Shen Qi drew a figure on the blackboard, consisting of a circle and many curves, which looked like a large octopus.

"From the point of view of modern differential geometry, two-dimensional Riemannian geometry is Gauss's intrinsic differential geometry. Combined with the graphics on the blackboard, can anyone give me an explanation of the implementation of non-Euclidean geometry in constant curvature space on a two-dimensional manifold?" Shen Qi began to discuss with the students the knowledge of professional non-Euclidean geometry.

"I calculated that the Gaussian curvature K of the surface is constant α and that when the α is greater than or equal to 0, this abstract surface can be defined on the entire (u,v) plane. Lukka was the first to speak.

"What about when the α is less than 0?" Shen Qi asked.

"The surface is defined by a 4/α of U squared plus V squared less than negative one. Applejack replied.

"Luca and Jack's answers add up to perfection. Shen Qi expressed satisfaction that Luka and Applejack have always been positive and quick-witted.

"I have a different view on this issue. Not to be outdone, Stephen, the son of an American farmer, voiced his opinion.

"It seems to me that non-Euclidean geometry is similar to calculus, and there is more than one correct answer, and I think this figure on the blackboard can be explained in this way......" John, the son of a British financial tycoon, disagreed.

The atmosphere of the discussion in the tutorial class was very lively.

Students make progress in discussions and learn from others.

After the tutorial, Shen Qi asked Applejack, Luca, Stephen, and John to stay.

"In this non-Euclidean geometry tutorial class I have led, the four of you are the best. Shen Qi admires outstanding students, and students who can stand out in Princeton must be outstanding: "Tell me, in addition to non-Euclidean geometry, what other courses are you good at?"

"Coordinate geometry, metric geometry, projective geometry. "Luca is a geometry controller, and he is obsessed with all kinds of geometry.

"Actually, my math is only at an amateur level, and I am better at socializing with girls, and if I socialize with girls, it is a course. Luca coquettishly styled his hair.

"Luca, how are you dating with your Alabama girl?" asked Shen Qi.

When it comes to pickling girls, no one is better at it than Italian men, Italian men have a sculptural three-dimensional silhouette, in addition to being handsome, another trick of their pickling girls is thick skin.

"I haven't been dating an Alabama girl for a long time, and my current girlfriend is a Colorado girl. Lukka said triumphantly.

"I've never been to a Colorado girl!" Stephen from Colorado was particularly angry, and he said enviously and condemningly: "The characteristics of Italian men, see one and love one, love one and last one!"

"That's the end of the girls' topic, Stephen, what about you, what do you like?" Shen Qi asked Stephen.

"I prefer discrete math to geometry, and I won't give up trying to become an IT engineer in Silicon Valley. "Once an American boy admits to death, he really doesn't cry when he sees the coffin.

Discrete mathematics is widely used in computer science, and the farmer's son, Stephen, wants to go in this direction.

"What about you, John?" Shen Qi continued to ask.

"Non-Euclidean geometry is a course for me, and I need to study it for credit and diploma. Game theory and financial mathematical analysis methods are a profession for me, a profession for the rest of my life. John said that sooner or later he would have to return to England to take over the family's financial services company, and that his studies at Princeton were purposeful.

Finally, it was Applejack's turn, the poorest in the family, to speak: "I also like game theory, as well as probability theory, co-ordination, computational mathematics, and I like practical courses." ”

Shen Qi: "No one is interested in number theory, or the history of mathematics?"

"Pure number problems are useless, can you make money, can't! History of mathematics, leave it to history lovers to study. "Applejack is very down-to-earth, he only loves practical subjects that make money.

"Well, I hope you guys do well in the upcoming final exams. Shen Qi announced his disbandment.

Each of the four second-graders has their own characteristics.

Italians like girls, he has a high talent for geometry, picking girls is his main business, and geometry is a hobby.

The Englishman studied for the benefit of his family with a very purposeful course, and from the age of one he planned a lifelong career path.

The Colorado farmer's son wants to pursue his ambitions in the IT industry, and he sees little future in farming, and Silicon Valley is the core of the future.

The son of the owner of the Chinatown fruit shop wants to make a fortune in order to change the family's social status and quality of life.

Today's young people either spend their days drinking and indulging, or they are working hard to make money and become CEOs, both fame and fortune.

There are fewer and fewer young people who can calm down and study pure theory, and even in Princeton, where academics are paramount, the atmosphere of hedonism, pragmatism, and utilitarianism is gradually rising.

It's not difficult to learn math, but it's hard to train young people.

"So, it's still the older generation of scholars like Professor Mueller and Professor Faltings who are really scholars, and Princeton will have to play sooner or later if he continues to degenerate like this. Shen Qi pinched his fingers, thirty years later, all the academic masters of the older generation were gone, and Princeton was over.

In the past two decades, only one of the twenty Fields Medal winners has been trained by Princeton himself, and that is Tao Zhexuan, a graduate and doctoral student in the Department of Mathematics at the University.

Relatively young Filipino laureates such as Okunkov and Lindon Strauss did not have the experience of studying at Princeton, and after they became famous, they were recruited to teach and do research in Princeton.

So beneath the surface of Princeton's prosperity, there are crises and hidden dangers.

Shen Qi had already discovered the crisis and hidden dangers of Wang Zitun, and he didn't want to talk about it.

"Thirty years of Hedong, thirty years of Hexi, Wang Zitun, you should be alert, now start from the doll, it's still too late to ......"

Before the Christmas holidays, Shen Qi completed the proof of the "Muller-Shen Near Forcing Theorem".

Shen Qi showed the paper to Professor Mill: "From the above proof, I get xα*→x0*, then x0* is the focus of {xn*}∞n=1, so there is a subcolumn {xnk*}∞k=1 of {xn*}∞n=1, so that each closed convex set of X* is an approximate tight Chebyshev set. ”

Professor Müller looked at it and said, "Sure enough, you can't get a rigorous proof of this complementary theorem from the Horsdave topological pathway, and you have used the separation method, which is very creative, and has a strong ability to derive logic." ”

The "Müller-Shen theorem" has a certain connection with the "Müller-Shen theorem" that has been recognized by the IMU, and it is also independent.

The proof of the "Müller-Shen proximity theorem" was basically completed by Shen Qi alone.

Professor Müller, who had not contributed much, was embarrassed to put his surname on it, and he suggested: "It is more appropriate to name the new theorem 'Shen's proximity theorem', because I was not involved in the proof of this theorem." ”

See, this is the ideological consciousness of the older generation of scientific research workers at the University.

Shen Qi strongly suggested adding Professor Mueller's surname, the two worked together so happily that they didn't have to care about these small details.

Professor Müller insisted on not getting into the pit, and he believed that "Shen's proximity theorem" was the most reasonable way to name it.

Unable to resist the German old man, Shen Qi revised the paper and changed the "Müller-Shen proximity theorem" to "Shen's proximity theorem".

The paper containing "Shen's Approximate Forcing Theorem" is entitled "Analysis of the Problem of Closed-Convex Sets in Barnach Spaces", which is a companion to the Annals of Mathematics paper on "Miller-Shen's theorem".

The Annals of Mathematics is published only twice a year, one in May and one in November, so it is called the "Annal".

The next issue of the Annals of Mathematics will not be published until May next year, so Shen Qi submitted the paper "Analysis of the Problem of Closed-Convex Set Problems in Banach Space" to "Mathematical Inventions". The editorial office of "Mathematical Inventions" in Germany is one of the four majors, and it is published every quarter.

It's another year of Christmas holidays, Shen Qi filled up his North American version of Malibu and made a body cleaning and beauty.

Zhou Yu'an departed before Ouye, he was already on the flight to New York's JFK Airport and was about to land.

Shen Qi started the car and went to New York to pick up the airport.