universal data

Chapter 360 Test
363 chapter

When the news reached Cheng Nuo, he had already received an offer from Princeton.

As one of the top universities in the world in the Department of Mathematics, becoming one of the students is probably the dream of many mathematics lovers.

The number of its 13 Fields Medal winners ranks third in the world.

Many mathematics masters also teach in this school, and the atmosphere of academic exchanges is strong.

But it is not bad with MIT, which is also in the United States.

In the newly released 2022 QS World University Rankings by Subject, the mathematics majors of the two are third and fourth!
Although there are only six Fields Medals, that is only because of its relatively late establishment.And there are three Fields Medalists currently teaching at the same two universities.

Overall, but general from the mathematics major, the two universities Lu Benwei.

For Cheng Nuo, it didn't matter which of the two universities he chose.

However, if you choose MIT, you will get an additional position as an assistant to the Fields Medal winner.Coupled with Professor Fang's suggestion, Cheng Nuo chose this option after thinking about it for a while.

Of course, the premise is to stand out from the competition group of 12 people.

Cheng Nuo regained that confidence.

Those competitors are at most at the level of doctoral students.If this person still can't handle it, then why should he be a king? !
…………

After a week of preparations, under the leadership of Professor Fang, Cheng Nuo came to a room where the test was conducted.

When Cheng Nuo walked in, the other eleven people had already arrived.

Cheng Nuo glanced lightly.

There are seven boys and three girls, generally three or four years older than Cheng Nuo.

Especially the two boys, the tops of their heads have become slightly bald, so they are not easy to provoke at first glance.

The other three female doctors are almost in the same dress.Thick lenses, a ponytail tied behind the head, and a plain face.

Cheng Nuo walked up to him with a solemn expression and sat down.

While Cheng Nuo was looking at other opponents, the others were also looking at Cheng Nuo.

After all, they are real competitors. Twelve to one is quite cruel.

For Cheng Nuo, what surprised them the most was naturally his age.

At this age, you should still be studying for an undergraduate degree, right?How do you come here to compete with a group of PhD students?

Could it be... through the back door?
But it shouldn't be, if you come in through the back door, it's still useless for an undergraduate to face a group of doctoral students!
Although I was puzzled in my heart, no one was free to ask this.

The high-spirited but confident eyes looked at each other one by one.

Suddenly, the door was pushed open, and a potbellied man in a suit walked in with a briefcase.

He glanced at the classroom and found that everyone was full, so he took out a stack of test papers from his briefcase and handed them out one by one.

Cheng Nuo took a look at the test paper.

There are only two questions in the entire test paper.

The fewer the questions, the higher the difficulty of the questions. This is a generally accepted theorem.

After distributing the test papers, the man with a pot belly coughed and said slowly, "The open-book exam, the exam time is four hours, you can hand in the papers in advance!"

After speaking, he moved a chair to the front of the room, crossed his legs and played with his mobile phone.

Cheng Nuo shrugged, spread the test paper on the table in front of his chest, and read it carefully.

Since it is this kind of test, the questions used for the test must be quite different from the test questions.

Difficulty, at least equal to the level of a doctoral dissertation.

After all, this is the assistant for selecting Professor Fresnel.

The first question: [Assume (N, g) is an n+1-dimensional Riemannian manifold, M is its n-dimensional submanifold, and assume that ψ is a given smooth function on N.Is there such an embedding φ: M→N such that f(x)=ψ.]

Not only are there few topics, but the questions are also short.

But the difficulty is comparable to a lot of nonsense outside, setting up scenarios, and making up stories about math problems, which are not on the same plane at all.

The first time he saw the title, Cheng Nuo had a feeling: This is a stubble!
Obviously, this is a topic in the field of Riemannian manifolds.

Since Professor Fresnel's main focus is in the field of geometry, it is reasonable to ask this question.

What is a Riemannian manifold?

This means that in differential manifolds and Riemannian geometry, a Riemannian manifold is a differential manifold with a Riemannian metric, in other words, this manifold is equipped with a symmetric positive definite second-order covariant tensor field, that is Equip the tangent space of each point with a positive definite quadratic form.After the measurement is given, we can measure length, area, volume, etc. like in elementary geometry.

There is a natural metric ds^2=(dx_1)^2++(dx_n)^2 in n-dimensional Euclidean space.Its matrix representation is the identity matrix.

A submanifold in Euclidean space naturally induces a metric, of course.In the differential geometry of curves and surfaces, we regard curves and surfaces as submanifolds of three-dimensional space, so they are naturally endowed with a metric structure.

Looking at the questions on the test paper, Cheng Nuo thought deeply.

The other contestants took out their mobile phones to search for information in a hurry after reading the questions, but Cheng Nuo didn't have to.

One is that it is impossible to find the correct answer on the Internet, and the other is that all the information about the Riemannian manifold has been printed in his mind.

Cheng Nuo was not unprepared for a week of preparations.

1 minute, 2 minutes, 3 minutes...

In Cheng Nuo's mind, thoughts were racing.

Groups of formulas are combined and connected in series, gradually forming a complete proof chain.

Ten minutes later, Cheng Nuo's closed eyes slowly opened.

Then, take a pen and start writing.

Cheng Nuo plans to solve this problem with the predetermined curvature problem of the hypersurface of the Riemannian manifold.

[The main curvature of the hypersurface φ(M) under the induced metric is k=(k1, k2, k3...), f is a symmetrical function, especially, if f(k)=∑ki or f(k) =∏ki.】

[Assuming N=R^n+1, when N is a curved Riemannian manifold, there is an n-dimensional Riemannian manifold (M, dσ^2) and a differentiable function h:I→R^2, so that N=I*M, And the measure of N can be written as ds^2=dt^2+h^2...]

…………

As time ticked by, Cheng Nuo also wrote the formula line by line on the test paper.

The train of thought was in his head, so Cheng Nuo's writing was extremely smooth.

In the eyes of outsiders, Cheng Nuo seemed as if he hadn't thought about it, and the formulas jumped out of the paper.

[There is an n-dimensional manifold M and diffeomorphism, where I=(a, b) is the development interval of R, a, b∈R...]

Done, perfect! !

Excited, he subconsciously snapped his fingers.

Then, several other people in the classroom looked at him suspiciously.

Cheng Nuo clasped his hands together, and after several people turned their heads away, he shook his head and smiled softly.

To be honest, for this topic, if the elaboration process of this topic is expanded into a thesis, it is not a problem to participate in the master's graduation defense.

In other words, it took Cheng Nuo more than half an hour to complete the research content of a doctoral student for half a month to a month.

This is hard power.

Cheng Nuo turned his mouth slightly and looked at the second question.

(End of this chapter)