Staring at the young man in front of him with some dark circles under his eyes, Feverman looked at it for a long time.

Finally, he sighed lightly and said, "You won." ”

With only six hours of sleep a day, and fourteen hours spent researching, Feverman admits that he had never been so crazy in his youth.

He clearly remembers that when he was the craziest and most serious in his research, he only worked more than ten hours a day for more than half a month when he was studying the local solvability of non-degenerative linear partial differential equations.

In comparison, his level of liver burst can be said to be nothing at all.

Fourteen hours a day, lasting more than a month and a half.

It's terrible to think about it, this state, if it is put on him, I'm afraid he will have to die suddenly, right?

There is nothing more desperate in the world than a genius who is better than you and works harder than you.

But it also made Feverman sigh a little.

It's good to be young, you can burst your liver unbridledly.

At his age, even if he wants to stay up late, he can't do it.

Now he finally knew why the boy in front of him could stand at the top of the mathematical world at the age of twenty.

Extreme talent + extreme focus + extreme effort, who will succeed if he doesn't succeed?

However, this kind of pathological research method is unacceptable, and it is still necessary to find an opportunity to persuade it.

At the very least, he didn't want to see such a genius who could advance mathematics die young, or be sick in his later years.

......

On the other side, Deligne, who was immersed in the original manuscript, looked up at Xu Chuan, did not speak, and lowered his head to read the manuscript again.

Xu Chuan's research method, which almost overdrew his future, had already been criticized many times when he was still a student.

If you don't find a way to solve the problem, that's fine.

But when he finds an idea, he, the student, will lock himself in his room and study it until he solves the problem or proves that his idea is wrong.

This research method, in Deligne's view, is a joke on one's own health.

He has advised him many times, hoping that he can relax when researching the problem, not stay up all day and devote all his time to research.

】

You can go out and play football, basketball, concerts, opera, etc.

But it didn't work.

Geniuses are stubborn, and just as Perlehman rejects all fame and fortune, his student's thirst for knowledge and problem-solving have reached an almost pathological level.

......

Xu Chuan didn't care what the two elders thought of him.

In fact, his research on the problem, although somewhat overdrawn, has not reached the point of being truly morbid.

In addition, he is young and runs and exercises every day, and the consumption of staying up late to burst his liver is not as serious as the two of them think.

Compared with the concerns of the two elders, he is now more curious about where Fevermanka is.

"May I ask, where are you stuck in the study of the equation problem?"

Xu Chuan's curious question, in his opinion, it is obviously easier for Feverman to put forward ideas.

Hearing the inquiry, Feverman came back to his senses and said, "I can't construct a regular Borel measure μ and a monotonically descending sequence of smooth functions, and that's where I get stuck." ”

"I've tried using Dirichlet functions, I've tried using Riemannian functions, and even analytic functions, but I can't build a regular Borel measure."

After thinking about it, he added: "Do you have any good advice?" If so, please let me know. ”

Xu Chuan was stunned for a moment, he didn't expect Feverman to get stuck in this kind of place: "If you just build a regular Borel measure μ and a monotonically descending smooth function sequence, why don't you try to use the high-dimensional residual bud smooth function?" ”

Hearing this, Feverman was a little confused, thought for a while, and after confirming that he had never heard of this kind of function, he asked hesitantly: "High-dimensional residual bud function?" What is that? ”

On the side, Deligne also looked up curiously, not only Feverman, even he had never heard of the name of this function.

Being stared at by the two, Xu Chuan was stunned again, the memory in his mind quickly flipped, and then he wanted to slap himself in annoyance.

It's 2018, and the high-dimensional residual bud function, which is applied to the identification of extreme points and singularities, has not yet appeared.

It would not be until two years later that this function would be officially proposed by him and applied to the physical discoveries of the time.

He had memories of the future, but neither Feverman nor Deligne.

However, since this function has been released in advance, there is no choice but to launch it in advance.

Fortunately, this research result was developed by himself in the future, not by others.

Otherwise, he really thought about whether to write it out or not.

After all, in his opinion, sending out the research results of others in the future directly in advance is tantamount to a kind of plagiarism, even if it is not related to the original owner's mind.

It's no wonder he finds it easier for Feverman to come up with an idea, while Feverman himself is stuck on the issue.

The reason why he feels that it is easier is because he has more knowledge in the next ten years, and some of the current problems have already been solved in the future.

Exhaled, Xu Chuan dragged a blackboard out of the corner of his study, which he specially 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: "Let f:(r,0)→r be a smooth function, and if 0 is an ak-shaped singularity of y=f(x), then there must be a differential homomorphic 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 of the high-dimensional residual bud function in his mind.

“.... For the mapping bud f:(u,p)→(r2,0), where ur2,f is at p point a—equivalent to 115 singularity (standard type is f(x1,x2)→(x1,x1x22+x42+x52)), the sufficient necessary condition is kf=1, hessλ(p)

From the side, Feverman and Deligne watched intently.

From curiosity at first, to surprise, to shock.

As the equations on the blackboard gradually became complete, both of them saw the value of this function from inside.

Especially Feverman, there was not only surprise and surprise in his eyes, but also incomprehensible confusion.

Judging from the data on the blackboard, this kind of 'high-dimensional residual bud function' is not a very complicated thing, and it can even be said to be very basic.

The positive definite nature of the matrix is mainly used, using two mathematical methods: Holwitz's theorem and the wave front with the surface as the wave front in the three-dimensional Euclidean space r3.

Through these two methods, some equivalent class mapping buds were made.

However, it is precisely this seemingly basic thing that can be perfectly integrated with the Dirichlet function to construct a regular Borel measure in the three-dimensional surface μ and a monotonically descending smooth function sequence.

The basic structure, the basic application, can solve the problem perfectly.

It's just that this kind of mathematical method doesn't seem to have been developed specifically for the sake of mathematics.

Looking at the equations on the blackboard, a strong sense of disobedience rose in Fefferman's heart.

He was not a pure mathematician compared to Bidrinje.

Because he also has some development in physics, and he is also a distinguished professor at the Fermi National Accelerator Laboratory, which specializes in calculating various physical data for Fermilab, he also has some understanding of physics.

From the equations on the blackboard, Fefferman was keenly aware of the physical use of these formulas, which in his opinion did not seem to have been developed for mathematics, but more like they were tailored for physics.

Of course, it can also be applied to mathematics.

For example, now, it is just right to solve the isospectral problem for him.

......

In front of the blackboard, Xu Chuan dropped the last stroke, then stopped the chalk in his hand and turned to look at Feverman and Deligne.

"This is the 'high-dimensional residual bud function', which is a method of calculating the extreme points at which a smooth function is constructed. Perhaps it can be applied to constructing a regular Borel measure in a 3D surface. ”

Feverman and Deligne nodded in agreement.

It's not hard for them to understand these things on the blackboard.

Deligne looked at the equation on the blackboard and pushed his glasses, and said, "This method is not difficult to build, it is very basic, it just requires a certain amount of skill, and it is not irreplaceable." ”

"For example, starting from Morse's lemma, a similar effect can be achieved by using the qualitative properties of the potential function near the non-degenerate stationary point."

On the side, Fefferman added: "Or use extreme value theory and singularity theory, it should be possible." ”

After Xu Chuan broke this key, the two of them had some ways to solve this problem from other angles.

But on this road, they are latecomers after all.

Compared with Xu Chuan, who opened up the road, there is no doubt that it is easier for latecomers to complete their ideas.

After expressing his thoughts, Feverman continued: "But what puzzles me is that this method does not seem to have been built specifically for mathematics, but more like a deduction for the study of physics. ”

As he spoke, he looked at Xu Chuan and said affirmatively, "Is that so?" Composed. ”

Xu Chuan smiled and said, "That's right, last year, I went with Witten to complete some physics experiments, and this method of deriving functions is part of the mathematical results derived from the calculation of physical particles. ”

"It can steadily reduce the interval values and construct the skull singularity and dovetail singularity, so as to provide analyzability for some of the original unanalyzable differential equations."

Hearing this, Feverman showed an expression that was indeed so.

On the side, Deligne smiled: "Young people's thinking is faster than our two old men, I have communicated with Feverman twice before, but I have not found a solution to this problem. ”

"And as soon as you came out, you provided a complete idea."

"After reading Fefferman, he can solve the isospectral problem in three months this time and complete the phased results of the ns equation."

Feverman glanced at Deligne, always feeling that he was laughing at himself, and said, "Please don't take me with you when talking about old men, I'm still very young." ”

"But I do have to thank Xu this time, it seems that I will try to burst 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 'burst liver', right?" ”

......