"Problems with algebraic geometry?"

Chen Zhou smiled softly and said, "Then you should ask my mentor, you just said, he is a master in the field of algebraic geometry." ”

After speaking, Chen Zhou looked at his watch.

This Senior Sister Noether has already delayed him for more than ten minutes.

If she doesn't say the purpose of the coincidence later, Chen Zhou plans to leave immediately.

When Notte saw Chen Zhou's action of looking at his watch, he naturally understood what Chen Zhou meant.

No longer rounding around, Knott said, "You know the Atin L-function, right?"

Chen Zhou frowned slightly: "Atin L function?"

Nutt nodded: "Yes, Artin's L-function." ”

"Of course I do. Chen Zhou said puzzledly, "But if your question is related to Artin's L-function, then you should ask Professor Artin, I believe he knows his father's work better." ”

Knott shook his head: "Professor Atin is not suitable for us, and he will not help us. ”

Chen Zhou was a little confused at this moment, he looked at Nott and said: "Professor Atin is not suitable for you, am I suitable for you? If Professor Atin will not help you, can I, as a student of Professor Atin, help you? Also, do you mean?"

Faced with Chen Zhou's series of questions, Noether did not feel impolite, but a smile appeared on the corner of his mouth.

She said slowly, "Do you know the two major mathematical problems that Professor Atin's father, Professor Emile Atin, left to future generations?"

Chen Zhou was stunned for a moment, and said softly: "What is the linear representation of the Artin L function of the Galois group? and given the number a, find a is the frequency of the original root of the p mode of different prime numbers?"

"That's right!" Hearing Chen Zhou's words, Noether's expression became excited, "These two major mathematical problems are not only the mathematical problems left by Professor Emil Artin to future generations, but also the two most important problems in the field of algebra!"

Chen Zhou glanced at Nott, but he didn't quite understand why this person was so excited.

Could it be that the senior sister in front of her really has something to do with the algebra queen?

But isn't this what Professor Emile Atin left behind?

Chen Zhou couldn't see the answer.

However, Chen Zhou still agrees with what Knott said.

In particular, the L-function does occupy a very important position in modern mathematics.

Starting from Euler's consideration of the function ζ(S)=∑n=1→∞n^(-S) and proving its value at S=2 1+1/2^2+3^2+......=π^2/6.

Later, in his famous paper, Riemann proposed that this function satisfies three conditions.

One is that it has an expression ∑n=1→∞n^(-S)=p∏prime1/1-p^(-S).

One is its value at 1-S and S, which is symmetrical and satisfies a certain functional equation.

The last one is that its trivial zero points are distributed on the line Re(S)=1/2.

The first two are easy to prove by elementary methods, while the third is the famous Riemann hypothesis.

Nowadays, this function is often referred to as the Riemann ζ function.

It is also a special case of a certain class of functions, which is called an L-function.

The L function has properties similar to those of the three conditions mentioned above, and at the same time their values at special points have Euler-like expressions.

Don't think this vague statement looks like elementary algebra.

In fact, its meaning is profound.

As for the reason......

It contains three of the millennial puzzle of the seven-million-dollar prize proposed by the Clay Institute in the early 21st century – the Behe and Swinaton-Dyer conjectures, the Hodge conjecture and the Riemann conjecture.

In addition to this, there are many other famous conjectures.

In a sense, behind this formulation of the L-function lies a series of incomparably magnificent mathematical structures.

Behind these structures, there is not only the meaning of the problem itself, but also many powerful tools for solving it.

In addition, there are generally two types of L-functions of different origins, namely the Motivic L-function and the self-defending L-function.

The Atin L function is also included in this.

The Motivic L-function, on the other hand, has its origins in algebraic number theory and algebraic geometry.

As we all know, one of the core problems of algebraic number theory is solving the one-element polynomial equation with integer coefficients.

For each prime p, the case of modulo p can be considered, and a univariate polynomial equation over a finite field can be obtained.

In principle, it can be easily solved.

How to relate the solution of modulo p to the integer solution is an important issue in number theory.

Gauss and Euler's famous quadratic reciprocal law is the solution to this problem in the special case of a quadratic polynomial.

Later, with the important discovery of class domain theory in the early 20th century, this problem was solved for univariate polynomial equations of a larger class.

But this type of equation is not limited by the number of polynomials, but by the implicit symmetry of the equation.

More precisely, depending on its Galois group.

I have to say that the development of mathematics really depends on some great gods.

Not only Gauss, Eula Riemann, but Galois's revolutionary work in the early 19th century was the first to introduce group theory.

Group theory is used to accurately measure the symmetry of polynomials.

Thus, for the first time, mathematicians were able to bypass tedious calculations and use deeper abstract properties to deal with more concrete problems on the surface.

It also marked the beginning of modern algebra.

The complexity of the unary polynomial lies in the complexity of the Galois group.

Analogous theory, on the other hand, deals with the case of commuting the Galois group.

As for the non-exchange situation, it is much more complex, and it has become an important goal of the modern Langlands Program.

The Langlands program is one of the three major reviewers of Chen Zhou's paper, Professor Langlands.

It can be said that, to a certain extent, the L function has guided the development of modern algebra.

As a leading algebraist, Professor Emil Artin has left behind two difficult problems that can indeed be said to be two of the most important problems in the field of algebra.

However, how much does this have to do with my current self?

Chen Zhou said: "It is indeed two very important problems, but the solution of these two problems is not so easy. Good luck if you're researching them. ”

Nott ignored Chen Zhou's words, she stared at Chen Zhou and said, "Don't you think it's very attractive to solve such a problem?"

Chen Zhou frowned and looked at Nott, is this to win himself over?

Seeing that Chen Zhou did not speak, Noether continued: "Even, we can solve a series of problems of the L function based on this!

Chen Zhou grinned, this senior sister, I'm afraid I didn't wake up, right?

The Langlands Program?BSD Conjecture?Hodge Conjecture?Riemann Conjecture?

This series of ...... Issue?

Chen Zhou really wanted to ask her, did she solve mathematical conjectures?

If not, he can tell her some experience.

Mathematical conjecture is really not mathematical fantasy, it is a series of problems that can be solved casually.

It is the brainchild of mathematicians, and it needs mathematical inspiration.

It's far from being as simple as it gets.

"This ......," Chen Zhou said hesitantly, "It's good for you to study, don't count me." ”

Noether was stunned for a moment, and then said, "Aren't you interested?"

Chen Zhou shook his head and said truthfully: "Interest is interest, but solving problems is not just about interest." ”

After all, this series of questions really fascinated Chen Zhou.

It would be too false to say that I am not interested.

I believe that no mathematician in the world will be interested in the Riemann conjecture, the BSD conjecture, or the Hodge conjecture.

Hearing Chen Zhou's words, Noether breathed a sigh of relief silently, this is the person he fancies.

After a pause, Noether said again: "These two major problems are not only proposed by Professor Emil Artin alone, nor are they only his research topics. ”

These two difficult questions are also the subject of research by Professors Emile Noether, Richard Bruul and Helmut Hassel. ”

"Especially Professor Emil Nott, as the queen of algebra, she has long been foresighted in the study of these two issues!"

Noether's voice slowly became excited again.

Especially when it comes to Emile Knott, the queen of algebra, her body seems to be trembling.

Chen Zhou, who noticed this, also had his own answer in his heart.

It seems that his previous guess was correct.

The Noether sister in front of her has an extraordinary connection with the algebra queen in the history of mathematics.

At the same time, Chen Zhou probably guessed Nott's intention to talk to him for a long time.

Sure enough, without waiting for Chen Zhou to ask, Noether calmed down himself: "I'm sorry, I was a little out of shape just now." You're probably thinking, what is my relationship with Professor Emil Knott?"

"I do wonder what the relationship between you is, as far as I know, Professor Emile Notte is never married?" Chen Zhou nodded, but he didn't hide his thoughts.

Hearing this, Noether smiled slightly and explained, "Emile Notte is my great-grandmother. ”

Chen Zhou didn't react at first, but then he understood.

Professor Emil Knott also has three younger brothers.

Presumably, the senior sister in front of her is someone's descendant, right?

Chen Zhou didn't expect that his blind guess when they met for the first time was really right.

Could it be that in the United States, it is so easy to meet a family of mathematics?

His own mentor, Professor Artin, yes, and now the identity of this senior sister has also been confirmed.

Chen Zhou thought for a moment and said, "So, is this the reason why you want to study these problems?"

Nott nodded, her expression was very heavy: "Since the death of my great-grandmother, although the Nott family has not produced a famous enough mathematician, the Nott family has never given up the glory of mathematics. ”

"From the time I was born, my father told me that the children of the Noether family had to regain their former glory in mathematics. ”

"So, our family's task, or my task, is to solve these mathematically left problems. ”

"That's why I chose the field of algebra to study and study. My supervisor, Professor Michelle, and I have been trying to solve these problems. ”

Speaking of this, Noether's expression changed for a moment, and he said in a firm tone: "I also believe that we can finally solve these remaining mathematical problems, and I can also restore the Noether family to its former glory in mathematics!"

After listening to this, Chen Zhou looked at the delicate girl in front of him, and didn't know what to say.

At least, Chen Zhou still admires the courage to take on the family mission.

Anything else, judging by the question Noether asked herself last time, the girl's mathematical talent was not bad.

Although it is not the top and not as good as itself, if you have this strong heart, it is enough to achieve certain results in mathematics.

As for the problem in her mouth, it is not just a matter of talent.

While Chen Zhou was thinking, Noether spoke again: "Chen Zhou, I solemnly invite you to join me and my supervisor's research group, and work with us to study these problems. You don't have to turn me down right away, I want you to seriously consider my invitation. ”

Nott's tone was sincere, and his eyes were sincere.

The freshman ball was not the first time she had seen Chen Zhou, she had seen Chen Zhou's report before.

It was also from that report meeting that Knott met Chen Zhou.

This young student made a great impression on her.

That's why I had a question about the freshman prom.

It was a question consultation, and it was also a test of strength.

After this period of time, Knott heard the professors' comments on Chen Zhou.

She finally made up her mind and invited Chen Zhou.

That's where we are today.

Chen Zhou asked a little puzzled: "Why me?" I still think Professor Atin can help you more, right?"

Noether shook his head again and said the same thing: "Professor Artin is not suitable for us, and he will not help us. ”

Chen Zhou: "Why?"

Knott was silent for a while before he spoke: "Because of the relationship between Professor Emile Artin and Professor Emile Nott." ”

Chen Zhou was stunned for a moment, and then said, "I'm sorry, I didn't mean to listen to privacy." ”

Knott chuckled softly, "Besides, Professor Artin is an old pedant, and you must be much more interesting than him." Moreover, as peers, it is definitely more convenient for us to communicate. ”

Chen Zhou also said with a smile: "Academic exchanges, what's interesting or not, I think these mathematics professors are sometimes quite cute." ”

Nott was slightly stunned, and looked up at Chen Zhou, is this person really not understanding, or?

Chen Zhou looked at his watch again, then turned to Nott and said, "I probably can't agree to your invitation." However, if you have any questions, need to communicate, or suggest, you can send me an email. ”

After speaking, leaving a sluggish Nott, Chen Zhou went straight back to his dormitory.

Looking at Chen Zhou's back, it took a while for Nott to come back to his senses.

Although Chen Zhou rejected her, she didn't plan to give up just like that.

Like her, even if she knows that she is gifted in mathematics, she may not be outstanding.

But he still resolutely embarked on the road of mathematics.

Chen Zhou is a person she is very optimistic about.

Her intuition told her that Chen Zhou's mathematical talent was extremely terrifying!