Muller-Shen Theorem?

"Ou Ye's TOEFL score is slightly lower than mine, her GRE score is slightly higher, and our GPAs are the same at 4.0."

"My research experience is not as Ou Ye's. I have only published one paper, which I co-authored with Ou Ye."

"If Ou Ye performs normally during the interview, then I shouldn't have a chance."

Zhou Yu'an explained the situation in China to Shen Qi.

"Oh, I see." Shen Qi had guessed just that. Given Ou Ye's limited verbal skills, she might be too nervous to speak if she encountered a strict interviewer.

After ending his call with Zhou Yu'an, Shen Qi called Ou Ye to confirm the situation.

As expected, Ou Ye's written grades were excellent, but she messed up the interview.

Throughout the entire interview, the Princeton interviewers did almost all the talking, with Ou Ye only occasionally offering a few words in response.

The interviewer lost his temper and gave Ou Ye the lowest possible interview score.

"The questions he asked were so strange, I, I..." Ou Ye was on the verge of tears on the other end of the phone.

"Don't be impatient, don't panic, let me think." Shen Qi mustn't panic if his woman is anxious; if both of them panic, then there's no hope.

"Are you confident about applying to the Mathematics Department at Princeton University again next season? I guess it's still a bit uncertain; you're stuck at the interview stage." Shen Qi seemed somewhat serious.

"Then... what should we do?" Ou Ye lacked social experience and was at a loss.

"Besides Princeton, did you apply to any other American universities?" Shen Qi asked in as calm a tone as possible.

Ou Ye: "No."

"You... sigh, this is so Ou Ye." Shen Qi was at his wit's end with Ou Ye. He paced back and forth in the apartment, thinking of a solution, "There's still time. Little Ye, listen to me, send application essays to other prestigious universities on the East Coast of the United States and around Princeton."

Ou Ye: "But I want to go to Princeton University."

"The situation is special, so let's settle for second best, my Little Yezi! The Mathematics Department at New York University and Columbia University are also very strong, and New York is very close to Princeton, about the same distance as driving from the capital to Langfang. I can visit you in New York often, it's no big deal. Many people who work in the capital still live in Langfang. In the US, as long as you have a car, this is not a problem. I bought a Chevrolet, so the transportation issue is solved." Shen Qi came up with a compromise.

Suddenly, Shen Qi had a brilliant idea. "That's right, our senior, Gong Changwei, is a professor in the Mathematics Department at Columbia University. You could apply to his graduate program! Senior Changwei is an expert in number theory, which is a perfect match for your major!"

"Hurry up, Ou Ye, submit your application to Columbia University's Mathematics Department right now! If you're lucky, you can go to New York to study next January, stay with Senior Changwei for a year or two, and then come to Princeton for your PhD. Once you're in the US, everything will be easy. I'll help you arrange everything for your PhD at Princeton. I'm just asking you Ou Ye, are you going to listen to me?"

"I'll listen to you."

The young couple had reached an agreement: Ou Ye would first pursue a master's degree at Columbia University, and then come to Princeton for her doctorate.

There's nothing I can do; Ou Ye just couldn't pass the interview.

This indirect strategy to salvage the relationship is the best strategy Shen Qi can think of at the moment.

So here's the question: Columbia University, located in New York City, is also Ivy League school. Does their interview guarantee a passing grade for Ou Ye?

This requires further investigation.

"Professor Muller, I don't know if there's such a system where you personally interview a Chinese student? She's an exceptionally talented mathematical genius." Shen Qi wasn't giving up. He had two plans in mind and immediately approached Professor Muller, hoping that Professor Muller would directly recruit Ou Ye to Princeton.

"I'm very sorry, Shen Qi, there is no such rule, at least not at Princeton. New York University and Columbia University have more flexible admissions systems, you can let her try." Professor Muller explained that for the admission of international undergraduates to Princeton University for graduate studies, the university professors only conduct written academic evaluations of the students and do not participate in the interviews; the interviews are handled by professional interviewers.

Shen Qi had Gong Changwei's contact information, and the two exchanged business cards when they held the "Ivy League Number Theory Symposium" last year.

Before contacting Gong Changwei, Shen Qi first spoke with Sun Erxiong, asking Sun Erxiong to also put in a good word for him.

Upon hearing this, Gong Changwei immediately made a decision. "Ou Ye is a good student. I remember her. She and Shen Qi jointly published the Proof of Diophantine Equation Wash Conjecture. No problem, let Ou Ye come to Columbia University to study for my master's degree. I will interview her personally, via video interview."

Shen Qi was relieved. With Gong Changwei personally conducting the interview, at least he wouldn't deliberately make things difficult for Yan Shi Da Mei.

Moreover, Ou Ye's academic achievements speak for themselves, and as Calculation Ji specializing in number theory, she is very well-suited to study under number theory expert Gong Changwei.

In September, Shen Qi became a doctoral student in the Mathematics Department at Princeton University.

Meanwhile in China, Ou Ye and Zhou Yu'an are entering their senior year of college. They are making preparations to enroll at Columbia University and Princeton University respectively in January of next year.

While waiting for Ou Ye, Shen Qi decided to do some meaningful and interesting academic research.

This interesting idea came from an unexpected discovery.

Shen Qi originally wanted to find a paper by Faltings published in the early 1980s, in which Faltings proved a theorem, thus laying the foundation for his proof of the Mordell Conjecture a year later.

It was because of his proof of the Mordell Conjecture that Faltings became a sensation in the 1980s and won the Fields Medal.

Faltings currently works at Princeton and is the editor-in-chief of the Annals of Mathematics.

Shen Qi sought out Faltings's old paper in order to conduct meaningful research, as he needed to cite Faltings's paper as source material.

In this mathematics journal published in the early 1980s, Shen Qi discovered an interesting secret: another paper.

This interesting paper has two authors: Allen Muller and Tony Manning.

Shen Qi didn't know who Tony Manning was, but he did know that Allen Muller was his master's and doctoral advisor, Professor Muller.

In 1982, Mathematical Research Allen Muller and Tony Manning co-authored this paper on the study of Banach space.

Banach space has been a popular research area for decades.

In their paper on Banach space, Allen Muller and Tony Manning claimed to have proven a remarkable theorem, which they named the "Muller -Manning theorem".

However, in Shen Qi's memory, he had never heard of any mathematical theorems related to "Muller -Manning". The combination of these two names seemed more like a name reaction in Organic Chemistry synthesis.

The Muller -Manning theorem is described as follows:

Let X be Banach space, and X* be the conjugate space of X. Then the following arguments are equivalent:

(1) If x*∈S(X*) attains its norm on S(X), then x* is an ω* concave point of the unit ball B(X*);

(2) X is a strongly smooth space;

(3) Every ω* closed convex set of X* is an approximate compact Chebyshev set.

"So Professor Muller, you and this Mr. Manning unilaterally claimed to have proven this theorem?" Shen Qi asked with great curiosity.

It's really intriguing. More than thirty years have passed, and the IMU has not acknowledged the so-called "Muller -Manning theorem."

What's going on? Mathematicians worldwide have spent over thirty years on this, and still haven't been able to verify the proof of the Muller -Manning theorem?

"This was the result of my early research with Manning. At the time, we thought it was definitely correct, but now I find that it is full of flaws," Professor Muller said with a heavy heart, as if he had done something foolish when he was young.

Shen Qi expressed his opinion. "I think there is nothing wrong with the description of this theorem. Of course, before it is thoroughly proven, it can only be considered a hypothesis. If I were to complete the proof, I think I would have a new approach to fill the gaps, to make the proof process as rigorous and perfect as possible, and ultimately gain the recognition of IMU."

Muller said, "Then give it a try. I'll work with you to redo the proof—a 21st-century proof. If we succeed, it will be rewritten as the 'Muller -Shen Theorem'."

"What about Mr. Manning? If I replace him, will he have any objections? Or would it be more appropriate to change it to the MMS theorem?" Shen Qi suggested, respecting the academic status of the original author.

"That damned bastard... he died ten years ago," Professor Muller suddenly said through gritted teeth, sounding sorrowful and resentful.

"May Mr. Manning rest in peace in heaven," Shen Qi said with regret, noting that a great mathematician had tragically died ten years ago.

"Heaven? That bastard must be in hell! There's no MMS theorem, only the Muller -Shen theorem. Let's begin this project that spans centuries, Shen Qi."