Chapter 363

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

As one of the world's top universities in mathematics, it is probably the dream of many mathematics enthusiasts to become one of them.

Its 13 Fields Medal winners are the third largest in the world.

Many math experts also teach in this school, and the atmosphere of academic exchange is strong.

But the Massachusetts Institute of Technology, which is in the same country as it, is not bad.

In the latest QS World University Rankings by Subject 2022, the mathematics majors of the two are third and fourth!

Although there are only six Fields Medals, that is only because of its late establishment. And the Fields Medal that is still teaching is the same for the two universities, with three each.

Overall, but from the mathematics major general, two universities Ashimoto reed.

For Cheng Nuo, it doesn't matter which of the two universities to choose.

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

Of course, the premise is to stand out from the 12-person competitive group.

Cheng Nuo has that confidence again.

Those competitors are, at most, at the level of doctoral students. If this person can't figure it out, then what else should he be to force the king?!

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

After a week of preparation, 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's eyes glanced lightly.

Seven men and three women, generally three or four years older than Cheng Nuo.

Especially the two boys, the top of their heads has become slightly bald, and they are not easy to provoke at first glance.

The other three female doctors, almost all dressed in the same way. Thick lenses, a ponytail tied at the back of the head, and a face without makeup.

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

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

After all, it is a real competitor, twelve into one, which can be described as quite cruel.

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

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

Could it be...... Came in through the back door?

But it shouldn't be, if you come in through the back door, it's useless to let an undergraduate face a group of doctoral students!

Although I was puzzled, no one was idle to ask this.

The scrappy but confident eyes looked at each other one by one.

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

He glanced at the classroom and found that there were all the people, so he took out a stack of test papers from his briefcase and handed them out one by one.

Taking the test paper, Cheng Nuo glanced at it.

There are only two questions in the whole test paper.

It is a well-established theorem that the fewer the questions, the more difficult the questions are.

After handing out the test papers, the pot-bellied man coughed and slowly spoke, "Open book exam, the exam time is four hours, you can submit the paper in advance!"

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

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

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

The level of difficulty is at least the same as that of a doctoral dissertation.

After all, this is the assistant to the selection of Professor Fresnel.

Question 1: [Suppose (N,g) is an n+1 Riemannian manifold and M is its n-dimensional submanifold, and suppose ψ is a given smooth function over N.] Is there such an embedding φ: M→N, such that f(x)=ψ.]

Not only are there few questions, but even the questions are short.

But the difficulty is not at the same level at all, compared to a lot of nonsense outside, setting up scenarios, and making up stories.

At the first glance of the title, Cheng Nuo had a feeling: this is a hard stubble!

Obviously, this is a topic in the field of Riemannian manifolds.

Since Professor Fresnel is mainly in the field of geometry, it makes sense to come up with this question.

What is a Riemann manifold?

This means that in differential manifolds and Riemannian geometry, a Riemannian manifold is a differential manifold with a Riemannian metric, in other words, the manifold is equipped with a symmetrically definite second-order covariant tensor field, i.e., a positive definite quadratic form on the tangent space of each point. Once we have given the measurements, we can measure length, area, volume, etc., as in elementary geometry.

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

The submanifold in Euclidean space naturally induces a metric. In the differential geometry of curves and surfaces, we treat curved surfaces as submanifolds of three-dimensional space, so we naturally give them a metric structure.

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

Other contestants are all taking out their mobile phones and hurriedly searching for information after reading the questions, but Cheng Nuo doesn't have to.

The first is that it is impossible to find the correct answer on the Internet, and the second is that all the information about the Riemann manifold has been imprinted in his mind.

A week of preparation, Cheng Nuo is not unprepared.

One minute, two minutes, three minutes......

In his mind, Cheng Nuo's thoughts were racing.

A group of formulas are combined with each other to form a complete chain of proofs.

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

Then, start writing.

In this problem, Cheng Nuo is going to solve the problem of the predetermined curvature of the hypersurface of the Riemann manifold to solve it.

【The principal curvature of hypersurface φ(M) under induction measure is k=(k1,k2,k3......) , f is a symmetric function, especially, if f(k)=∑ki or f(k)=∏ki.]

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

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

Time ticked by, and Cheng Nuo also wrote lines of formulas on the test paper.

The idea is in his head, so Cheng Nuo's writing is extremely smooth.

In the eyes of outsiders, Cheng Nuo seems to have not thought about it, and one formula after another jumps on the paper.

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

That's it, perfect!!

Excited, he subconsciously snapped his fingers.

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

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

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

In other words, Cheng Nuo spent more than half an hour to get the content of a doctoral student's research for half a month to a month.

That's hard power.

The corners of Cheng Nuo's mouth turned up slightly, and he looked at the second question.