Research starts with PhD students

Chapter 130 Opportunity? Not Existing! Purely Relying on Talent! NS Equation and Singularity Problem

"Your answer is really..."

Qiu Chengwen laughed for a long time and then made a comment, "It doesn't have much content, but it's quite correct. Research is like this."

It’s normal for others to think about it, because research is a process of putting an elephant in a refrigerator.

Open the door, put the elephant in, and then close the door.

As for whether you can get in, it depends on your personal ability and opportunities, and even on the research itself.

If the research itself is problematic, the elephant will naturally never be "locked up".

Qiu Chengwen exclaimed, "You are so talented!"

This is a review from the heart.

Before, Qiu Chengwen felt that he saw his younger self in Zhang Shuo. He believed that doing research is a combination of talent and opportunity. Talent is very important, but opportunity is also important.

When there is no opportunity, a problem cannot be solved.

Kabila's conjecture was suddenly inspired after he got married, and he then completed the subsequent proof.

That moment of inspiration is very important.

Now Qiu Chengwen no longer thinks so, because Zhang Shuo has completed so much research.

The research related to the numerical simulation of the NS equations has reached the world's top level in terms of mathematics, scientific research and application.

Zhang Shuo has also completed many cutting-edge research projects, including the Monge-Ampere equation, the Gerbov conjecture, and algorithms for high-energy physics.

There are so many achievements that there is no way to talk about "opportunities".

It's like buying a lottery ticket. If you win occasionally, you can say it's good luck, but if you win all the time, can you still say it's luck?
Therefore, Zhang Shuo relies on his talent.

Qiu Chengwen continued chatting with Zhang Shuo with a sigh. In the past thirty years, he has devoted most of his energy to education and has done a lot to promote the development of domestic mathematics.

He established awards specifically for mathematical research; founded a foundation to support young mathematicians; and took the lead in organizing the influential Conference of Chinese Mathematicians.

Of course, the work of the Center for Mathematical Sciences is also a focus, and he hopes to train more outstanding young mathematicians.

Qiu Chengwen talked about the domestic mathematical work and the basic science conference held in the capital in July. "You didn't attend this conference."

"Among the judges invited at that time, there were five Fields Prize winners, two Nobel Prize winners, two Turing Award winners, and many experts who have won international awards."

“We have established a very important international award at the conference, called the Best Paper Award.”

"More than 100 judges spent a month discussing and evaluating the most outstanding papers in the past five years that best represent the direction of mathematical development."

"In order to ensure objectivity and fairness, we set up a review committee and finally selected 86 best papers."

Qiu Chengwen added, "The conferences held in the capital, the selection meetings and discussions are all convened by domestic university professors, so it is unlikely that domestic scholars will be discriminated against."

“You may not expect the result…”

Zhang Shuo listened carefully.

“Of the final 86 papers, only eight were research by professors at domestic universities.”

"Many professors who have won awards in China were not selected, which means that their research results are not at the forefront of world mathematics and have not been recognized by international peers."

Qiu Chengwen was talking about the problems of the domestic academic evaluation mechanism.

Many mathematicians lack innovative research and only conduct research in an "inductive" direction.

For example, if a certain direction can get more funding, then we can do research in that direction.

For example, if it is easy to publish papers in a certain field, you can continue to delve deeper into it.

Some innovative directions are simply not being worked on because project applications are not approved. Reviewers do not understand cutting-edge research and will not approve them. Even if one is interested in doing research, there is no funding. Even if one does achieve results in one's own research, it is difficult to evaluate their innovativeness.

These have led to a serious lack of cutting-edge innovation in the field of mathematics.

Qiu Chengwen shook his head and changed the subject, "Among the eighty-six papers selected, there is also one of yours. Can you guess which one it is?"

Zhang Shuo thought about it carefully and said the name of the paper, "PDE transformation method and parameter analysis to prove asymptotic solutions?"

"This is it."

Qiu Chengwen smiled and said, "Your paper was submitted by a French professor. We discussed at the time that this paper has great application value and the methods used are very innovative."

“Now I think, if the conference is held this month, and you review the papers, your paper will not be selected.”

"You and Professor Sun's proof of the Jebov conjecture, as well as the three papers on the NS equation, are all eligible for the award, but we will combine the three papers into one and give you the best paper award."

Zhang Shuo smiled and said, "It's a pity that I missed the award."

He was just joking.

Qiu Chengwen smiled and said, "Don't feel sorry, your research will never lack awards."

"Not to mention the smaller domestic awards, you will definitely be included in the Natural Science Award at the end of this year."

"You will definitely win one of the several awards at the next International Congress of Industrial and Applied Mathematics. The Collatz Prize is more likely to be yours. The last winner, Maria Colombo, won the award for her work on singularity analysis of partial differential equations."

"There is also the International Congress of Mathematicians. If I can't win the Fields, I can still win the highest award in applied mathematics in the future."

He was talking about the 'Gauss Prize'.

The Gauss Prize is the highest award in international applied mathematics, specifically used to reward achievements in applied mathematics.

……

In the afternoon, Zhang Shuo stayed in the Center for Mathematical Sciences, where he had a long discussion with Qiu Chengwen and several professors.

In the eyes of others, Zhang Shuo's main field is computational mathematics, especially the calculation of partial differential equations, and the topic of discussion has always been around the research of partial differential equations. Qiu Chengwen talked about the research content in the field of partial differential equations that received the most attention from the audience - the NS equation argument.

"Professor Zhang Shuo, you have completed the research on the numerical simulation of the NS equation. The next step is to try to prove the NS equation."

“It’s a good direction, but also a difficult direction.”

"There has been no progress in this direction for a long time. I don't expect to complete all the demonstrations, but a small progress is good."

Qiu Chengwen spoke very seriously, and they also talked about the research direction of the NS equation proof.

Over the past few decades, there have been two main directions in the argumentation of the NS equations.

The first direction is to study the partial regularity of the equation. This method studies the size of the singular set of weak solutions based on the idea of ​​infinite time singularity and fractal idea.

The second direction is to attach certain criteria to known weak solutions and prove their uniqueness and smoothness.

Since the new century, many people have conducted research in the second direction and published a large number of papers. Most of the results prove the existence of smooth solutions to the equation.

This proof is incomplete because they do not take into account the problem of singularities.

Qiu Chengwen said, "You have done a thorough research on the numerical method of the NS equation. You can use this as a breakthrough point to continue your research from the direction of proving the weakness."

When it comes to the argument for the solution to the NS equations, everyone else is speechless.

This question is too difficult.

Zhang Shuo shook his head and said, "After I completed the numerical simulation research, I realized that there was no hope of completing the demonstration from the perspective of weak solution."

"Oh?"

Qiu Chengwen immediately became interested.

The others who looked over also looked a little surprised. It was rare to see someone deny Qiu Chengwen's point of view to his face.

Zhang Shuo said, "The weak solution is not a numerical solution, but it is not a strict exact solution either. The reason why the NS equation studies the weak solution is because it is too difficult to solve."

"However, proving weak solutions is similar to numerical simulations, and there is a big problem, which is that we cannot be sure whether there is a singularity."

“Even if we do more proofs and prove the uniqueness of more known weak solutions, we cannot get around the singularity problem.”

“So if we want to complete the proof, we must start from the singularity problem.”

He was very sure.

When he was setting up the tasks related to the proof of the NS equation, he determined that the singularity problem was the core of the NS equation. The difficulty of studying the singularity problem of the NS equation and completing the entire proof was not much different.

Weak solution research is only a necessary choice due to the difficulty in finding a solution.

Qiu Chengwen smiled bitterly and said, "What you said is correct, but the singularity argument is... too difficult."

"It's very difficult to find a solution, let alone prove the singularity. Many people have thought about this direction, but they have no way to get started."

In other words, it just can't start.

Some scholars doing physical engineering research add specific boundary conditions to the NS equations and combine them with clear physical laws to start studying the singularity problem of the NS equations.

For example, a certain physical phenomenon will 'explode' when it reaches a certain limit.

This can be regarded as the singularity of the NS equation.

The problem is that this is not the case from a mathematical perspective. The physical explosion may be affected by the boundary coefficient. If we return to the NS equation with conventional boundary conditions, the "explosion point" may be continuous.

On the other hand, the NS equations with conventional boundary conditions are too difficult to solve, making it even more difficult to study singularities.

In short, there is nothing wrong with studying from the perspective of singularity, but there is simply no starting point.

……

Zhang Shuo discussed with Academician Qiu Chengwen and others for a long time. Seeing that it was almost five o'clock, he decided not to disturb them any further.

Qiu Chengwen and his team sent Zhang Shuo to the school gate.

Qiu Chengwen said with some regret, "When I invited you in the beginning, I hoped you could come and give a lecture."

"Many students want to listen to your lectures. You don't have to talk about professional things, just talk about your research and encourage students."

"You are the idol of many students..."

Zhang Shuo readily promised, "Next time I come, I will definitely give a lecture."

"Then it's settled?"

"That's a deal!"

Zhang Shuo smiled and shook hands with Qiu Chengwen, then shook hands with several other professors, and then got into the car.

He was riding in a special car sent by the Second Academy of Aerospace Science and Technology.

The next destination is the headquarters of the Second Academy of Aerospace Science and Technology, but due to time constraints, he decided to go directly to the hotel to rest and go to the Second Academy of Aerospace Science and Technology after breakfast tomorrow.

On the way, Zhang Shuo thought of the discussion on the proof of the NS equation with Qiu Chengwen, and suddenly became very interested in the singularity problem.

The argument for the NS equation is generally conducted from the perspective of weak solutions, mainly because it is very difficult to solve. From a purely mathematical perspective, that is, without setting special boundary conditions, it is impossible to address the singularity problem of the equation.

“How do I start doing research from the perspective of singularity argument?”

Zhang Shuo was still thinking but had no idea. Suddenly, an idea came to his mind, "Maybe, I should settle down and do an A-level research?"

"The proof of the NS equation is really good. Anyway, the next step is the experiment, and there is nothing else that must be studied."

"Take it slow, accumulate little by little, and the research will eventually be completed..."

(End of this chapter)