From Almighty Scholar to Chief Scientist

Chapter 289 Can You Change Theme?

"P-adic theory, you can call it p-adic number theory, is a relatively important basic theory in our number theory. At the same time, it is also very compatible with other fields in mathematics, and it can even be used as a basis for your graduate students in the future. a research direction."

Speaking of this, Lin Xiao smiled slightly: "At the last International Congress of Mathematicians, there was a 31-year-old Fields Medal winner named Peter Schultz, who studied the p-adic theory. He used a very It is a wonderful method, introducing some very complex geometric problems into the p-adic theory to achieve simplification, and then solve a lot of problems."

"For example, he transformed arithmetic algebraic geometry into p-adic domains by introducing quasi-complete spaces, which can simplify arithmetic problems on local domains as combinations of specific features and feature domains."

"Using this technique, he succeeded in generalizing the near-purity theorem in Hodge's theory."

"So studying this theory, maybe you may get a Fields Medal."

Hearing Lin Xiao's words, the students present rolled their eyes one after another.

Look at what you said, it's as if you can win a prize as long as you study this thing.

If it was so easy, would they still be able to sit here?

Of course, Lin Xiao's introduction still gave them some interest in these students. It's something that even a Fields Medalist has studied. Wouldn't it be better for them to study it?

In particular, Lin Xiao also mentioned Hodge's theory. Although they don't all know Hodge's theory, as students of mathematics, they all know Hodge's conjecture.

Well, something that can be linked to the Millennium Prize puzzle must be a good thing.

So these students all showed serious expressions.

Seeing their expressions, Lin Xiao smiled slightly, which aroused his interest, which would be convenient for his subsequent lectures, so he stopped talking and started a formal lecture on the p-adic theory.

"p-adic number is a kind of complete number field expanded from the rational number field. This kind of expansion is different from the common number system expansion from the rational number field to the real number field. It is specific in the defined concept of "distance". We generally use Qp to express……"

The class began, and the students present also began to think seriously.

Although most of the students present had done previews before, for students studying mathematics, knowledge that they could not understand even after reading books was very common.

And this p-adic theory, of course, also makes many students feel more difficult to understand.

After all, compared to the classic theorems in number theory they have learned before, p-adic theory, which even has a slightly strange name, still has some difficulties for them to understand it.

Of course, it is the teacher's turn to play a role at this time.

With Lin Xiao's detailed description, these top students in mathematics can gradually understand.

So, in the first 30 minutes of this class, Lin Xiao led these students to understand what p-adic theory is.

"Now, all students should have a basic understanding of what p-adic numbers are."

"P-adic numbers have two main properties."

"The first one is algebraic properties."

"In algebra, Qp is the fraction field of Zp, and to be precise, Qp=Zp[1/p]..."

"Everyone should remember that in our field of number theory, the algebraic properties of p-adic numbers are more important. After you go back, you should study this knowledge carefully and consolidate it. The exam will be passed~"

Speaking of this, Lin Xiao smiled slightly.

Seeing his smile, the students present all trembled, quickly picked up a pen, and wrote down this point.

None of the students present knew that as long as Lin Shen smiled when they talked about the places that might be tested in the exam, their life and death would be unpredictable.

Because this means that Lin Xiao will often come up with a final question in this aspect. Although the difficulty will not be as difficult as the previous question, the scoring rate will definitely not be high.

While watching them taking notes, Lin Xiao comforted with a smile: "Everyone, don't be nervous, after all, I'm not a devil."

However, every student here did not believe it, they rolled their eyes one after another, and then added another key mark in this place, and wrote the words "very important" by the way, so as not to overlook it in the review later.

Seeing that no one believed him, Lin Xiao shrugged and continued the lecture: "Then it is the second property, that is, the topological property. The topological property is not the point. As I said before, learning from our current In mathematics, it is best to specialize in one direction. If you are interested in developing topology, you can study it, but for now, I will just talk about it briefly.”

"The topological properties of p-adic are mainly manifested as the norm on Qp, |·|p is a hypermetric norm. It not only satisfies the triangle inequality, but also satisfies stronger relations..."

"This shows that if Qp is imagined as a geometric space, then the length of one side of the triangle is always less than or equal to the longer of the other two sides, that is to say, all the triangles are acute-angled isosceles triangles. This is different from the actual Euclidean geometry The spaces are completely different. Hence Qp and R have very different topological properties...huh?"

When he said this, Lin Xiao frowned suddenly and stopped his narration.

But the students present were puzzled when they heard Lin Xiao say "huh?" and stopped talking.

What's going on here?
However, after hesitating for a moment, Lin Xiao continued to narrate: "The topology on Qp is a completely disconnected Hausdorff space. At the same time, Qp is obtained from the completion of Q, so Q is dense in Qp, not only that. , any given... hmm?"

As soon as he said this, Lin Xiao suddenly stopped again, looked up at some mathematical formulas that he listed on the PPT that stated the topological properties of p-adic numbers, put one hand on his chin, and fell into a state of contemplation.

And this made the students present even more curious.

What did Lin Xiao think of?

"You said, Lin Shen won't have an epiphany again, right?"

Below, a student whispered.

The others nodded thoughtfully: "It seems so..."

After all, Lin Xiao's epiphany is famous all over the world.

"This is another epiphany..."

"Maybe it's Hodge's conjecture? Didn't Lin Shen say that this p-adic theory is related to Hodge's theory before class?"

"Although Hodge's conjecture is related to Hodge's theory, Hodge's theory includes more content, right? I remember that Hodge's theory mainly talks about a method of using partial differential equations to study the cohomology group of a smooth manifold M. The Hodge conjecture is just included, right?"

"Goofy, you even know this? Don't fuck up, don't fuck up~"

……

Just as the students below were all looking at Lin Xiaona staring at the ppt and thinking, Lin Xiao finally came back to his senses.

Remembering that he was still in class at this time, he came back to his senses and apologized: "I'm sorry, I remembered something else just now."

"Let's continue."

Afterwards, he accelerated the lecture. Of course, he was almost finished here. He quickly finished the topological structure, and then gave them a problem according to the usual practice, and asked them to do it by themselves.

Then, Lin Xiao sat on the desk, found out the paper and pen, and began to calculate.

The reason why he paused twice just now is because he saw a problem in this p-adic theory that could help him solve the Hodge conjecture he was currently facing.

"Transforming arithmetic algebraic geometry into p-adic fields by introducing quasi-complete spaces, and applying them to Galois representations, can be used to develop a new cohomology theory..."

"And it can be entirely Motive coherent!"

Lin Xiao wrote down several seemingly complicated formulas on paper, and then began to try to move towards the cohomology direction.

But after a while, he frowned again.

"How to prove that there is a class of finite non-divergent Galois extension L/Kp whose ring is O` and residual field is k` for which there exists A`∈H1(E*o′, Z/2(1)) respectively? "

"If this problem is not solved, there will be certain problems in the process of Galois' expression..."

After thinking for a while, he simply logged into his mailbox, attached his thoughts to it, and sent it to Peter Schultz.

He certainly has Peter Schulz's contacts.

However, because he used the computer on the multimedia, and the projection was directly projected on the screen of the blackboard, all the students present saw it.

After seeing Lin Xiao attaching his ideas, all the students present were at a loss.

What the hell is this?

They didn't know anything except a p-adic at the beginning, and because Lin Xiao sent it to Peter Schultz, his email was also in English, which made the students feel even more Confused.

So this is what top mathematics experts usually study?
However, this is not over yet. When they finally saw that Lin Xiao had attached the name of Peter Schultz, they were even more shocked. Lin Xiao's email was actually sent to a Fields Medal winner?

What is a network?This is called networking!
And these have nothing to do with them for the time being, they can only lower their heads and continue to work hard on their questions.

In this way, time passed quickly.

The get out of class bell rang, and 10 minutes later, the class bell rang again, and Lin Xiao continued to teach.

Soon, when the class was almost over, Lin Xiao gave the students a period of time for self-study, while he continued to enter the mailbox, and was surprised to find that Peter Schultz replied so quickly.

When he opened the email, Peter Schultz sent an attachment directly. After he downloaded the attachment, he read it.

[Professor Lin, hello!I'm very glad to receive your letter. I didn't expect you to be interested in my original research. After reading your letter, I think your research should be the Hodge conjecture, right?

Regarding your question, how to prove this problem about Galois representation, I happened to study it recently when I was studying Hodge's theory.

Note first that A`∈H1(E*o′, Z/2(1)) can be set as the class of Hlet(E, Z/1), since it is invertible in the residual field, this group will be E on The Z/2 parameterization of...

Br(S′)[2]→Br(S′Kp )[2]=Z/2, here, we need to continue to classify it into the p-adic field, and then use the method of number theory to solve it, I believe that in On this issue, there is no one more than you, Professor Lin.

In fact, in the process of studying Hodge's theory, I also thought about Hodge's conjecture. I don't know if you have read the paper by Rosenson Andreas in 2016, where the problem of how to obtain the correct integral Hodge Odd conjecture, made a conjecture, I recommend you to take a look, in a word, cohomology and Hodge conjecture are closely related, perhaps Motive is the most critical factor to solve Hodge conjecture!
...]

After reading this reply, Peter Schultz basically has no secrets, and has given Lin Xiao a lot of inspiration.

And the paper recommended by Schultz, Lin Xiao naturally read it.

And now, he has the confidence to really solve Hodge's conjecture.

At least, it is an important phased result of Hodge's conjecture.

Thinking of this, he took a long mouthful, and the corner of his mouth curled up.

Maybe, when we go to the International Congress of Mathematicians, we can change the topic of the report?

(End of this chapter)