Prince Town's Prosperity and Hidden Risks

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

"From the perspective of modern Differential Geometry, two-dimensional Riemannian Geometry is Gauss intrinsic Differential Geometry. Combining the diagram on the blackboard, can anyone give me an explanation of how to realize Non-Euclidean Geometry in a space with constant curvature on a two-dimensional manifold?" Shen Qi began to discuss professional Non-Euclidean Geometry with his students.

"I've calculated it. The Gauss curvature K of this surface is a constant α. When α is greater than or equal to 0, this abstract surface can be defined on the entire (u, v) plane," Luca spoke first.

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

"The domain of this surface is u² plus v², which is less than negative 4/α," A-Jie answered quickly.

"The answers from Luca and Jack combined are perfect," Shen Qi said with satisfaction. Luca and Jack were always proactive and quick-witted.

"I have a different opinion on this issue," Stephen, the son of an American farmer, chimed in.

"In my opinion, Non-Euclidean Geometry is similar to calculus; there's more than one correct answer. I think this figure on the blackboard can also be interpreted like this..." John, the son of a British financier, disagreed.

The discussion atmosphere in the Recitation Class was very lively.

Students made progress through discussions and learned valuable lessons from others.

After the Recitation Class ended, Shen Qi asked A-Jie, Luca, Stephen, and John to stay behind.

"In this Non-Euclidean Geometry tutorial class I'm teaching, the four of you are the most outstanding." Shen Qi admired excellent students; those who could stand out at Princeton must have extraordinary abilities. "Tell me, besides Non-Euclidean Geometry, what other courses are you good at?"

"Coordinate Geometry, metric geometry, Projective Geometry." Luca is a geometry enthusiast; he's wildly obsessed with all kinds of geometry.

"Actually, my math skills are just amateurish. I'm much better at interacting with girls. If interacting with girls is a course in itself, then so be it." Luca casually adjusted his hair.

"Luca, how's your relationship with your Alabama girl going?" Shen Qi asked.

When it comes to picking up girls, no one is better than Italy men. Italy men have sculpted, three-dimensional features. Besides being handsome, another of their secrets to picking up girls is their shamelessness.

"I haven't dated any Alabama girls in ages. My current girlfriend is a Colorado girl," Luca said Yangyang.

"I've never even managed to date a Colorado girl!" Stephen, from Colorado, said with a mix of envy and condemnation, "It's typical of Italy men—they fall in love with every girl they see and sleep with every one they love!"

"Let's leave the girls' conversation here. Stephen, what about you? What do you like?" Shen Qi asked Stephen.

"I prefer discrete mathematics to geometry, and I won't give up my efforts to become an IT Engineer in Silicon Valley." Once an American boy makes up his mind, he really won't give up until he's faced with a fatal mistake.

Discrete mathematics is widely used in Computer Science, Stephen, the farmer's son, wants to pursue a career in this field.

"And you, John?" Shen Qi continued to ask.

"Non-Euclidean Geometry is a course for me; I need to learn it for credits and a degree. Game theory and financial Mathematical Analysis are a career for me, my career for the rest of my life," John said. He plans to return to the UK sooner or later to take over his family's financial services company, and his studies at Princeton are very purposeful.

Finally, it was A-Jie's turn to speak, as he was the poorest in the family. "I also like game theory, as well as Probability Theory, operations research, and computational mathematics. I prefer practical courses."

Shen Qi: "Is nobody interested in number theory or the History of Mathematics?"

"Pure number problems are useless. Can they make money? No way! Let history enthusiasts study the History of Mathematics." A-Jie was very practical; he only loved practical subjects that could make money.

"Alright, I hope you all do well in your upcoming final exams," Shen Qi announced, dismissing the class.

These four second-grade students each have their own unique characteristics.

Italy love girls. He has a great talent for geometry. Chasing girls is his main job, and geometry is just a hobby.

The British are highly purposeful in pursuing advanced studies that will benefit their families, following a step-by-step approach and planning their lifelong career path from the age of one.

The son of a Colorado rancher was eager to pursue his ambitions in the IT industry. In his view, farming and horse raising had no future, and Silicon Valley was the core place that represented the future direction of development.

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

Young people these days either indulge in extravagant lifestyles or strive to make money, become CEOs, and reap both fame and fortune.

Young people who can settle down to study pure theory are becoming increasingly rare. Even at Princeton, where academia is paramount, hedonism, pragmatism, and utilitarianism are gradually on the rise.

Learning math isn't difficult; the challenge lies in mentoring young people.

"So, it's the older generation of scholars like Professor Muller and Professor Faltings who are truly dedicated to scholarship. If Princeton continues to decline like this, it's bound to fail sooner or later." Shen Qi calculated that in thirty years, all the older generation of academic masters would be gone, and Princeton would be finished.

Of the twenty Fields Medal winners in the past two decades, only one was trained by Princeton itself: Terence Tao. Terence Tao's graduate and doctoral students studied in Princeton University's Mathematics Department.

Younger Fields Medal winners, such as Okonkov and Lyndon Strauss, did not study at Princeton; they were recruited to teach and conduct research at Princeton after they became famous.

Therefore, beneath the surface of Princeton's prosperity lie crises and hidden dangers.

Shen Qi had long since discovered the crisis and hidden dangers in Prince Town, but he simply chose not to speak of them.

"Thirty years east of the river, thirty years west of the river, things change. Prince Town, you should be on your guard. It's not too late to start from the children..."

Before the Christmas holidays, Shen Qi completed the proof of the "Muller -Shen approximation theorem".

Shen Qi showed the paper to Professor Muller for review. "From the above proof, I obtained that xα*→x0*, then x0* is the focus of {xn*}∞n=1, therefore there exists a subsequence {xnk*}∞k=1 of {xn*}∞n=1, such that every closed convex set of X* is an approximate compact Chebyshev set."

After reviewing it, Professor Muller said, "As expected, a rigorous proof of this supplementary theorem cannot be obtained through the Hustav Topology approach. Qi, you used the separation method, which is very creative and demonstrates strong logical deduction ability."

The "Muller -Schen approximation theorem" and the "Muller -Schen theorem" which has been recognized by the IMU are related to each other, but they are also independent.

The proof of the "Muller -Shen approximation theorem" was basically completed by Shen Qi alone.

Professor Muller, who hadn't contributed much, felt embarrassed to put his surname on the list. He suggested, "It would be more appropriate to name the new theorem 'Shen Proximity Theorem,' since I wasn't involved in its proof."

Look, this is the ideological awareness of the older generation of scientific researchers at Princeton University.

Shen Qi strongly suggested adding Professor Muller's surname; the two were working so well together that there was no need to worry about such minor details.

Professor Muller insisted on not falling into the trap, believing that "Shen Proximity Theorem" was the most reasonable naming method.

Unable to persuade the German professor, Shen Qi revised his paper, changing "Muller -Shen Approximation Theorem" to "Shen Proximity Theorem".

The paper containing the Shen Proximity Theorem " is titled "Analysis of the Problem of Closed Convex Sets in Banach space," which is a companion piece to the " Muller -Shen theorem" paper Annals of Mathematics.

The Annals of Mathematics is published only twice a year, once in May and once in November, hence the name "annual journal".

The next issue of the * Annals of Mathematics * wouldn't be until May of next year, so Shen Qi submitted his paper, "Analysis of the Problem of Closed Convex Sets in Banach space," to * Inventiones Mathematicae *. * Inventiones Mathematicae *, whose editorial office is located in Germany, is one of the four major journals, publishing one issue per quarter.

Another Christmas holiday has arrived. Shen Qi filled up the gas tank of his North American version of the Malibu and gave it a thorough cleaning and detailing service, both inside and out.

Zhou Yu'an departed before Ou Ye; he was already on a flight bound for New York's Kennedy Airport and was about to land.

Shen Qi started the car and headed to New York to pick someone up from the airport.