Chapter 363: test

Chapter 363
When the news reached Cheng Nuo, he had already received an offer from Princeton.
As one of the top universities in the world for mathematics, it is probably the dream of many mathematics enthusiasts to be one of them.
The number of 13 Fields Medal winners ranks third in the world.
Many mathematics experts also teach in this school, and there is a strong academic exchange atmosphere.
But the Massachusetts Institute of Technology in the United States is not bad.
In the latest QS World University Rankings in 2022, the math majors of the two are ranked third and fourth!
Although there are only six Fields Medals, it is only because of its late establishment. In addition, the two universities that are still teaching the Fields Medal are the same, and both have three.
In general, but from the mathematics major, Lu Benwei from two universities.
For Cheng Nuo, it doesn't matter which university the two universities choose.
However, if you choose MIT, you will get an additional position of Assistant to the Fields Medal winner. Coupled with the professor's suggestion above, Cheng Nuo chose this option after thinking about it.
Of course, the premise is to stand out from the 12-person competitive group.
Cheng Nuo had that confidence again.
Those competitors, at most, have the level of a doctoral student. If this person can't figure it out, then what is he going to be a king? !
…………
After preparing for a week, 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 were already there.
Cheng Nuo glanced lightly.
Seven men and three women are generally three or four years older than Cheng Nuo.
Especially the two boys, the tops of their heads have become slightly bald, and they are not easy to provoke characters.
The other three female doctors are almost dressed up. Thick lenses, ponytail pierced behind the head, plain face.
Cheng Nuo walked up and sat down solemnly.
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, it can be said to be quite cruel.
For Cheng Nuo, what surprised them the most was their age.
At this age, you should still be undergraduate, right? How come to compete with a group of doctoral students?
Could it be... through the back door?
But it shouldn't be. If you come in through the back door and let an undergraduate face a group of doctoral students, there is still nothing to do!
Although I was puzzled in my mind, no one was free to ask this.
Highly motivated but confident eyes, they looked at each other one by one.
Suddenly, the door was pushed open, and a man with a big belly in a suit walked in with a briefcase.
He glanced around the classroom and found that everyone was there, so he took out a stack of test papers from his briefcase and distributed them one by one.
Cheng Nuo glanced at the test paper.
There are only two questions in the whole test paper.
The fewer questions, the more difficult the questions, which is a generally accepted theorem.
After sending out the test papers, the man with a bowel movement coughed and slowly said, "Open-book exams. 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 and played with the phone with his legs folded.
Cheng Nuo shrugged, spread the test papers on the table in front of his chest, and read them carefully.
Since it is this kind of test, the test questions must be quite different from the test questions.
The difficulty is at least equal to the level of a doctoral dissertation.
After all, this is an assistant in selecting Professor Fresnel.
The first question: [Assume (N, g) is an n+1-dimensional Riemannian manifold, M is its n-dimensional submanifold, and suppose 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 questions, but the questions are also short.
But the difficulty is comparable to a lot of nonsense outside. The math topics for setting up scenarios and making up stories are completely out of the same plane.
At the first glance of the subject, Cheng Nuo had a feeling: This is a stubborn problem!
Obviously, this question in the field of Riemannian manifolds.
Since Professor Fresnel focuses on the field of geometry, it is reasonable to come up with this topic.
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 Equipped with a positive definite quadratic form on the tangent space of each point. After giving the measurement, we can measure the length, area, and volume as in elementary geometry.
In n-dimensional Euclidean space, there is a natural metric ds^2=(dx_1)^2++(dx_n)^2. Its matrix representation is the identity matrix.
Submanifolds in Euclidean space naturally induce a metric. In the differential geometry of curves and surfaces, we all regard curves and surfaces as submanifolds in three-dimensional space, so we naturally give a metric structure.
Looking at the questions on the test paper, Cheng Nuo thought deeply.
The other contestants took out their mobile phones and hurriedly searched for information after reading the questions, but Cheng Nuo didn't need to do this.
One is that it is impossible to find the correct answer online, and the other is that all the information about the Riemannian manifold has already been printed in his mind.
Cheng Nuo was not unprepared for the week of preparation.
One minute, two minutes, three minutes...
In his mind, Cheng Nuo's thoughts flew.
A group of formulas are combined and connected in series to gradually form a complete proof chain.
Ten minutes later, Cheng Nuo's closed eyes slowly opened.
Then, start writing.
For this problem, Cheng Nuo is going to solve the problem of the predetermined curvature 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 symmetric function, in particular, if f(k)=∑ki or f(k) =∏ki.】
[Assuming N=R^n+1, when N is a curved Riemannian manifold, there are n-dimensional Riemannian manifold (M, dσ^2) and a differentiable function h:I→R^2, such that N=IM, And the measure of N can be written as ds^2=dt^2+h^2……]
…………
As time went by, Cheng Nuo also wrote line-by-line formulas on the test paper.
The idea is in his head, so Cheng Nuo writes extremely smoothly.
In the eyes of outsiders, ~EbookFREE.me~ Cheng Nuo seems to have not thought about it, and all the formulas come 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...]
Get it done, perfect! !
Excited, he snapped his fingers subconsciously.
Then, several other people in the classroom looked at him with suspicious eyes.
Cheng Nuo folded his hands together and shook his head and smiled slightly after the few people turned their heads.
To be honest, this topic, if the elaboration process of this topic is expanded into a dissertation, it will not be a problem to participate in the graduation defense of master students.
In other words, it took Cheng Nuo more than half an hour to study the content of a doctoral student for half a month to a month.
This is hard power.
The corner of Cheng Nuo's mouth was slightly tilted, looking towards the second question.