Research starts with PhD students

Chapter 31 B-level research, the system actually encourages more krypton coins?
Although there are no explicit regulations in academia, the funding for scientific research projects is often linked to their difficulty.

The higher the cost, the higher the difficulty.

vice versa.

The smoothness proof of the Monge-Ampere equation is undoubtedly a very important and high-end research, and its difficulty is very clear from its research history.

The Monge-Ampere equation originated from the optimal transmission problem proposed by Monge more than two hundred years ago. Later, two French mathematicians, Monge and Ampere, started this theoretical direction together.

In the 1980s, Brenier further elaborated on the relationship between optimal transmission and the Monge-Ampere equation.

His student Villani applied optimal transport theory to differential geometry and statistical physics, and won the Fields Medal for his proof of nonlinear Landau damping and his study of the convergence of the Boltzmann equation to the equilibrium state.

Villani's student Alessio Figali studied the regularity theory of optimal transport mappings and its intrinsic connection with the Monge-Ampere equation. He also won the Fields Medal for his a priori estimate of the second-order derivative W of the solution of the Monge-Ampere equation and his application to geometric inequalities.

In other words, the research on the Monge-Ampere equation has involved two Fields Medalists, which shows the difficulty and importance of the research.

The Monge-Ampere equation has a wide range of applications, and with the continuous development of computer technology, its applications will only become more extensive in the future.

From the most challenging transmission problems to medical imaging, wireless communications, automotive industry, deep learning, and more.

The shadow of Monge-Ampere equation is everywhere in modern technology.

The Monge-Ampere equation has always been a very difficult problem to solve due to its completely nonlinear characteristics.

This is also the direction in which most scholars study the Monge-Ampere equation - in order to make it easier to solve, it is necessary to study its existence, uniqueness and smoothness (regularity).

For nonlinear partial differential equations, advanced research papers all deal with problems related to the three major properties.

The proof of the smoothness of the Monge-Ampere equation will definitely be able to reach the level of a Young Elite Scientist project or above.

“In fact, it’s not just because of the difficulty.”

Luo Yongjun was exposed by Zhang Shuo for the problems of the project, so he just gave up. He opened a paper document and said depressedly, "My luck is not good either. I just applied for the project, and the project funds have not yet been received, and others have completed the same research."

Zhang Shuo looked at the information carefully, and then looked at Luo Yongjun with sympathy and pity.

The author of this paper is Professor Chen from the University of Science and Technology. The journal in which it was published is the Annals of Mathematics, which is considered the "most difficult to publish in". The content of the paper is a study on the smoothness of the Monge-Ampere equation.

The first person to make a breakthrough in the study of the smoothness of the Monge-Ampere equation was Louis Caffarelli, who proved that when two regions are uniformly convex and the density function is smooth, the optimal transport solution is smooth.

There are some restrictions here: the two regions are uniformly convex and the density function is smooth.

Over the next twenty years, relevant scholars believed that these conditions (especially the uniform convexity of the region) were indispensable.

The research results of Professor Chen's team removed the uniform convexity conditions of the two regions and even reduced the smoothness requirements for the boundaries, proving the overall smoothness of the Monge-Ampere equation under natural boundary conditions.

This is a major step forward in the study of the Monge-Ampere equation, which is equivalent to expanding the scope of a theorem to a wider field.

So Luo Yongjun had no room for research.

Luo Yongjun didn't think so. He said confidently, "Professor Chen's research is indeed a great breakthrough, but I later thought about it and felt that the smoothness requirements of the boundary could be further reduced."

"I modified the project content to continue to expand on the research of Professor Chen's team."

“Mathematics has been improved little by little, especially in the field of nonlinear partial differential equations.”

Zhang Shuo pursed his lips and asked, "Any progress?"

Luo Yongjun was silent in an awkward manner, and after a long while he spoke with a guilty conscience, "There is a little bit of progress!"

"Can you tell me?"

Zhang Shuo said nonchalantly.

While opening the system, he created a task in order to see the difficulty of the research and also to find out whether Luo Yongjun's research was feasible.

【Task 1】

[Research Project Name: Smoothness Demonstration of Monge-Ampere Equation (Further Reducing the Smoothness Requirements of Boundaries) (Difficulty Assessment: B). ]

[Progress: 0.002%]

(The task can be canceled. Currently, the number of scientific research coins required to cancel the task is 0.)
(The remaining progress requires research coins: 500.)
“Being able to establish a mission shows that the research is feasible.”

"Difficulty level B? 500 research coins required?"

"Is the difficulty level of B-level research all 500 points, or does different research difficulties require different amounts of research coins?"

"besides……"

“Pure mathematics is much more difficult than algorithms!”

A small breakthrough in the study of the Monge-Ampere equation has reached a level of difficulty that requires a program of 500 research coins.

But it makes sense if you think about it carefully. The existence proof of smooth solutions of the NS equation in three-dimensional space is one of the seven mathematical conjectures of the millennium.

Zhang Shuo thought about it and shook his head, then patiently listened to Luo Yongjun's explanation. "This is what I think."

"Starting from the analysis of boundary functions, we set a third-order bounded region, where a, b, c, d, and e represent functions..."

"We can analyze them separately..."

"So we have a representation point that contains a positive function..."

"Look at Professor Chen's proof. This part is..."

"To make further changes..."

Luo Yongjun observed Zhang Shuo's expression as he spoke. He found that Zhang Shuo could completely understand him, so he couldn't help but speed up his explanation until he talked about the latest changes.

Zhang Shuo pointed at the last transformation and asked, "Is this step using my substitution transformation method? The transformation of an equation seems to have become a little more complicated and not easy to understand."

Luo Yongjun nodded in agreement. "Yes, it is indeed a bit complicated, but it is just not easy to understand. However, after my feeling changed, it seems to be related to the ordered proof part..."

"And then?" Zhang Shuo asked.

"Gone."

"Gone?"

Luo Yongjun pursed his lips tightly and said, "That's what I thought of."

"So, this is what you have achieved in one year..."

Zhang Shuo complained, and took a look at the system. He took a deep breath and quickly changed his tone, "Awesome!"

"what?"

This change was a bit too fast, and Luo Yongjun didn't know how to react. He asked in confusion, "Are you praising me or being sarcastic?"

"Of course it's a compliment!"

Zhang Shuo's voice was full of sincerity, because he found that the task progress had reached '46.304%'.

Luo Yongjun has completed nearly half of his research.

Zhang Shuo suppressed his surprise and immediately said, "Teacher Luo, I think there is nothing wrong with your research. This way of argumentation will work."

"Stop comforting me."

Luo Yongjun didn't know whether his research direction was correct or not, and he didn't even know whether he should continue. He said a little unconfidently, "I used your substitution transformation method to make the transformation, and then I didn't know whether I should study the complex equation after the transformation or the original equation."

“No matter which one I study, I don’t know how to proceed next!”

He said this with a long sigh.

It was rare to see Luo Yongjun struggling with his research, so Zhang Shuo simply took the draft Luo Yongjun was using for his explanations and said, "Let me help you think about it."

Luo Yongjun waved his hand indifferently.

He was well aware of the difficulty of the research, but he didn't think that Zhang Shuo could help him think of the next direction.

Zhang Shuo returned to his seat and after some careful thought, he understood the reason for Luo Yongjun's distress.

Regardless of whether a transformation is made or not, the next step is to prove the density function.

But, I don’t know where to start.

This was not a problem that could be figured out in a short time, so he simply used his research coins to buy inspiration.

【Scientific research coin - 5.】

"The minimum purchase progress is 1%? You used 5 points at a time?" Zhang Shuo grinned hard, feeling a little heartbroken. It is not easy to save up scientific research coins.

But it's worth it to help Luo Yongjun find his direction.

Anyway, it’s just five days of minimum living allowance.

After spending the research coins to purchase progress, inspiration suddenly emerged in my mind and my research ideas were opened up at once.

Before he could digest the content, he found that the system had a new prompt--
[The number of scientific research coins consumed to purchase the task progress reaches 10, and a new task column is opened: Task . ]

"..."

Zhang Shuo was shocked. "So, upgrading depends on coins. More coins can unlock new quests?"

"Under normal circumstances, shouldn't I be encouraged to use my brain more and rely more on myself to solve research problems?"

"The system actually encourages me to spend more coins?"

(End of this chapter)