Great Country Academician

Chapter 201 Smooth Manifolds—Communication with Fefferman

When Fefferman came to his door, Xu Chuan smiled awkwardly. For more than a month, he was obsessed with his own research and forgot about it.

After assuring Fefferman that he would start classes this week, the dean of the math department was also relieved.

However, Xu Chuan's research for more than a month also aroused his curiosity.

The professor who looked a bit like Peng Yuyan when he was young asked curiously: "What have you been researching recently?"

Xu Chuan shook his head and said, "I didn't study anything, I was 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 all zero?"

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

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

"From this point of view, if you want to study this problem, you may start with defining an affine flat manifold as a manifold with torsion-free flat affine connections on a tangent bundle (torsion-free flflat affiffine connection )."

Fefferman thought for a while, and said doubtfully: "The finite expansion of the free product of the sum 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 around"

In the office, Fefferman, who was supposed to inform Xu Chuan not to forget his professorship at Princeton, fell into communication and thinking.

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

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

Sure enough, communicating with top mathematics professors is a pleasant thing. The thoughts and ideas that can be gained from one exchange are enough to think for a long time, and even benefit for a lifetime.

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

Then submit your course to Princeton.

It has been a month and a half and almost two months since the beginning of the Spring Festival, but he hasn't had his first class yet.

If classes are not held again, Princeton University may feel that its 600,000-meter gold has been lost in Carnegie Lake.

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 Department of Mathematics.

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

But it has been more than a month since the school started, and this young professor seems to have forgotten what it was like.

time flies.

With the arrival of Wednesday, Xu Chuan's first class in Princeton finally arrived.

For the first class, the schedule he hung up 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 aspect of knowledge.

As the grandson of the pope in algebraic geometry, although Xu Chuan is not only good at algebraic geometry, but in any case, the first class must be this.

When Xu Chuan walked into the big classroom of the school in formal attire, he was a little taken aback.

In the amphitheater that can accommodate more than 300 people, not to mention the empty seats, there are even two students squeezed into each other's seats.

The original corridors were not vacant, and there were no seats, so these students sat directly on the ground.

The crowded classrooms made Xu Chuan wonder whether there are so many mathematics students in the Princeton mathematics department.

In fact, he guessed right, the students who came here were not only from the Princeton Department of Mathematics, but also some students from New York and Philadelphia.

Some of these students came here voluntarily after being notified by their friends, while others were pulled over.

A math class by a mathematician who proved Hodge's conjecture is a great attraction for students who want to go further in mathematics.

Not only the students, but some professors from Princeton walked into the lecture hall 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 didn't think too much about his popularity in the mathematics world, Xu Chuan still walked onto the podium calmly with a smile on his face.

There is no such thing as stage fright in him today, let alone a lecture theater with three or four hundred people, even a math lecture meeting with thousands of people, he can handle it with ease.

Walking up to the podium, Xu Chuan looked at the students in the audience, those eyes full of expectation made him seem to return to the past.

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

"Let me introduce myself. My name is Xu Chuan and I come from Huaguo. 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 almost forgot my identity as a professor."

Hearing this, a burst of laughter erupted in the classroom.

"Is it because being a professor for the first time is too nervous to start a class?" A student asked jokingly in the audience.

Xu Chuan shrugged and said, "That's not true. Giving you a lecture is no more difficult than standing in the auditorium and explaining Hodge's conjecture to those masters 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 the ridicule of this student, Xu Chuan didn't know whether it was intentional or unintentional, but he had to suppress the ridicule, otherwise he might not be easy to lead in class in the future.

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

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

And his answer was exactly the same.

It means that you people are scumbags, and it is easy to give you lessons.

Hearing what he said, the student who asked the question seemed to have noticed that there was something wrong with his question, and he touched 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."

"On this, I would like to thank a great mathematician who gave me the original inspiration."

In English, he and she have two completely different pronunciations. Hearing what Xu Chuan said, the students in the classroom immediately 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 me a manuscript paper before his death. On that manuscript paper, I saw some of her thoughts on the fields of 'Weyl-Berry conjecture' and 'incomparable decomposition of differential algebraic varieties' which I proved earlier. .”

"Based on this, I refined her idea and solved the problem of 'Irreducible Decomposition of Differential Algebraic Varieties'."

"Based on this, I made an extension again, and finally created the tool of 'algebraic variety and group mapping', and 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 researching Hodge's conjecture."

After the words fell, some students noticed some details sensitively.

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

Xu Chuan nodded: "That's true, what's the problem?"

Hearing this, everyone in the auditorium couldn't help swallowing.

What is the problem?

This is a big problem!

From July to November when Hodge's conjecture was published, 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 silent as a chicken, and then there were loud exclamations and noises.

"How can this be!"

"Five months to prove the Hodge conjecture, this is definitely a joke."

"God can't even do it."

"."

More than four hundred people would rather believe that Xu Chuan was joking with them than believe it was true.

The two years previously speculated by everyone to solve the Hodge conjecture are already terrifying. After all, it took Wiles eight years to prove Fermat's last theorem, and Perleman to prove the Poincaré conjecture for seven years.

In comparison, two years can be said to be too short.

But now, this has been shortened to five months, and no one can believe it is true.

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

On the podium, Xu Chuan didn't care about his own words. Five months proved that Hodge's conjecture was too outrageous, and no one would believe it when he said it. Everyone would just regard it as a class fun.

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

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

"And today's lesson is the part of beating the foundation"

Being a professor for more than ten years in the previous life was not for nothing. His abundant energy allowed him to smoothly change the topic to the formal classroom and start lecturing.

In the 90 minutes of class time, no one left early.

Only a few people couldn't hold back and ran to the toilet a few times during the lecture, but they came back soon.

The first day of lectures went smoothly, and Xu Chuan flipped through books and PPTs very quickly. He directly flipped through nearly half of a hundred-page Algebraic Geometry textbook.

Of course, this also has something to do with the students that this class is aimed at today.

This kind of big class is aimed at a lot of students, undergraduates and graduate students can listen to it, but in order for most people to understand it, the knowledge explained must not be very advanced.

Really top-notch knowledge can only be learned in small classes and following a mentor.

However, this kind of big class is not without its merits. In the big class, some ideas and ideas taught in lectures are more critical and important than those advanced knowledge.

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

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

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

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

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 the class today actually didn't have much hope in this regard.

Even if the lecturer is a top-notch expert who proved Hodge's conjecture, it is his first time teaching a class and he has almost zero teaching experience.

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

He can understand complex problems and ideas in simple language, which is especially valuable for a professor.

How to let students understand and absorb knowledge and understand problems in the fastest way, the difference between good tutors and poor tutors is often here.

And this young professor, even teaching for the first time, has done just that.

Surprised both the students in the classroom and some of the math professors.

Ninety minutes passed, but Xu Chuan did not procrastinate. He ended the class on time and left three questions as homework.

Princeton's major courses can assign homework, of course, you don't need to assign homework.

This depends on the wishes of the lecturer.

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

"A wonderful lecture. I was still worried about whether you would be nervous or not. Now it seems that I am thinking too much. In terms of teaching, you are far better than I imagined."

As the get out of class ended, Princeton Mathematics Department Dean Fefferman came out from the corner of the lecture theater, stood in front of Xu Chuan and congratulated him with a smile.

He listened to the 90-minute lecture in its entirety. Although those basic things were not very useful to him, he saw the teaching ability of this young professor and friend.

It is undoubtedly worthwhile for Princeton to keep him with an annual salary of 600,000 meters.

Xu Chuan smiled and said, "Thank you."

After a word of praise, the dean of the mathematics department did not leave, but continued: "After communicating with you last time, I have some new ideas about smooth fashion. I don't know if you have time recently. I want to talk to you again."

Xu Chuan said quickly, "I'm extremely happy."