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 different mathematical concepts are linked together, revealing the profound 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 discrete group canonical action (properaction) in an affine space, then Chen's conjecture is true."
"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 a torsion-free flat affine connection on the tangent bundle (torsion-freeflflataffiffiffineconnection)."
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 on the schedule 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, which 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 a lecture lecture with thousands of people, let alone a lecture theater with three or four hundred 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 at 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 I am too nervous to start a class as a professor for the first time?" a student asked jokingly in the audience.
Xu Chuan shrugged and said, "That's not true. Giving lectures to you is no more difficult than standing in the auditorium and teaching the Hodge conjecture to those big names in mathematics."
In American classrooms, the relationship between students and teachers is more relaxed than in China. It is not uncommon for students to tease and question teachers.
Regarding this student's teasing, Xu Chuan didn't know whether it was intentional or unintentional, but he had to suppress the teasing, otherwise it might be difficult to teach him in class in the future.
After all, he is so young that most of the students here are older than him.
In this case, even if he is the professor, he will give students some unreliable or other ideas.
And his answer was exactly the same.
It means that you people are all scum and it is easy to teach you a lesson.
Hearing what he said, the student who asked the question seemed to notice that there was something wrong with his question, and he rubbed his head in embarrassment and stopped talking.
"Professor, can you tell us how you proved Hodge's conjecture?" A student in the classroom asked curiously.
Xu Chuan smiled and said, "Of course."
"For this, I want to thank a great mathematician who gave me the original inspiration."
In English, he and she have two completely different pronunciations. When Xu Chuan said this, the students in the classroom suddenly became interested and looked up at the podium.
Xu Chuan continued: "In July last year, Professor Mirzakhani, the first female winner of the Fields Medal in mathematics, left us forever." "And this professor left a manuscript for me during his lifetime. , on that manuscript, I saw some of her thoughts on the fields of 'Weyl-Berry conjecture' and 'irreducible decomposition of differential algebraic varieties' that I had previously proved."
"On this basis, I improved her idea and solved the problem of 'irreducible decomposition of differential algebraic varieties'."
"Based on this, I made another extension and finally created the 'algebraic variety and group mapping' tool, which finally solved the Hodge conjecture."
"So I am very grateful to Professor Mirzakhani. Without her, perhaps I would never have embarked on the path of studying the Hodge conjecture."
As soon as he finished speaking, some students sensitively noticed some of the details.
"Professor, you said that you only received Professor Mirzakhani's posthumous 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 theater couldn't help but swallow.
What's the problem?
This is a big problem, okay!
From the beginning of July to the publication of the Hodge conjecture in November, the interval was only five months.
Five months, Hodge’s conjecture was proven, what the hell is this?
For a moment, the huge lecture hall was as silent as a chicken, and then there were noisy exclamations and noises.
"How is this possible!"
"Five months proved Hodge's conjecture, which is definitely a joke."
"Even God comes and can't do it."
"."
More than 400 people would rather believe that Xu Chuan was joking with them than believe that it was true.
The two years that everyone had 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 Poincaré conjecture.
In comparison, two years can be said to be too short.
But now, it has been shortened to five months. No one can believe this is true.
They would rather believe that this was just a joke made by Xu Chuan. It actually took two years to prove Hodge's conjecture.
On the podium, Xu Chuan didn't care about his words. Five months proved that Huo Qi's conjecture was too outrageous. No one would believe it if he told it. Everyone would just treat it as a piece of fun in class.
But he followed this path and changed the topic to today's class.
"Although the time to prove Hodge's conjecture was short, I laid the foundation for more than ten years before that. High-rise buildings are not built in a day. Only by laying the foundation can we pave the way up step by step."
"Today's lesson is about laying the foundation"
Being a professor for more than ten years in his previous life was not for nothing. His abundant energy allowed him to smoothly change the topic to a formal classroom and start giving lectures.
During the ninety-minute class time, no one left early.
Only a few people couldn't bear it and ran to the toilet a few times during the lecture, but they came back quickly.
The first day of lectures went smoothly. Xu Chuan flipped through books and PPTs very quickly. He flipped through nearly half of a hundred-page algebraic geometry textbook.
Of course, this also has something to do with the students this class is for today.
This kind of large course is aimed at a wide range of students. Both undergraduates and graduate students can take it. In order for most people to understand it, the knowledge explained must not be very advanced.
Real top knowledge can only be learned through small classes and following tutors.
However, this kind of large class is not without its merits. In large classes, some of the ideas and ideas taught in lectures are more critical and important than advanced knowledge.
Knowledge can be learned, but ideas and ideas can only come from creation.
In mathematics, listening to a top expert’s understanding of a certain basic field can be of 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 useful for undergraduates, but for graduate students and doctoral students, they have already learned it.
But sometimes, different professors explaining the same knowledge point will bring different insights.
This depends on the professor's own knowledge reserve, teaching ability, and teaching experience.
Most of the students who came to attend the class today actually didn’t have much hope in this aspect.
Even if the professor is a top expert who has proven Hodge's conjecture, it is his first time teaching and he has almost zero teaching experience.
But what is surprising is that this extremely young professor seems to have more than ten years of experience in teaching.
He can explain things in simple terms and make you understand complex issues and ideas in simple language, which is especially valuable for a professor.
How to allow students to understand and absorb knowledge and understand problems in the fastest way is often the difference between good tutors and poor tutors.
And this young professor did this even if he was teaching for the first time.
The students in the classroom and some mathematics professors were surprised.
Ninety minutes passed. Xu Chuan did not delay the class and ended the class on time, leaving three questions as homework.
Princeton's major classes can assign homework, of course, or they don't have to.
This depends on the instructor's wishes.
As for whether to do it or not, it depends on the students' own wishes. Anyway, the lecturer will not check it.
"A wonderful lecture. I was worried before that you would be nervous. But now it seems that I was overthinking it. You are far better at teaching than I thought."
As get out of class ended, Fefferman, head of the mathematics department at Princeton, walked out of the corner of the lecture theater, stood in front of Xu Chuan and congratulated him with a smile.
He completely listened to the ninety-minute lecture. Although the basic things were not of much use to him, he saw the teaching ability of this young professor and friend.
Princeton's annual salary of 600,000 yuan is undoubtedly worth keeping him.
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 wonder if you have time recently. I want to communicate with you again.”
Xu Chuan said quickly: "I am extremely happy."