"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 harmonious with other fields in mathematics. It can even be used as a basis for your graduate studies 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. He studied p-adic theory. He used a very
It is a wonderful method that introduces some very complex geometric problems into p-adic theory to achieve simplification, and then solves many problems."
"For example, he transformed arithmetic algebraic geometry into p-adic fields by introducing quasi-complete spaces, which can simplify arithmetic problems in local fields into specific characteristics and combinations of characteristic fields."
"Using this technology, he successfully generalized the near-purity theorem in Hodge's theory."
"So by studying this theory, you might get a Fields Medal."
Hearing Lin Xiao's words, the students present rolled their eyes.
From what you said, it's as if you can win a prize just by studying this thing.
If it was really that easy, would they still be sitting here?
Of course, Lin Xiao's introduction still gave these students some interest. Wouldn't it be better for them to study something that a Fields Medalist has studied?
In particular, Lin Xiao also mentioned the Hodge theory. Although not all of them know the Hodge theory, as mathematics students, they all know the Hodge conjecture.
Well, something that can be linked to the Millennium Prize problem must be a good thing.
So these students all showed serious expressions.
Seeing their expressions, Lin Xiao smiled slightly and aroused interest, which would facilitate his subsequent lectures, so he stopped talking and started a formal lecture on p-adic theory.
"The p-adic number is a type of complete number field that is expanded from the rational number field to the real number field. This expansion is different from the common number system expansion from the rational number field to the real number field. It is specific to the defined concept of "distance". We generally use Qp to
express……"
When the class started, the students present also began to think seriously.
Although most of the students present have done previews before, it is very common for students studying mathematics to find knowledge that they cannot understand even if they read a book.
Of course, this p-adic theory makes it difficult for many students to understand.
After all, compared with the classical theorems in number theory that they had studied before, it was still somewhat difficult for them to understand a theory like p-adic theory, which even had a slightly strange name.
Of course, it is the teacher's turn to play a role at this time.
As Lin Xiao explained in detail, these top students in mathematics gradually began to 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."
"Algebraically, Qp is the fractional 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 relatively important. After you go back, you should study this knowledge and consolidate it, and you will be able to pass it in the exam~"
Speaking of this, Lin Xiao smiled slightly.
Seeing his smile, the students present shivered and quickly picked up their pens to write this down.
None of the students present knew that as long as Lin Shen smiled when he mentioned the possible test points, their life or death would be uncertain.
Because this means that Lin Xiao will often ask a final question in this area. Although the difficulty will not be as difficult as the previous question, the scoring rate will definitely not be high.
Watching them taking notes, Lin Xiao comforted him with a smile: "Don't be nervous, everyone. After all, I'm not a devil."
However, every student here didn't believe it. They rolled their eyes one after another, and then added another important mark in this place, and wrote the four words "very important" by the way, so as not to ignore it during review later.
Seeing that no one believed in him, Lin Xiao shrugged and continued the lecture: "Then it is the second property, which is the topological property. The topological property is not the point. I have also said before that learning from our current
In mathematics, it is actually 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 represented by the norm on Qp, |·|p is a hypermetric norm. It not only satisfies the triangle inequality, but also satisfies a stronger relationship..."
"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, which means that all triangles are acute isosceles triangles. This is different from the actual Euclidean geometry
The spaces are completely different. Therefore Qp and R have completely different topological properties... huh?"
When he said this, Lin Xiao suddenly frowned and stopped talking.
The students present were confused when they heard Lin Xiao say "huh?" and stopped talking.
What's going on here?
However, after Lin Xiao hesitated for a moment, he continued to talk: "The topology on Qp is a completely disconnected Hausdorff space. At the same time, Qp is obtained by the completion of Q, so Q is dense in Qp. Not only that
, any given... hmm?"
Just as he said this, Lin Xiao suddenly stopped again, looked up at some mathematical formulas he listed on the PPT that stated the topological properties of p-adic numbers, held his chin with one hand, and fell into a state of deep thought.
And this made the students present even more curious.
What was Lin Xiao thinking about?
"You think Lin Shen won't have another epiphany, right?"
After all, Lin Xiao's epiphany was famous all over the world.
"This is another epiphany..."
"Maybe it's the Hodge conjecture? Didn't Lin Shen say before class that this p-adic theory is related to the Hodge theory?"
"Although the Hodge conjecture is related to the Hodge theory, does the Hodge theory include more content? I remember that the Hodge theory mainly talks about a method of using partial differential equations to study the cohomology group of the smooth manifold M.
Hodge guesses it's just included in it, right?"
"Gouzi, you even know this? Stop rolling it, stop rolling it~"
…
Just as the students below were looking at Lin Xiao 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 just remembered other things."
"Let's continue."
Afterwards, he accelerated the lecture. Of course, he was almost finished at this point. He finished the topology very quickly, and then gave them a question as usual and asked them to do it by themselves.
Then, Lin Xiao sat on the desk, found paper and pen, and began to calculate.
The reason why he paused twice just now was because he saw a problem in this p-adic theory that could help him solve the Hodge conjecture he was currently facing.
“By introducing quasi-complete spaces, arithmetic algebraic geometry is transformed into p-adic fields and applied to Galois representation, which can be used to develop a new cohomology theory..."
"And it can definitely be Motive cohomology!"
Lin Xiao wrote down several seemingly very complicated formulas on paper, and then began to try to move towards cohomology.
But after a moment, he frowned again.
“How to prove that there is a kind of finite non-divergent Galois expansion L/Kp, whose ring is O` and the residual domain is k`, for which A`∈H1(E*o′,Z/2(1)) exists respectively?
"
"If this problem is not solved, there will be certain problems in the process of Galois expression..."
After thinking for a moment, he simply logged into his mailbox, attached his ideas to it, and sent it to Peter Schulz.
Of course he had Peter Schulz's contact information.
However, because he was using a multimedia computer and the projection was projected directly onto the blackboard screen, all the students present could see it.
When they saw Lin Xiao attaching his thoughts, the students present were all at a loss.
What the hell is this?
Apart from the p-adic they met at the beginning, they didn't know anyone else after that. Moreover, because Lin Xiao sent it to Peter Schulz, his email was also in English, which made the students present feel even more
Confused.
So is this what top math experts usually study?
However, this was not over yet. When they finally saw that Lin Xiao 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 connection? This is called a connection!
But 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 bell rang for the end of get out of class, and ten minutes later, the bell rang again, and Lin Xiao continued to lecture.
Soon, when the class was almost over, Lin Xiao left the students some time for self-study, and he continued to enter the mailbox. He was surprised to find that Peter Schulz responded so quickly.
When he opened the email, Peter Schulz directly sent an attachment. After he downloaded the attachment, he started reading it.
[Hello, Professor Lin! I'm very happy to receive your letter. I didn't expect that you would be interested in my original research. After reading your letter, I think you are currently studying the Hodge conjecture, right?
Regarding your question, how to prove this problem about Galois representation, I happened to do some research recently when I was studying Hodge theory.
First of all, note that A`∈H1(E*o′,Z/2(1)) can be set as the class of H1et(E,Z/2). Since it is reversible in the residual domain, this group will
The Z/2 parameterization of...
Br(S′)[2]→Br(S′Kp)[2]=Z/2. At this point, we need to continue to classify it as p into the field, and then use number theory methods to solve it. I believe that in
On this issue, no one has more knowledge than you, Professor Lin.
In fact, in the process of studying Hodge's theory, I also thought about Hodge's conjecture. I wonder if you have read the 2016 paper by Rosensson Andreas, where he faced the problem of how to obtain the correct integral of Hodge.
Qi conjecture, made a conjecture, I recommend you to take a look. In short, cohomology and Hodge conjecture are closely connected. Perhaps Motive is the most critical factor in solving the Hodge conjecture!
...]
After reading this reply, Peter Schulz basically had nothing to hide and gave Lin Xiao a great inspiration.
Lin Xiao has naturally read the paper recommended by Schultz.
But now, he has the confidence to truly solve Hodge's conjecture.
At least, it is an important stage result of Hodge's conjecture.
Thinking of this, he made a mouthful and then raised the corners of his mouth.
Perhaps, when I go to the International Congress of Mathematicians, I can change the topic of my report?
