Chapter 201 Communication Between Smooth Manifolds and Fefferman

Chapter 201 Smooth Manifolds—Communication with Fefferman

Xu Chuan smiled awkwardly when Fefferman came to his door. For more than a month, he had been so absorbed in his research that he had forgotten about it.

After reassuring Fefferman that he would start classes this week, the math department chair was relieved.

However, Xu Chuan's research over the past month also aroused his curiosity.

The professor, who looked a bit like Peng Yuyan when he was young, asked curiously: "What are you studying recently?"

Xu Chuan shook his head and said: "I'm not studying anything, I'm just reading some books on manifolds."

Fefferman looked over in surprise: "Manifold? Which direction are you going to study?"

Xu Chuan: "Currently it is a smooth manifold."

Hearing this, Fefferman's eyes brightened: "What do you think about the problem that the bounded cohomology of affine flat manifolds is zero?"

Xu Chuan thought for a while and said: "This question is related to Chen's conjecture. As we all know, Euler numbers are topological invariants of manifolds, and affine flatness is a purely geometric concept. The wonderful thing about this conjecture is that it combines these two Very disparate mathematical concepts are linked together, revealing the deep connections between them."

"If the affine flat structure on the affine flat manifold M is complete, that is to say, M is the quotient space of a certain discrete group proper action in an affine space, then Chen's conjecture holds."

"From this point of view, if you want to study this issue, you may start by defining the affine flat manifold as a manifold with torsion-free flflat affiffiffine connection on the tangent bundle. ).”

Fefferman thought for a while and said with hesitation: "The finite expansion of the free product of the harmonic group of the affine flat structure on the amenable group?"

Xu Chuan thought for a while and said: "Start from the direction of the finite-dimensional Lie group with an affine flat structure, and then turn it."

In the office, Fefferman, who was originally telling Xu Chuan not to forget his position as Princeton Professor, fell into communication and thinking.

It wasn't until more than an hour passed that the two of them stopped still.

Fefferman left with satisfaction, and Xu Chuan also gained a lot of ideas.

Sure enough, communicating with top mathematics professors is a pleasant thing. The ideas and ideas you can gain from one communication are enough to think about for a long time, or even benefit for a lifetime.

Xu Chuan really wanted to sort out some thoughts and ideas now, but before that, he still had one thing to do.

Then submit your course to Princeton.

It has been a month and a half and almost two months since the Spring Festival started, but his first class has not yet started.

If classes don’t start again, Princeton University will probably feel like its 600,000 yuan was lost in Lake Carnegie.

After Xu Chuan submitted his course report to the school, Princeton immediately put it on the curriculum.

This news also caused a lot of discussion in the Princeton Mathematics Department.

After all, many students had been looking forward to this young professor who proved Hodge's conjecture to explain this knowledge.

But more than a month after school started, the young professor seemed to have forgotten about this matter.

time flies.

As Wednesday arrives, Xu Chuan’s first class at Princeton finally arrives.

The first class he took was algebraic geometry.

As the most popular research field in the history of mathematics, it can be said that almost everyone who studies mathematics has been exposed to this knowledge.

As a disciple of the Pope of Algebraic Geometry, although Xu Chuan is not only good at algebraic geometry, but no matter what, the first lesson must be this.

When Xu Chuan walked into the school's large classroom dressed in formal attire, he was a little startled.

In the lecture theater that can accommodate more than 300 people, there are no seats, and there are even two students squeezed into the same position.

There were no vacant seats in the original corridor, so these students sat directly on the ground.

The crowded classroom made Xu Chuan doubt whether there were so many mathematics students in the Princeton mathematics department.

In fact, his guess was correct. The students who came here were not just Princeton’s mathematics department, but also some students from New York and Philadelphia.

Some of these students came here on their own after being notified by their friends, while others were dragged here.

A mathematics class taught by a mathematician who proved Hodge's conjecture is very attractive to students who are interested in going further in mathematics.

Not only students, but also some Princeton professors walked into this lecture theater and sat quietly in the corner waiting for the class to start.

Although he was a little surprised by the number of people in the classroom and did not expect much of his popularity in the mathematics community, Xu Chuan still walked onto the podium calmly with a smile on his face.

Stage fright does not exist for him today. He can handle it with ease, not to mention a lecture hall with three or four hundred people, or even a math lecture with thousands of people.

Walking onto the podium, Xu Chuan looked at the students in the audience. Those expectant eyes made him feel like he was back in time.

After coming back from his thoughts, Xu Chuan adjusted his headset and spoke gently.

"Let me introduce myself. My name is Xu Chuan. I come from China. You can call me Professor or Professor Xu."

"Today is my first class in Princeton. To be honest, I have to thank Professor Fefferman for this. If he hadn't reminded me, I would have almost forgotten my identity as a professor."

Hearing this, a burst of laughter broke out in the classroom.

"Is it because you are so nervous about being a professor for the first time that you dare not teach?" A student asked jokingly in the audience.

Xu Chuan shrugged and said, "That's not the case. Giving you a lecture is no more difficult than standing in the auditorium and giving those math masters a lecture on the Hodge conjecture."

In the classroom in the United States, the relationship between students and teachers is more relaxed than in China, and it is not uncommon for students to tease and question teachers.

Xu Chuan didn't know whether this student's teasing was intentional or unintentional, but he had to suppress this teasing, otherwise it might be difficult to lead the class in the future.

After all, he is too young, so young that the students present are basically older than him.

In this case, even if he is a professor, he will give students some unreliable or different thoughts.

And his answer is exactly the same.

It means that you are all scum, and it is easy to teach you.

Hearing him say this, the student who asked the question seemed to notice that there was something wrong with his question, and he touched his head a little embarrassedly and stopped talking.

"Professor, can you tell us how you proved the Hodge conjecture?" A student in the classroom asked curiously.

Xu Chuan smiled and said, "Of course."

"For this, I want to thank a great mathematician. She gave me the initial inspiration."

In English, he and she are two completely different pronunciations. Hearing Xu Chuan say this, the students in the classroom suddenly became interested and looked up at the podium.

Xu Chuan then continued: "In July last year, Professor Mirzakhani, the first female Fields Medal winner in mathematics, left us forever."

"And this professor left me a manuscript paper before her death. On that manuscript paper, I saw some of her ideas in the fields of 'Weyl-Berry conjecture' and 'irreducible decomposition of differential algebraic clusters' that I had proved before."

"On this basis, I perfected her ideas and solved the problem of 'irreducible decomposition of differential algebraic clusters'."

"Based on this, I extended it again and finally created the 'algebraic cluster and group mapping' tool, and finally solved the Hodge conjecture."

"So I am very grateful to Professor Mirzakhani. If it weren't for her, I might never have embarked on the path of studying the Hodge conjecture."

After the words fell, a student sensitively noticed some of the details.

"Professor, you said that you only received Professor Mirzakhani's manuscript in July last year, and then you started to study the Hodge conjecture, right?"

Xu Chuan nodded: "That's true, is there any problem?"

Hearing this, everyone in the lecture hall couldn't help swallowing their saliva.

What do you mean by is there any problem?

This is a big problem!

From July to November, the Hodge conjecture was publicly published, and the interval was only five months.

Five months, proved the Hodge conjecture, what the hell is this?

For a while, the huge lecture hall was as quiet as a chicken, and then there were noisy exclamations and noises.

"How is this possible!"

"Proving the Hodge conjecture in five months, this is definitely a joke."

"Even God can't do it."

"."

More than 400 people would rather believe that Xu Chuan was joking with them than believe that this was true.

The two years that everyone speculated before to solve the Hodge conjecture was already terrifying. After all, it took Wiles eight years to prove Fermat's Last Theorem and Perelman seven years to prove the Poincare conjecture.

In comparison, two years is too short.

And now, it has been shortened to five months, and no one dares to believe it is true.

They would rather believe that this is just a joke by Xu Chuan, and it actually took two years to prove the Hodge conjecture.

On the podium, Xu Chuan didn't care about his words. It was too outrageous to prove the Hodge conjecture in five months. No one would believe it if they told it out. Everyone would just treat it as a fun in the classroom.

But he followed this path and turned the topic to today's class.

"Although the time to prove the Hodge conjecture was short, before that, I laid the foundation for more than ten years. High-rise buildings are not built in one day. Only when the foundation is laid can we lay the foundation step by step."

"And today's class is the part of hammering the foundation"

The professor for more than ten years in the previous life was not in vain. His abundant energy allowed him to smoothly turn the topic to the formal class and start the lecture.

In the 90-minute class, no one left early.

Only a few people couldn't stand running to the toilet several times during the lecture, but they came back soon.

The first day of lectures went smoothly. Xu Chuan flipped through the book and the PPT very quickly. He flipped through nearly half of a hundred-page algebraic geometry textbook.

Of course, this is also related to the students this class is for.

This kind of large class is for a wide range of students, including undergraduates and graduate students. In order to make most people understand, the knowledge explained must not be very advanced.

The real top-level knowledge can only be learned in small classes and by following the instructor.

However, this kind of large class is not without merit. In large classes, some ideas and thoughts of the lecturer are more critical and important than those advanced knowledge.

Knowledge can be learned, but ideas and thoughts can only come from creation.

In mathematics, listening to a top expert's understanding of a basic field can bring great help to students.

Sometimes, a word from these experts can bring great gains and give them a deeper understanding.

As for the basic knowledge in the textbooks, it may be very useful for undergraduates, but for graduate students and doctoral students, those have already been learned.

However, sometimes, different professors will bring different insights when explaining the same knowledge point.

This depends on the professor's own knowledge reserves, teaching ability, and teaching experience.

Most of the students who came to the class today actually did not have much hope in this regard.

Even though the lecturer is a top expert who proved the Hodge conjecture, it is his first time to teach after all, and his teaching experience is almost zero.

But surprisingly, no one expected that this extremely young professor seemed to have more than ten years of experience in teaching.

He can explain things in simple language, and let you understand complex problems and ideas, which is particularly valuable for a professor.

How to let students understand and absorb knowledge and understand problems in the fastest way, good tutors and bad tutors often differ here.

And this young professor did this even in his first class.

The students and some mathematics professors in the classroom were surprised.

After 90 minutes, Xu Chuan did not delay the class, and finished the class on time, and then left three questions as homework.

Princeton's large classes can assign homework, of course, or not.

This depends on the professor's own willingness.

As for whether to do it or not, it also depends on the students' own willingness. Anyway, the professor will not check it.

"A wonderful lecture, I was still worried about whether you would be nervous or something, but now it seems that I thought too much. In terms of teaching, you are much better than I thought."

As the class was over, Fefferman, the head of Princeton's mathematics department, walked out from the corner of the lecture hall, stood in front of Xu Chuan and smiled and congratulated him.

He listened to the entire 90-minute class. Although the basics were not very useful to him, he saw the teaching ability of this young professor and friend.

It was undoubtedly worth it for Princeton to retain him with an annual salary of 600,000 MiG.

Xu Chuan smiled and said, "Thank you."

After complimenting him, the head of the mathematics department did not leave, but continued, "After communicating with you last time, I have some new ideas about smooth pop. I don't know if you have time recently. I want to communicate with you again."

Xu Chuan quickly said, "I am very happy."