his messy hair. He had been studying mathematics for more than a month. His hair was long enough to cover his ears, so he had to find 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. He found the manuscript paper he had compiled from his previous research on the problem of 'irreducible decomposition of differential algebraic varieties'. He copied it in his hand and prepared to work together. Take it with you.
Although the mathematical tools made for Hodge's conjecture are more important than this, they still need to be brought over and let two instructors help check them. But those things are still lying in a mess throughout the dormitory, on the table, on the floor, on the bed, everywhere, and there is no time to sort them out.
On the contrary, the mathematical tools suitable for the problem of 'irreducible decomposition of differential algebraic varieties' have been sorted out before and can be taken away directly now.
Mathematics tutor Professor Deligne has a lot of experience in differential equations. You can give him a look first to see if there is anything that needs to be modified before submitting the manuscript.
After all, he is only one person, and what he considers may not be comprehensive. Sometimes he can see something different from other perspectives.
...
Carrying a manuscript paper to solve the problem of 'irreducible decomposition of differential algebraic varieties', Xu Chuan walked through the Princeton campus and quickly rushed to the Institute for Advanced Study.
I knocked on the door of the tutor's office and walked in. Both tutors Witten and Deligne were there.
Seeing his slovenly appearance, Deligne couldn't help but frowned and asked, "How long has it been since you went 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 kind of thing is it?" Edward Witten asked curiously from the side. He didn't care about Xu Chuan's image.
It is actually normal for scientific researchers to look like this. Purely theoretical calculations may be slightly better. Except for the weirdo Perleman, there are still very few mathematicians who will look like this.
But in many other disciplines, various experiments are often required. When he was there, he dealt with many staff.
Sometimes when certain equipment is being repaired, the staff often make themselves look unkempt, which is normal.
However, Deligne said before that Xu Chuan was studying the manuscript left to him by Professor Mirzakhani, which made him a little curious.
As a student, does he still have some relationship with Professor Mirzakhani?
"Um."
Xu Chuan nodded and continued: "Some ideas on algebraic varieties are related to the problem of 'irreducible decomposition of differential algebraic varieties'."
Hearing this, Professor Deligne raised his eyelids, leaned forward slightly, and asked with interest: "Can I see the manuscript?"
"The manuscript is still in my dormitory, but I brought some of my own research today and asked two teachers to help me see if there are any flaws in it."
With that said, 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 that 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 instructors took the manuscript from Xu Chuan with some curiosity and began to read it.
The student in front of them has very strong mathematical ability. They all know that more than 99.99% of the Fields Medal will be awarded to him one year later.
Although I am a little younger, in the subject of mathematics, older is not always better.
Between the ages of twenty-five and forty-five, it is a golden career to study mathematics. No matter how young you are, the basic knowledge in your mind is insufficient and you cannot lay the foundation. No matter how old you are, your thinking begins to solidify and age, and it is difficult to achieve anything. Such results.
Of course, this age does not apply to everyone, especially geniuses with excellent mathematical talents.
For example, genius mathematicians such as Schultz and Terence Tao, who are favored by God, both made huge contributions to the mathematics community in their early twenties.
There is no doubt that Xu Chuan is also such a genius, and even more so than Schultz and Terence Tao. After all, the first two did not solve world-class mathematical problems before they were eighteen or nineteen years old.
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 theory algebraic variety correlation method."
On the first piece of manuscript paper, the eye-catching title occupying the top layer caught the eyes of Deligne and Professor Witten. The two of them were shocked. They raised their heads and looked at each other, and then looked down at the proof. process.
The irreducible decomposition problem of differential algebraic varieties is another world-class mathematical problem after Weyl-berry conjecture.
After studying at Princeton for more than a year, has their student finally focused his attention on mathematics?
Compared with the Weyl-berry conjecture, the difficulty of the irreducible decomposition problem of differential algebraic varieties is not much different, 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 Weyl-berry conjecture, the completeness of the irreducible decomposition problem 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, no one has ever broken through it.
The result of the irreducible decomposition problem of differential algebraic varieties has been defined a long time ago, and the irreducible decomposition of differential algebraic varieties exists.
It's just that mathematicians have not yet been able to find a way to a final definition.
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 originate from the Ritt-Wu zero-point decomposition theorem, and are partially solved by the Ritt-Wu zero-point decomposition theorem.
However, the Ritt-Wu zero-point decomposition theorem still has certain limitations in these two issues.
One is the need to further obtain irreducible decompositions, and the other is the failure to provide an algorithm to decompose the solution set of differential algebraic equations into irreducible differential algebraic varieties.
If these two problems can be solved at the same time, the systematic difficulty can surpass the Weyl-Berry conjecture. However, the difficulty of the irreducible decomposition problem of a single differential algebraic variety is indeed not as difficult as the Weyl-Berry conjecture.
But solving these two problems is easier said than done.
In particular, the irreducible decomposition problem of differential algebraic varieties is not much less difficult than the Weyl-berry conjecture when taken alone.
Although it had been proved by Ritt et al. as early as the 1930s: "Any difference algebraic variety can be decomposed into the union of irreducible difference algebraic varieties."
But today, nearly a century has passed, and no one has yet been able to provide an algorithm to decompose the solution set of differential algebraic equations into irreducible differential algebraic varieties.
In the past seventy or eighty years, it is not that no one has tried to solve this problem.
Ritt and others, including those who proved that "any difference algebraic variety can be decomposed into the union of irreducible difference algebraic varieties", also tried to extend the Ritt-Wu zero-point decomposition theorem to algebraic difference equations.
But the result obtained can decompose the differential algebraic variety into the form zero(s)=u/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 problem of the century.
.......
In the office, Deligne and Witten were immersed in the manuscripts in their hands.
Xu Chuan, on the other hand, skillfully took out a copy of the latest issue of "Annual Review of Mathematics" from his tutor's office and started reading it.
In the Institute for Advanced Study in Princeton, there are many top journals of this kind. Almost any professor, whether in 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 unsolicited by journals. Deligne and Witten are naturally the latter.
This has something to do with the fact that these two top bosses are academic editors of various top journals.
After all, in academia, peer review is generally a voluntary labor without any monetary remuneration.
In this case, the journal will naturally pay something else in order to find suitable reviewers. For example, previous reviewers’ submissions were exempted from publication fees, and journal articles were given as free gifts.
Of course, in addition to these, there are also some other invisible benefits, such as improving personal reputation, always updating one's grasp of current scientific research hot spots, etc.
After all, you are reviewing the latest academic papers during peer review. You can obtain different ideas, techniques and perspectives from the reviewed manuscripts, broaden your horizons, and learn from the mistakes made by other researchers. Help improve your own research and more.
.......
The three of them, two old and one young, were immersed in their own manuscripts and papers. They didn't know how long it had passed before the office became active again.
"It's really exciting. I didn't expect that Bruhat decomposition and Weyl groups could be introduced into domain theory in this way." In the office, Deligne sighed after reading the manuscript paper in his hand.
The problem of irreducible decomposition of differential algebraic varieties is a difficult problem in differential equations and algebraic geometry, but it is not oriented to the most cutting-edge mathematics. On the contrary, it was born on the basis of both.
This is like opening a channel on the ground floor of two mathematical buildings to connect the two.
Although everyone knows that this is completely possible 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, a drill or a chisel, these mathematical tools that were commonly used in the past cannot carve out a hole in them.
Now, Xu Chuan constructed a new tool, dug a hole in the originally indestructible wall, successfully connected the two buildings, and further decomposed differential algebraic varieties into irreducible differential algebraic varieties, thus giving The process of irreducible decomposition of differential algebraic varieties.
In this tool, Deligne saw some techniques and shadows of Weyl-berry conjecture, in addition to some algebraic groups, subgroups and torus.
I just don’t know how many of these things belong to Professor Mirzakhani and how many belong to his student.
After all, he had never read Professor Mirzakhani's manuscript, and he didn't know how much was in that manuscript.
But no matter what, there is a high probability that a difficult problem in the palace of mathematics can be solved again.
He didn't say for sure, but he was at least 80-90% sure.
Of course, the manuscript paper in hand is not 100% perfect. There are still some places that can be slightly adjusted, but these are just minor details.
As for whether there are other major flaws, he can't tell now. After all, this is not a simple problem. The difficulty is there. Simply going through it is not enough for him to guarantee that there will be no problems.
.......
