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Chapter 442 Maybe This Is A Coincidence (Updated)
Chapter 442 Maybe This Is A Coincidence (Updated)
Chen Zhou returned to the dormitory, put his backpack on the chair, reached out and opened a page of draft paper.
What was written on the draft paper, if that Senior Sister Nott was there, she would have cried out in surprise.
Because, this is also the content of the draft paper, which is the research content of "the linear representation of the Artin L function of the Galois group".
This is also the reason why Chen Zhou hesitated when Professor A Ting said that he would assign him a sub-topic for research.
Compared with Professor Artin's sub-project, it will be more interesting to study "Linear representation of Artin's L function of Galois group".
"This Nott-senpai is really good at finding topics..."
"Maybe it's just a coincidence?"
Chen Zhou picked up the draft paper, read it back and forth, and shook his head helplessly.
If the subject hadn't crashed, Chen Zhou might have thought about it more.
But the topic I am interested in is actually invited to study together.
Then Chen Zhou had no choice but to refuse.
It's not that Chen Zhou thinks cooperation is not good, it's just that he prefers to conduct research independently now.
Especially this kind of interesting subject.
Unless Yang Yiyi is studying with him, Chen Zhou will not be used to other people.
As for this topic, if Nott and her mentor have taken the lead.
Then Chen Zhou wouldn't care, on the contrary, he would congratulate this Senior Nott.
After all, when it comes to mathematical research, nothing is certain.
Gently put down the draft paper, Chen Zhou took his backpack away and sat on a chair.
Then find a new draft paper, pick up a pen, and start sorting out the research content involved in this topic.
Of course, the priority of this topic is far lower than Gechai's research and the glueball experiment topic.
Maybe Chen Zhou will raise its priority after Brother Guess is resolved.
As Knott said, the series of questions here are simply too fascinating.
[For each univariate polynomial, we can define L functions, which are usually called Dedekind ζ functions...]
After writing this paragraph, Chen Zhou drew a circle on the Dedekind ζ function with a pen, and habitually took a pen to click on the side.
Then, next to this circle, write down the Riemann zeta function.
The Riemann zeta function is a special case of polynomials of degree [-] in one variable.
However, the Dedekin ζ function, like the Riemann ζ function, can be proved to satisfy the first two conditions of this function by the method of elementary proof.
Thinking of this, Chen Zhou's thoughts spread.
A natural generalization of the Dedekind zeta function considers the case of multivariate polynomials.
And here we enter the realm of algebraic geometry.
The zero points of multivariate polynomials define a geometric object, that is, an algebraic variety.
The study of algebraic varieties is called algebraic geometry.
Speaking of it, although algebraic geometry is an ancient subject, it also experienced a spectacular development in the 20th century.
In the early 20th century, the Italian school made great progress in the study of algebraic surfaces.
Its imprecise foundation, however, prompted Oskar Zariski and André Weil to restructure the foundations of algebraic geometry as a whole.
Weil even pointed out the surprising connection between algebraic geometry and number theory and topology.
After that, Grothendieck, known as the emperor of algebraic geometry, went further to rebuild the foundation of algebraic geometry with a more abstract and essential method in order to understand Weil's conjecture, and introduced a series of powerful tools.
In particular, his cohomology theory finally prompted his student, Professor Deligne, one of Chen Zhou's three reviewers, to completely prove Weil's conjecture.
And for this, won the Fields Medal.
In fact, Grothendieck's cohomology theory is rooted in algebraic topology.
Moreover, Grothendieck constructed a series of cohomology theories at the same time, which have very similar properties.
But originate from a very different structure.
Grothendieck tried to find out their common essence, and thus proposed the theory of Motive.
This theory is not complete because it is based on a series of conjectures.
Motive theory is also called the standard conjecture by Grothendieck.
If the standard conjecture is proved, then a complete Motive theory is obtained.
It derives all cohomology and at the same time can prove a series of seemingly unrelated problems.
For example, the importance of Hodge's conjecture, one of the seven millennium puzzles, is that it can lead to the standard conjecture.
It has to be said that the proof of the standard conjecture is probably the most important thing in algebraic geometry.
However, the difficulty of proving the standard conjecture is top-notch.
If you really want to compare, from Chen Zhou's point of view, the difficulty of the standard guess is one level higher than that of brother guess.
Withdrawing his thoughts, Chen Zhou returned to the draft paper in front of him, picked up a pen, and began to write:
[About Motivic L function and automorphic L function, every Motivic L function is given by Motivic.
For these functions, it is easy to verify that it satisfies the first condition of the Riemann zeta function, but the second condition cannot prove the general situation.
A known example is the case of elliptic curves on rational numbers, which is a corollary of the proof of Fermat's Last Theorem (Taniyama-Shimura conjecture). 】
Chen Zhou remembers seeing in the literature that the complete situation of the Taniyama-Shimura conjecture was proved by several students of Professor Wiles in 2001.
It has to be said that Professor Wiles' students have buff bonuses when facing the deduction of Fermat's last theorem.
Chen Zhou made a mark next to the Gushan-Shimura conjecture, and continued to write:
[For almost all L functions, the third condition, the Riemann hypothesis, is unknown.
The only exception is the case of Motive in the finite field. At this time, the L function satisfies the conditions of the Riemann hypothesis, which is the Weil conjecture. 】
Next to Weil's conjecture, Chen Zhou wrote down the word "Deligne".
Although it seems that the problems here have been solved a lot.
But in fact, the unresolved problems are the real big ones.
For the problem of the special value of the Motivic L function, it is generally believed that a generalization of Motive is needed.
This is a bigger and more distant dream.
Mathematicians call it a mixed motive.
Its existence can derive a series of extremely beautiful equations, and promote Euler's formula for Riemann ζ.
The famous Bellingson conjecture, the BSD conjecture, one of the seven millennium problems, etc., all belong to the list that can be derived.
To some extent, mixed motive is comparable to, or even surpasses, the standard conjecture.
Because the current mathematics community does not know how to construct it.
Of course, although the current mathematics world cannot construct a mixed motive, it can construct a weakened variant of it, that is, a derived category.
Russian mathematician Vladimir Voevodsky won the Fields Medal in 2002 because of such a structure.
Thinking of this, Chen Zhou's inner longing is extremely longing. If the standard conjecture is resolved, then the mixed motive theory can be constructed.
How many Fields Medals can I win?
I'm afraid I will become the first mathematician to win a prize and become a billionaire?
But soon, Chen Zhou woke up.
I didn't go to bed at night, so let's stop dreaming.
The most important thing is to be honest, down-to-earth, and do your own research step by step.
Chen Zhou stopped thinking about it, and continued to sort out the research content involved in this topic on the draft paper.
[Each Motive can give a series of representations of Galois groups and Hodge structures in complex geometry, which completely determine the L function, so considering them is a more fundamental problem...]
In fact, Motive is more essential than the L function, but it is difficult to calculate it directly.
An alternative is to consider different expressions of Motive.
Judging from the existing examples, the class field theory has solved the situation of exchanging Galois groups.
That is, a simple, but fundamental idea, is that representations of groups are more fundamental than groups themselves.
So what needs to be considered is not the Galois group itself, but its representation.
In this way, all commutative Galois groups are equivalent to one-dimensional Galois representations, and non-commutative ones are equivalent to high-dimensional representations.
Thinking of this, Chen Zhou frowned slightly. He turned on the computer and began to search for documents.
If you look at it in this way, you must consider their internal symmetry.
Surprisingly, these symmetries stem largely from a completely different class of mathematical objects, namely automorphic forms.
The origin of automorphic forms can be traced back to the 19th century, and the mathematician Poincaré was the pioneer in this direction.
Chen Zhou quickly entered the content he wanted to find on the computer.
Download the documents one by one.
Chen Zhou originally planned to come back and stay for a while before going to eat.
In this way, I fell into the world of mathematics unconsciously.
(End of this chapter)
Chen Zhou returned to the dormitory, put his backpack on the chair, reached out and opened a page of draft paper.
What was written on the draft paper, if that Senior Sister Nott was there, she would have cried out in surprise.
Because, this is also the content of the draft paper, which is the research content of "the linear representation of the Artin L function of the Galois group".
This is also the reason why Chen Zhou hesitated when Professor A Ting said that he would assign him a sub-topic for research.
Compared with Professor Artin's sub-project, it will be more interesting to study "Linear representation of Artin's L function of Galois group".
"This Nott-senpai is really good at finding topics..."
"Maybe it's just a coincidence?"
Chen Zhou picked up the draft paper, read it back and forth, and shook his head helplessly.
If the subject hadn't crashed, Chen Zhou might have thought about it more.
But the topic I am interested in is actually invited to study together.
Then Chen Zhou had no choice but to refuse.
It's not that Chen Zhou thinks cooperation is not good, it's just that he prefers to conduct research independently now.
Especially this kind of interesting subject.
Unless Yang Yiyi is studying with him, Chen Zhou will not be used to other people.
As for this topic, if Nott and her mentor have taken the lead.
Then Chen Zhou wouldn't care, on the contrary, he would congratulate this Senior Nott.
After all, when it comes to mathematical research, nothing is certain.
Gently put down the draft paper, Chen Zhou took his backpack away and sat on a chair.
Then find a new draft paper, pick up a pen, and start sorting out the research content involved in this topic.
Of course, the priority of this topic is far lower than Gechai's research and the glueball experiment topic.
Maybe Chen Zhou will raise its priority after Brother Guess is resolved.
As Knott said, the series of questions here are simply too fascinating.
[For each univariate polynomial, we can define L functions, which are usually called Dedekind ζ functions...]
After writing this paragraph, Chen Zhou drew a circle on the Dedekind ζ function with a pen, and habitually took a pen to click on the side.
Then, next to this circle, write down the Riemann zeta function.
The Riemann zeta function is a special case of polynomials of degree [-] in one variable.
However, the Dedekin ζ function, like the Riemann ζ function, can be proved to satisfy the first two conditions of this function by the method of elementary proof.
Thinking of this, Chen Zhou's thoughts spread.
A natural generalization of the Dedekind zeta function considers the case of multivariate polynomials.
And here we enter the realm of algebraic geometry.
The zero points of multivariate polynomials define a geometric object, that is, an algebraic variety.
The study of algebraic varieties is called algebraic geometry.
Speaking of it, although algebraic geometry is an ancient subject, it also experienced a spectacular development in the 20th century.
In the early 20th century, the Italian school made great progress in the study of algebraic surfaces.
Its imprecise foundation, however, prompted Oskar Zariski and André Weil to restructure the foundations of algebraic geometry as a whole.
Weil even pointed out the surprising connection between algebraic geometry and number theory and topology.
After that, Grothendieck, known as the emperor of algebraic geometry, went further to rebuild the foundation of algebraic geometry with a more abstract and essential method in order to understand Weil's conjecture, and introduced a series of powerful tools.
In particular, his cohomology theory finally prompted his student, Professor Deligne, one of Chen Zhou's three reviewers, to completely prove Weil's conjecture.
And for this, won the Fields Medal.
In fact, Grothendieck's cohomology theory is rooted in algebraic topology.
Moreover, Grothendieck constructed a series of cohomology theories at the same time, which have very similar properties.
But originate from a very different structure.
Grothendieck tried to find out their common essence, and thus proposed the theory of Motive.
This theory is not complete because it is based on a series of conjectures.
Motive theory is also called the standard conjecture by Grothendieck.
If the standard conjecture is proved, then a complete Motive theory is obtained.
It derives all cohomology and at the same time can prove a series of seemingly unrelated problems.
For example, the importance of Hodge's conjecture, one of the seven millennium puzzles, is that it can lead to the standard conjecture.
It has to be said that the proof of the standard conjecture is probably the most important thing in algebraic geometry.
However, the difficulty of proving the standard conjecture is top-notch.
If you really want to compare, from Chen Zhou's point of view, the difficulty of the standard guess is one level higher than that of brother guess.
Withdrawing his thoughts, Chen Zhou returned to the draft paper in front of him, picked up a pen, and began to write:
[About Motivic L function and automorphic L function, every Motivic L function is given by Motivic.
For these functions, it is easy to verify that it satisfies the first condition of the Riemann zeta function, but the second condition cannot prove the general situation.
A known example is the case of elliptic curves on rational numbers, which is a corollary of the proof of Fermat's Last Theorem (Taniyama-Shimura conjecture). 】
Chen Zhou remembers seeing in the literature that the complete situation of the Taniyama-Shimura conjecture was proved by several students of Professor Wiles in 2001.
It has to be said that Professor Wiles' students have buff bonuses when facing the deduction of Fermat's last theorem.
Chen Zhou made a mark next to the Gushan-Shimura conjecture, and continued to write:
[For almost all L functions, the third condition, the Riemann hypothesis, is unknown.
The only exception is the case of Motive in the finite field. At this time, the L function satisfies the conditions of the Riemann hypothesis, which is the Weil conjecture. 】
Next to Weil's conjecture, Chen Zhou wrote down the word "Deligne".
Although it seems that the problems here have been solved a lot.
But in fact, the unresolved problems are the real big ones.
For the problem of the special value of the Motivic L function, it is generally believed that a generalization of Motive is needed.
This is a bigger and more distant dream.
Mathematicians call it a mixed motive.
Its existence can derive a series of extremely beautiful equations, and promote Euler's formula for Riemann ζ.
The famous Bellingson conjecture, the BSD conjecture, one of the seven millennium problems, etc., all belong to the list that can be derived.
To some extent, mixed motive is comparable to, or even surpasses, the standard conjecture.
Because the current mathematics community does not know how to construct it.
Of course, although the current mathematics world cannot construct a mixed motive, it can construct a weakened variant of it, that is, a derived category.
Russian mathematician Vladimir Voevodsky won the Fields Medal in 2002 because of such a structure.
Thinking of this, Chen Zhou's inner longing is extremely longing. If the standard conjecture is resolved, then the mixed motive theory can be constructed.
How many Fields Medals can I win?
I'm afraid I will become the first mathematician to win a prize and become a billionaire?
But soon, Chen Zhou woke up.
I didn't go to bed at night, so let's stop dreaming.
The most important thing is to be honest, down-to-earth, and do your own research step by step.
Chen Zhou stopped thinking about it, and continued to sort out the research content involved in this topic on the draft paper.
[Each Motive can give a series of representations of Galois groups and Hodge structures in complex geometry, which completely determine the L function, so considering them is a more fundamental problem...]
In fact, Motive is more essential than the L function, but it is difficult to calculate it directly.
An alternative is to consider different expressions of Motive.
Judging from the existing examples, the class field theory has solved the situation of exchanging Galois groups.
That is, a simple, but fundamental idea, is that representations of groups are more fundamental than groups themselves.
So what needs to be considered is not the Galois group itself, but its representation.
In this way, all commutative Galois groups are equivalent to one-dimensional Galois representations, and non-commutative ones are equivalent to high-dimensional representations.
Thinking of this, Chen Zhou frowned slightly. He turned on the computer and began to search for documents.
If you look at it in this way, you must consider their internal symmetry.
Surprisingly, these symmetries stem largely from a completely different class of mathematical objects, namely automorphic forms.
The origin of automorphic forms can be traced back to the 19th century, and the mathematician Poincaré was the pioneer in this direction.
Chen Zhou quickly entered the content he wanted to find on the computer.
Download the documents one by one.
Chen Zhou originally planned to come back and stay for a while before going to eat.
In this way, I fell into the world of mathematics unconsciously.
(End of this chapter)
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