Chapter 183 Another World-Class Problem
Hodge conjecture, one of the seven millennium problems.
It is a major unresolved problem in algebraic geometry.
It was proposed by William Valence Douglas Hodge and is a conjecture about the relationship between the algebraic topology of non-singular complex algebraic varieties and the geometry described by the polynomial equations that define the 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.
It, together with Fermat's Last Theorem and Riemann's hypothesis, constitute the geometric topological carrier and tool of the 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 large.
After all, he studied physics in his previous life and was taught by Edward Witten. He is very familiar with both general relativity and M-theory.
Suddenly, the phone on the desk vibrated again. The loud ring interrupted Xu Chuan's thoughts. He picked up the phone and found that it was Professor Witten who called.
"Hello, tutor, what do you want to talk to me about?"
"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 anything?" 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 looked at the phone screen that lit up automatically. The date on it scared him.
August 27th.
He had stayed in the dormitory for more than a month without knowing it, which was far longer than the time he had asked Professor Deligne for leave before.
More importantly, Professor Deligne didn't even ask about this 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 tutor didn't even ask.
Shaking his head, Xu Chuan went to the bathroom to wash his face and tidied up his messy hair. He had been concentrating on studying mathematics for more than a month, and his hair had grown long enough to cover his ears. He had to find time to trim it.
Just as he stepped out of the dormitory and was about to close the door, Xu Chuan paused, turned around and went back into the room, found the manuscript he had sorted out for the previous study of the problem of "irreducible decomposition of differential algebraic clusters", copied it in his hand, and prepared to take it with him.
Although the mathematical tools made for the Hodge conjecture are more important than this, they also need to be taken over for the two tutors to help check. But those things are still lying in a mess in the whole dormitory, on the table, on the floor, on the bed, everywhere, and there is no time to sort them out.
However, the mathematical tools suitable for the problem of "irreducible decomposition of differential algebraic clusters" have been sorted out before, and now they can be taken away directly.
Professor Deligne, the mathematics tutor, has a lot of experience in differential equations, so you can show him first to see if there is anything that needs to be modified, and then submit it.
After all, he is just one person, and what he considers may not be very comprehensive. Sometimes, different things can be seen from other perspectives.
Carrying the manuscript paper for solving the problem of "irreducible decomposition of differential algebraic clusters", Xu Chuan crossed the Princeton campus and quickly rushed to the Princeton Institute for Advanced Study.
He knocked on the door of the tutor's office and walked in. Both tutors, Witten and Deligne, were there.
Seeing his sloppy appearance, Deligne couldn't help frowning and asked, "How long have you not been out?"
Xu Chuan scratched his head and smiled, "Maybe two months?"
"Are you studying the manuscript left to you by Professor Mirzakhani? What is it about?" Edward Witten asked curiously on the side. He didn't care about Xu Chuan's image.
It is actually normal for scientific researchers to look like this. Pure theoretical calculations may be slightly better. Except for the weird Perelman, there are still few mathematicians who will make themselves look like this.
But many other disciplines often have to do various experiments. When he was at CERN, he had dealt with many staff members.
Sometimes when some equipment is being repaired, the staff often make themselves dishevelled, which is normal.
But Deligne said before that Xu Chuan was studying the manuscript left to him by Professor Mirzakhani, which made him a little curious.
Is this student of his still related to Professor Mirzakhani?
"Yes."
Xu Chuan nodded, and then said: "Some ideas about algebraic clusters are related to the problem of 'irreducible decomposition of differential algebraic clusters'."
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. Please help me see if there are any flaws in it."
As he 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.
Professor Deligne needs no further explanation. He is the only two Grand Slam players in the mathematics world, and differential algebra and algebraic geometry are his main fields.
Although Witten is a physicist, he is also very good at mathematics. After all, he has won the Fields Medal. From his perspective, he might be able to find some loopholes.
The two mentors took the manuscript from Xu Chuan with some curiosity and started 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 a good 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, 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' - algebraic variety correlation method in field theory."
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 the 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 problem of irreducible decomposition 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 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 problems of irreducible decomposition of differential algebraic varieties and 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 will 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 was 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.
However, the results obtained can decompose the differential algebraic variety into the form 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 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, 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 the right reviewer. For example, the previous reviewer's submission is free of page charges, and the journal paper is given as a gift.
Of course, in addition to these, there are some other invisible benefits, such as improving personal reputation, constantly updating one's grasp of current scientific research hotspots, etc.
After all, peer review is the latest academic paper you review, and you can get different ideas, techniques and perspectives from the reviewed manuscripts, broaden your horizons, and learn from the mistakes made by other researchers, take them as a warning, and help improve your own research, etc.
Two old and one young, the three were immersed in their own manuscripts and papers, and it was unknown how long it had passed before the office became active again.
"It's really wonderful. I didn't expect that Bruhat decomposition and Weyl group could be introduced into field theory in this way." In the office, after reading the manuscript in his hand, Deligne sighed.
The problem of irreducible decomposition of differential algebraic clusters is a difficult problem in differential equations and algebraic geometry, but it is not aimed at the most cutting-edge mathematics. On the contrary, it was born on the basis of both.
This is like opening a passage on the ground floor of two mathematical buildings to connect the two.
Although everyone knows that this can be done as long as it does not affect the load-bearing wall.
But the difficulty lies in the fact that the materials used to construct the walls of the two buildings are too hard. Whether it is a hammer or a chisel, these commonly used mathematical tools cannot make a hole in it.
Now, Xu Chuan has constructed a new tool to make a hole in the originally indestructible wall, successfully connecting the two buildings, and further decomposing the differential algebraic cluster into irreducible differential algebraic clusters, thus giving the irreducible decomposition process of differential algebraic clusters.
In this tool, Deligne saw some techniques and shadows of the Weyl-Berry conjecture, as well as some algebraic groups, subgroups and tori.
I just don't know how many of these things belong to Professor Mirzakhani and how many belong to his student.
After all, he has not seen Professor Mirzakhani's manuscript, and he doesn't know how much is in that manuscript.
But in any case, a difficult problem in the hall of mathematics is likely to be removed.
He didn't say for sure, but he was at least 80% to 90% sure.
Of course, the manuscript in his hand was not 100% perfect. There were still some places that could be slightly adjusted, but these were just minor details.
As for whether there were other major flaws, it was impossible to determine now. After all, this was not a simple problem. The difficulty was there. Simply going through it once was not enough for him to guarantee that there were no problems.