Fefferman stared at the young man with some dark circles under his eyes for a long time.
Finally, he sighed softly and said, "You win."
He only sleeps six hours a day and devotes 14 hours to research. Fefferman thinks he has never been so crazy when he was young.
He clearly remembered that when he was at his craziest and most serious research, he was only studying the local solvability of non-degenerate linear partial differential equations, working more than ten hours a day for more than half a month.
In comparison, his level of liver-explosion can be said to be nothing at all.
Working fourteen hours a day lasted for more than a month and a half.
It's scary just thinking about it. If this state were to happen to him, he would probably die suddenly, right?
The most despairing thing in the world is that a genius who is better than you works harder than you.
But this also made Fefferman sigh.
It's good to be young, you can be wild and explosive.
At his age, even if he wanted to stay up late, he couldn't do it.
Now he finally understood why the young man in front of him could stand at the top of the mathematics world at the age of twenty.
Extreme talent, extreme concentration, extreme hard work, if he doesn't succeed, who will succeed?
However, this kind of pathological research method is unacceptable, and we still need to find opportunities to persuade it.
At least, he didn't want to see such a genius who could promote the development of mathematics die young, or suffer from illness in his later years.
...
On the other side, Deligne, who was immersed in the manuscript, looked up at Xu Chuan, said nothing, and then lowered his head to read through the manuscript.
Xu Chuan's research method, which almost overextended his own future, had been criticized many times when he was still his student.
If you don't find a way to solve the problem, that's fine.
But when he finds an idea, he, as a student, will lock himself in the room and study non-stop until he solves the problem or proves that his idea is wrong.
According to Deligne, this research method is making fun of his own health.
He had persuaded him many times, hoping that he could relax a little while studying the problem and stop staying up all day and night and devote all his time to research.
】
You can go out to play football, basketball, watch concerts, listen to operas, etc.
But it's no use.
Geniuses are stubborn. Just like Perleman who rejected all fame and fortune, his student's desire for knowledge and problem solving also reached an almost pathological level.
...
Xu Chuan didn't care what the two elders thought of him.
In fact, although his research on the problem was a bit overwhelming, it was not truly morbid.
Coupled with the fact that he is young and runs every day for exercise, staying up late and exploding the liver is actually not as serious a drain on the body as the two of them thought.
Compared with the two elders' concerns, he is now more curious about where Fefferman is stuck.
"Can I ask, where are you stuck in your research on the isospectral problem?"
Xu Chuan was curious about the question. In his opinion, it was obviously easier for Fefferman to put forward the idea.
After hearing the inquiry, Fefferman came back to his senses and said: "I cannot construct a regular borel measure μ and a monotonically decreasing smooth function sequence. This is where I get stuck."
"I have tried using the Dirichlet function, the Riemann function, and even the analytic function, but I cannot construct a regular borel measure."
After thinking for a while, he added: "Do you have any good suggestions? If so, please tell me."
Xu Chuan was stunned for a moment. He did not expect that Fefferman would get stuck in this place: "If you just simply build a regular borel measure μ and a monotonically decreasing smooth function sequence, why don't you try to use high-dimensional cobud smoothing
Where are the functions?”
Hearing this, Fefferman was a little confused. After thinking for a moment and confirming that he had never heard of such a function, he hesitantly asked: "High-dimensional cobud function? What is that?"
On the side, Deligne also raised his head curiously. Not only Fefferman, but even he had never heard of the name of this function.
Being stared at by the two people, Xu Chuan was stunned for a moment. The memories in his mind quickly turned over, and then he wanted to slap himself in frustration.
It is now 2018, and the high-dimensional cobud function, a function used to identify function extreme points and singular points, has not yet appeared.
It would be two years later before this function would be formally proposed by him and applied to the physical discoveries at that time.
He has future memories, but Fefferman and Deligne do not.
However, since this function has been released in advance, there is no other way but to take advantage of the situation and launch it in advance.
Fortunately, this research result was developed by himself in the future, not someone else's.
Otherwise he would really consider writing it out.
After all, in his opinion, publishing other people's future research results directly in advance is tantamount to plagiarism, even if the original owner has no relevant ideas in mind at this time.
It's no wonder that he found the idea proposed by Fefferman easier, but Fefferman himself was stuck on this problem.
The reason why he finds it easier is because he has more knowledge about the next ten years and some of the current problems will have been solved in the future.
After exhaling, Xu Chuan dragged a blackboard out from the corner of his study. This was something he specifically asked for from Princeton University, in order to facilitate daily mathematical research.
After pondering for a moment, he picked up the chalk and began to write: "Suppose f: (r, 0) → r is a smooth function. If 0 is an ak-type singularity of y=f(x), then there must be a diffeomorphism
Mapping φ: (r,0)→(r,0), such that f°φ=±xk 1 f(0)....."
"..."
On the blackboard, Xu Chuan slowly sorted out the construction and theorems about high-dimensional cobud functions in his mind.
"....For the mapping bud f: (u, p) → (r2, 0), where ur2, f is at p point a - equivalent to 115 singular points (the standard form is f (x1, x2) → (x1
,x1x22 x42 x52)) The necessary and sufficient condition is kf=1,hessλ(p)
This chapter is not finished yet, please click on the next page to continue reading the exciting content! On the side, Fefferman and Deligne watched intently.
From curiosity at the beginning, to surprise, and then to shock.
As the calculations on the blackboard gradually became complete, both of them saw the value of this function.
Especially Fefferman's eyes were filled with not only surprise and surprise, but also incomprehensible confusion.
Judging from the data on the blackboard, this 'high-dimensional cobud function' is not a very complicated thing, and can even be said to be very basic.
Mainly using two mathematical methods: the positive definiteness of the matrix using Horwitz theorem and the wave front in the three-dimensional Euclidean space r3 whose curved surface is the wave front.
Through these two methods, certain equivalence class mapping sprouts were made.
But it is this seemingly basic thing that can be perfectly integrated with the Dirichlet function to construct a regular borel measure μ and a monotonically decreasing smooth function sequence in a three-dimensional surface.
The basic structure and basic application can perfectly solve the problem.
However, this mathematical method does not seem to be developed specifically for mathematics.
Looking at the calculations on the blackboard, Fefferman felt a strong sense of violation in his heart.
Compared to Bideligne, he was not a pure mathematician.
Because he has also made some progress in physics, and is also a distinguished professor at Fermilab National Accelerator Laboratory, specializing in calculating various physical data for Fermilab, so he also has some understanding of physics.
From the calculations on the blackboard, Fefferman was keenly aware of the physical uses of these formulas. In his opinion, these formulas did not seem to be developed for mathematics, but more like they were tailor-made for physics.
Of course, it can also be applied to mathematics.
For example, now, it can solve the equal spectral problem for him.
...
In front of the blackboard, Xu Chuan put down the last stroke, then stopped the chalk in his hand and turned to look at Fefferman and Deligne.
"This is the 'high-dimensional cobud function'. It is a method of calculating the extreme points of a smooth function. It may be applied to constructing a regular borel measure in a three-dimensional surface."
Fefferman and Deligne nodded in unison.
It is not difficult for them to understand these things on the blackboard.
Deligne looked at the calculations on the blackboard, pushed up his glasses, and said: "This method is not difficult to construct. It is a very basic thing, but it requires a certain amount of skill, and it is not irreplaceable."
"For example, starting from Morse's lemma, similar effects can be achieved by using the qualitative properties of the potential function near the non-degenerate stationary point."
On the sidelines, Fefferman added: "It should also be possible to use extreme value theory and singularity theory."
After Xu Chuan figured out this key, the two of them had some ideas to solve the problem from other angles.
But on this road, they are latecomers after all.
Compared to Xu Chuan who blazed the trail, there is no doubt that it is easier for latecomers to complete their ideas.
After expressing his thoughts, Fefferman continued: "But what puzzles me is that this method does not seem to be specially constructed for mathematics. It is more like an inference for studying physics."
As he said that, he looked at Xu Chuan and said affirmatively: "Is that so? Xu."
Xu Chuan smiled and said: "Yes, last year, I went with Witten to complete some physical experiments. This function derivation method is part of the mathematical results derived from calculating physical particles."
"It can stably reduce the interval value, construct sharp singularities and swallow-tail singularities, etc., thereby providing analyzability to some differential equations that were originally unanalyzable."
Hearing this, Fefferman showed a true expression.
On the side, Deligne smiled and said: "Young people's thinking is faster than the two old men like us. I communicated with Fefferman twice before, but we still couldn't find a solution to this problem."
"And as soon as you came out, you provided a complete idea."
"After seeing Fefferman, I can solve the isospectral problem and complete the ns equation within three months."
Fefferman glanced at Deligne, always feeling that he was laughing at him, and said: "Please don't bring me when you talk about the old man, I am still very young."
"But I do have to thank Xu this time. It seems that I will try to make my liver explode when I go back."
After a pause, he looked at Xu Chuan and said with a smile: "By the way, you young people use the word 'explosive liver', right?"