Chapter 205: Interim Results of the NS Equation

Fefferman stared at the young man with dark circles under his eyes for a long time.

Finally, he sighed softly and said, "You win."

He only slept six hours a day and spent 14 hours on research. Fefferman thought that he had never been so crazy when he was young.

He clearly remembers that when he was the craziest and most serious in his 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 was nothing at all.

The work lasted fourteen hours a day 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.

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 not allowed, and we still need to find an opportunity 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 the manuscript.

Xu Chuan's research method, which almost overdraws his 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, a student, will lock himself in the room and study continuously 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 and play football, basketball, watch concerts, listen to operas, etc.

But it didn't work.

Geniuses are stubborn. Just like Perelman 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 big guys thought of him.

In fact, although his research on the problem was a bit overwhelming, it was not truly morbid.

In addition, he is young and runs and exercises every day. Staying up late and exploding the liver is actually not as serious a drain on the body as the two think.

Compared with the two big guys' 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 and a monotonically decreasing smooth function sequence. This is where I get stuck."

"I tried using the Dirichlet function, the Riemann function, and even the analytic function, but I couldn't construct a regular Borel measure."

After thinking about it, 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 build a regular Borel measure and a monotonically declining smooth function sequence, why don't you try to use a high-dimensional co-bud smooth function? Woolen cloth?"

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 again. The memories in his mind quickly turned over, and then he wanted to slap himself in annoyance.

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 memories of the future, 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.

It’s no wonder that he found Fefferman’s idea easier, while Fefferman himself was stuck on the problem.

The reason why he feels it is 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 out a blackboard from the corner of his study. This was something he specifically asked for from Princeton University to facilitate daily mathematical research.

After pondering for a while, 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 Map 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 type is f (x1, x2) → (x1, x1x22+ x42+x52)) The necessary and sufficient conditions are corankf=1, Hessλ(P)

On the sidelines, 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 not only full of surprise and surprise, but also full of 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 used are two mathematical methods: the positive definiteness of the matrix using Horwitz theorem and the wave front where the curved surface is the wave front in the three-dimensional Euclidean space R3.

Through these two methods, a certain equivalence class mapping bud was made.

But it is this seemingly basic thing that can be perfectly integrated with the Dirichlet function to construct a regular Borel measure and a monotonically declining 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 with Bideligne, he was not a pure mathematician.

Because he has also made some progress in physics, and is also a distinguished professor at the 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 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 constructing 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 side, Fefferman added: "Or it should 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 this problem from other angles.

But on this road, they are latecomers after all.

Compared with 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 to CERN 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 and construct pointed singular points and swallow-tail singular points, etc., thus providing analyzability to some previously unanalyzable differential equations."

Hearing this, Fefferman showed an expression that was indeed true.

On the side, Deligne said with a smile: "Young people's thinking is faster than the two old men like us. I communicated with Fefferman twice before, but we 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 explode my liver 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?"