From Almighty Scholar to Chief Scientist

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

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?"

"Gouzi, you even know this? Don't curl up, don't curl 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 can help him solve the Hodge conjecture he is 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 exist 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 the email 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 H1et(E, Z/2). Since it is invertible in the residual field, this group will 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?

Chapter 288 Be a Dog

A week passed quickly.

In Lin Xiao's room, he was still working on his work.

According to the exchange with Peter Schultz that day, Lin Xiao's thinking also got many different perspectives.

It's like writing. When you get a topic in hand, the content written by different people will be very different, no matter the angle of entry or the thoughts that the article wants to express, it will be very different.

The same goes for math problems.

Even if it is a question under that kind of exam-oriented education, there may be many different solutions, let alone a world-class question like Hodge's conjecture.

Therefore, Peter Schultz's different views also broadened Lin Xiao's thinking.

But it also made him gain a lot.

Until now.

[Assume (X, A) is a space pair, G is an arbitrary commutative group, denote that C(X, A) represents the singular chain complex of (X, A), and is a complete space composed of the closed p form of the differential manifold M For the subvariant space composed of proper p forms, it is the Motive motivation cohomology group. 】

【HdR(M)≌R^n...】

Looking at the final definition of motivational cohomology on the draft paper, a smile appeared on Lin Xiao's face.

Motive cohomology is the key to figure out the integral Hodge conjecture, and now, he finally got it done.

Of course, what he created is not the kind of universal cohomology pursued by motivation theory. Universal cohomology is an encapsulation of all cohomology sets, and his current motivational cohomology is only in line with Motive motivation theory. relevant.

However, it already has the rudimentary form of motivational theory—because of this motivational cohomology, it has communicated various cohomology theories such as singular cohomology and Durham cohomology, but it is only limited to these types for the time being. The real omnipotence, or in other words, the universal cohomology.

As for the real motivation theory, Lin Xiao faintly felt that perhaps more development was needed.

"In this way, I should call it Lin's motivational coherence for the time being?"

Smiling slightly, Lin Xiao picked up the pen and added the word "Lin's" in front of the name of the homology of the motive.

Nodding in satisfaction, he put down his pen.

It is one step closer to let the word "Lin's" spread all over the mathematics world.

I just hope that future students will not blame him.

But in retrospect, the multidimensional field theory in the physics world has probably caused headaches for many students. It is said that some university textbooks have already included part of the multidimensional field theory, especially those students who study particle physics. After all, the Standard Model under the multidimensional field theory has replaced the previous Standard Model, and the Standard Model under the multidimensional field theory has higher and higher requirements for mathematics, which of course hurts those physics students.

Of course, these have nothing to do with Lin Xiao.

Bring your attention back to your eyes, and with Lin's motivational coherence, it is basically enough.

"Then the next step is to figure out the Hodge conjecture in the integral form."

The corner of Lin Xiao's mouth curled up. With Lin's motivational coherence, he could complete this step with his eyes closed.

So he picked up his pen again and began to write.

"Firstly, Lin's motivational cohomology is expressed as a differential form..."

【ω=dα＋δβ＋γ】

[(α, β)=∫αΛ*β]

【…】

Until ten minutes passed.

"In summary, the Hodge conjecture in the integral form should be like this: Let X be a projective complex manifold, and then each cohomology class H^2k(X, Z) ∩ H^k, k(X) is the sum of the torsional and cohomological classes of algebraic rings with integral coefficients on X."

"Finally, it's done."

Looking at himself after a lot of mathematical formula calculations, and finally simplified the form of Hodge's conjecture, Lin Xiao showed a brighter smile than before.

"Hodge's conjecture in integral form, it turns out to be like this."

"It seems that it is indeed easier to understand than the previous form."

Lin Xiao touched his chin.

The reason why Hodge’s conjecture is converted into this integral form is naturally because the original form is too difficult to understand. Mathematics is the process of simplifying complex things, making complex things into simple things, and finally solving this problem is this process.

In a word, it is to pull those difficult problems to the same level as myself, and then use my rich experience to solve them.

"Well... Next, it's time to prove Hodge's conjecture in integral form."

By proving the Hodge conjecture in integral form, the final Hodge conjecture can be proved.

One of the Seven Millennium Problems...

Even Lin Xiao couldn't help but look forward to it at this time.

This is another truth in the universe, how can Lin Xiao not look forward to it?

Moreover, the role of Hodge's conjecture in promoting the great unification of mathematics also makes Lin Xiao very much looking forward to the rewards that will be brought to him after the proof.