Chapter 385

"Fritz John's Necessary Optimality Conditions on Riemannian Manifolds"!

This is the proposed topic of the project to be studied by Cheng Nuo in the next two months.

Professor Fresnel briefly told Cheng Nuo and Hull in the cubicle some things that needed attention, and then asked the two of them to go back with the documents to prepare, and the next day they officially began to study the topic.

Cheng Nuo naturally has no opinion.

He also wanted to take this time to learn some knowledge related to the topic.

While his mission may be to give Professor Fresnel a hand, it's always right to be well prepared.

Cheng Nuo sat on the desk, supporting his chin with one hand and flipping through the documents Fresnel gave him with the other.

The topic of Riemann manifold is one of the 50 national key mathematical research projects in 2022 directly approved by the Clay Institute of Mathematics in the United States.

These 50 mathematical research projects are among the top in the world in terms of project difficulty and importance.

In fact, as one of the most developed countries in the field of mathematics in the world, the Clay Institute of Mathematics in the United States plays a leading role in the forefront of mathematics in the world.

In addition to the deep pockets of the Clay Institute of Mathematics, each of the 50 national key mathematical research projects has been given $100,000 in financial support.

In addition, the mathematicians who are responsible for the research work of these 50 scientific research projects are all among the top mathematicians in the world.

Just like Cheng Nuo's current boss, Professor Fresnel, as a super expert in the field of geometry, the Clay Institute has entrusted him with the most difficult one of the three topics in the field of geometry among the fifty projects.

That is, the subject of the Riemannian manifold that Cheng Nuo got in his hands.

In the morning, Cheng Nuo was reading the documents and looking for relevant papers on the Internet to read.

It's hard!

This is the result given by Cheng Nuo's research in the morning.

He finally knew why the Clay Institute for Mathematics had entrusted this project to Professor Fresnel, because in today's mathematical community, there are probably no more than five mathematicians who can guarantee that they can solve this problem within two months.

And Professor Fresnel is obviously the safest one.

Not to mention the short time given for scientific research, there are too few papers and materials on the Internet about this area, which means that they are almost starting from scratch.

Riemannian manifolds are already extremely difficult in the field of geometry, and together with the relevant knowledge of function theory and differentiation, it is enough to drive most mathematicians in the world crazy.

Ask yourself, if this project is handed over to Cheng Nuo to complete by himself, it will start in at least three years.

"It seems that for the time being, I still have to hold Professor Fresnel's thigh firmly!" Cheng Nuo sighed and continued to bury his head in collecting information.

..................

The next day, Cheng Nuo came to the office early.

As soon as Professor Fresnel arrived, Cheng Nuo and Hull were called into the cubicle again.

"How's the preparation?" asked Professor Fresnel, coming up.

Hull smiled bitterly, "Teacher, there is indeed too little information about this on the Internet, and there are no books in the library that are too relevant, so ......"

Professor Fresnel waved his hand, as if anticipating this.

"At present, the mathematical research in this direction is indeed a blank, so we need to study and fill in!" Professor Fresnel's gaze slowly swept over the faces of the two, "That's why I said yesterday, you should be mentally prepared. It's an uphill battle!"

"Starting from scratch, there was no information to draw on, and the time frame was ...... It's only been two months!"

Professor Fresnel continued, "I'm not going to say anything about cheering and encouraging, I just hope that you two don't forget the purpose of being here, and if you want to quit, I'm always welcome." ”

With that said, let's now talk about the subject. ”

Professor Fresnel asked the two of them to find a place to sit down, moved over a laptop, opened a PowerPoint presentation, and pointed, "This is a short research process I did. ”

"I am in charge of this project, and the task of the two of you is to assist me and solve some difficult links. ”

Cheng Nuo and Hull nodded and said they knew.

With the abilities of the two of them, it is not enough to support the framework of this project.

Professor Fresnel continued, "The proposed name of this project is called Fritz John's Necessary Optimality Condition on Riemann Manifolds. First of all, we must understand what is a Riemann manifold and what is the necessary optimality condition of Fritz John!"

"Needless to say, the concept of the Riemannian manifold should be relatively new to you, and the Fritz John necessary optimality condition should be relatively new to you. He looked at Cheng Nuo first, "Cheng Nuo, do you understand this concept?"

Cheng Nuo replied without thinking, "The so-called Fritz John necessary optimality condition refers to minf(x),st. {g(x)≤0,h(x)=0,x∈M. ”

"Yes, that's the necessary optimality for Fritz John. As you can see, if you study the necessary optimality condition of Fritz John directly, not only are there many variables, the function equations are not easy to define, but also there are complex formulas in the derivation process. ”

"That's why we need to change our thinking. ”

Professor Fresnel flipped to the next page of the PPT, which only had one line of formula:

f:M→R，g:M→R^l，h:M→R^n

Cheng Nuo glanced at it and suddenly realized, "Lipschitz function?!"

Professor Fresnel glanced at Cheng Nuo with a hint of appreciation in his eyes, "To be precise, it's the local Lipschitz function!"

The Lipschitz function means that if f(x) satisfies on interval I that there are any two different real numbers x1 and x2 for the defined domain D: ∥f(x1)-f(x2)∥<=K∥x1-x2∥ is true, there must be f(x) consistent continuity over interval I.

In Cheng Nuo's heart, he already roughly understood what Professor Fresnel's problem-solving point was in this project.

Professor Fresnel continued his theory, "In this formula, we can think of M as an m-dimensional Riemannian manifold. ”

"You two should have read Aytonck's paper on the MP problem in Hilbert's space, right?"

Both nodded at the same time.

"Well, by analogy, we can extend the MP problem from linear space to differential manifolds, which are non-smooth, then we can have the following framework. ”

The next PPT is displayed in front of the two of them.

"The first step is to establish a non-smooth analysis tool on the Riemannian manifold, that is, to define the generalized directional derivative and the generalized gradient on the manifold. ”

"In the second step, we discuss the nature of generalized gradients. ”

"In the third step, on the basis of the first two steps, the Fritz John-type optimality condition for the Riemann manifold problem (MP) is discussed."

"Step 4,......"

The framework had already been set up by Professor Fresnel.

And Cheng Nuo felt empowered when he saw the orderly process steps.

It turns out that this project should be done this way!