Volume 1: Student Life Chapter 442 Maybe This Is Just a Coincidence
Chen Zhou returned to the dormitory, threw his backpack on the chair, and reached out to open a page of draft paper.
If senior sister Nott were here, she would definitely exclaim in surprise at what was written on the draft paper.
Because the content of this draft paper is about the research on "Linear representation of Artin's L function of the Galois group".
This is also the reason why Chen Zhou was a little hesitant when Professor Ating said he would assign him a sub-topic to research.
Compared with Professor Artin’s sub-topic, it would be more interesting to study “Linear representation of Artin’s L-function of the Galois group”.
"This senior sister Nott is really good at finding research topics..."
"Maybe this is just a coincidence?"
Chen Zhou picked up the draft paper, read it over and over again, then shook his head helplessly.
If the topics hadn't conflicted, Chen Zhou might have given it more thought.
But I was actually invited to study the topic that I was interested in.
Then Chen Zhou had no choice but to refuse.
It’s not that Chen Zhou thinks cooperation is bad, but he now prefers to conduct research independently.
Especially this kind of interesting topic.
Unless Yang Yiyi studies with him, Chen Zhou will not be used to other people.
As for this topic, it was probably taken up by Nott and her mentor.
Then Chen Zhou would not care. On the contrary, he would go and congratulate this senior sister Nott.
After all, when it comes to mathematical research, nothing is certain.
Gently putting down the draft paper, Chen Zhou took off his backpack and sat on the chair.
Then find a new piece of 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 Gochai's research and the jellyball experiment.
Perhaps Chen Zhou will raise its priority only after the Ge guess is solved.
As Nott said, the series of questions here are simply fascinating.
[For every univariate polynomial, we can define L functions, which are usually called Dedekind zeta functions...]
After writing this paragraph, Chen Zhou circled the Dedekind zeta function with a pen and habitually tapped a few times beside it.
Then, next to this circle, the Riemann zeta function was written.
The Riemann zeta function is a special case of a linear polynomial.
However, like the Riemann zeta function, the Dedekind zeta function can be proved to satisfy the first two conditions of this function using elementary proof methods.
Thinking of this, Chen Zhou's thoughts expanded.
A natural generalization of the Dedekind zeta function is to consider the case of multivariate polynomials.
And here, we enter the field of algebraic geometry.
The zeros of a multivariate polynomial define a geometric object, an algebraic variety.
The study of algebraic varieties is called algebraic geometry.
Speaking of which, although algebraic geometry is an ancient subject, it only 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.
However, its loose foundations prompted Oscar Zariski and Andrei Weil to reconstruct the entire foundation of algebraic geometry.
Weil further pointed out the surprising connection between algebraic geometry and number theory and topology.
Later, Grothendieck, known as the emperor of algebraic geometry, went a step further to understand Weil's conjecture, reconstructed the foundations of algebraic geometry with a more abstract and essential method, and introduced a series of powerful tools.
In particular, his cohomology theory eventually prompted his student, Professor Deligne, one of Chen Zhou's three reviewers, to completely prove the Weil conjecture.
And for this, he won the Fields Medal.
In fact, Grothendieck's cohomology theory is rooted in algebraic topology.
Moreover, Grothendieck also constructed a series of cohomology theories that have very similar properties.
But they originate from very different structures.
Grothendieck tried to find their common essence and thus proposed the Motive theory.
This theory is incomplete 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 the complete Motive theory will be obtained.
It leads to all cohomology and can prove a whole range of seemingly unrelated problems.
For example, the importance of the Hodge conjecture, one of the seven Millennium Problems, lies in the fact 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 extremely high.
If we really have to compare , from Chen Zhou's point of view, the difficulty of the standard conjecture is one level higher than that of Brother's conjecture.
Retracting his thoughts, Chen Zhou returned to the draft paper in front of him, picked up the pen, and began to write:
[Regarding Motivic L functions and automorphic L functions, each Motivic L function is given by Motivic.
For these functions, it is easy to verify that they satisfy the first condition of the Riemann zeta function, but the second condition cannot be proved in general.
A known example is the case of elliptic curves over rational numbers, which is a consequence of the proof of Fermat's Last Theorem (the Taniyama-Shimura conjecture).
Chen Zhou remembered reading in the literature that the complete case of the Taniyama-Shimura conjecture was proved by several students of Professor Wiles in 2001.
It has to be said that Professor Wiles' students all have a buff when facing the corollary of Fermat's Last Theorem.
Chen Zhou made a mark next to the Taniyama-Shimura conjecture and continued writing:
[For almost all L-functions, the third condition, namely the Riemann hypothesis, is unknown.
The only exception is the case where Motive is in a finite field, in which case the L function satisfies the conditions of the Riemann hypothesis, which is the Weil conjecture. 】
Chen Zhou also wrote the three words "Deligne" next to Weil's conjecture.
Although it seems that many of the problems here have been solved.
But in fact, the problems that have not been solved are the truly huge ones.
Regarding the special value of the Motivic L function, current general research believes that a generalization of Motive is needed.
This is a bigger and more distant dream.
Mathematicians call it a mixed motive.
Its existence enables the derivation of a series of extremely beautiful equations, generalizing Euler's formula for Riemann zeta.
The famous Beilinson conjecture and the BSD conjecture, one of the seven Millennium Problems, are all among those that can be deduced.
To some extent, mixed motive is comparable to or even better than the standard conjecture.
Because the current mathematical community does not know how to construct it.
Of course, although the current mathematical community cannot construct a mixed motive, it can construct a weakened variation of it, that is, the derived category.
Russian mathematician Vladimir Voevodsky won the 2002 Fields Medal for proposing such a construction.
Thinking of this, Chen Zhou was filled with anticipation. If he could solve the standard conjecture, he could then construct the mixed motive theory.
How many Fields Medals can I win?
I am afraid that I will become the first mathematician to win the award and become a billionaire?
But soon, Chen Zhou woke up.
It’s not time to go to bed yet, so I’d better not dream for now.
The most important thing is to be honest, down-to-earth, and do your 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 approach is to consider different expressions of Motive.
From the existing examples, class field theory has solved the case of commutative Galois groups.
That is, a simple but fundamental idea is that the representation of a group is more fundamental than the group itself.
Therefore, 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 his computer and began to look for literature.
Following this line of thought, we must consider their intrinsic symmetry.
Surprisingly, these symmetries arise largely from a completely different class of mathematical objects: automorphic forms.
The origin of automorphic forms can be traced back to the 19th century, and the mathematical god Poincare was a pioneer in this direction.
Chen Zhou quickly typed the content he wanted to find on the computer.
Then 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 unknowingly fell into the world of mathematics.
