Volume 1: Student Life Chapter 441: Nott's Mission
"A question about algebraic geometry?"
Chen Zhou smiled softly and said, "Then you should ask my mentor. You just said that he is a master in the field of algebraic geometry."
After saying that, Chen Zhou looked at his watch.
This senior sister Nott has already delayed him for more than ten minutes.
If she didn't tell him the purpose of their chance encounter later, Chen Zhou would leave immediately.
When Nott saw Chen Zhou looking at his watch, he naturally understood what Chen Zhou meant.
Without beating around the bush, Noether asked , "You know Artin's L function, right?"
Chen Zhou frowned slightly: "Ating L function?"
Nott nodded: "Yes, Arting L function."
"Of course I know that." Chen Zhou said puzzledly, "But if your question is related to Arting's L-function, then you should ask Professor Arting . I believe he knows more about his father's work."
Nott shook his head: "Professor Artin is not suitable for us, and he will not help us."
Chen Zhou was a little confused. He looked at Nott and said, "Professor Arting is not suitable for you. Am I suitable for you? If Professor Arting will not help you, then as Professor Arting's student, will I help you? And who are you referring to?"
Faced with Chen Zhou's series of questions, Nott did not feel it was impolite. Instead, a smile appeared on the corner of his mouth.
She said slowly, "Do you know the two major mathematical problems that Professor Artin's father, Professor Emil Artin, left to future generations?"
Chen Zhou was stunned for a moment, and then he said softly, "Linear representation of Artin's L function of the Galois group? And given a proof number a, find the frequency of a being a primitive root modulo different prime numbers p?"
"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 man was so excited.
Could it be that the senior sister Nott in front of me is really related to the Queen of Algebra?
But isn’t this what Professor Emil Artin left behind?
Chen Zhou couldn't figure out the answer.
However, Chen Zhou still agreed with what Nott said.
Especially the L function, which really occupies a very important position in modern mathematics.
It starts with Euler considering the function ζ(S)=∑n=1→∞n^(-S) and proving that its value at the point S=2 is 1＋1/2^2＋3^2＋…=π^2/6.
Later, Riemann proposed in his famous paper that this function satisfies three conditions.
One is that it has the expression ∑n=1→∞n^(-S)=p∏prime1/1-p^(-S).
One is that its value between 1-S and S has symmetry and satisfies a certain function equation.
The last one is that its trivial zeros are distributed on the straight line Re(S)=1/2.
The first two are easy to prove using elementary methods, and the third is the famous Riemann hypothesis.
Today, this function is often referred to as the Riemann zeta function.
It is also a special case of a certain type of function, which is called L-function.
L functions have properties similar to the above three conditions, and their values at special points have expressions similar to Euler's.
Don't think this vague statement looks like elementary algebra.
In fact, its meaning is extremely profound.
As for the reason...
It includes three of the seven million-dollar Millennium Problems proposed by the Clay Institute in the United States in the early 21st century - the Bayh and Swinnerton-Dyer conjectures, the Hodge conjecture, and the Riemann conjecture.
In addition, there are many other famous conjectures.
In a sense, behind this expression of L function, there is a series of extremely magnificent mathematical structures hidden.
Behind these structures, there are not only the meaning of the problems themselves, but also many powerful tools for solving them.
In addition, L-functions generally have two different origins, namely, Motivic L-functions and Automorphic L-functions.
Artin's L function is also included in this.
The Motivic L function originates from algebraic number theory and algebraic geometry.
As we all know, a core problem in algebraic number theory is to solve a univariate polynomial equation with integer coefficients.
For every prime number p, we can consider the case modulo p and obtain a univariate polynomial equation over a finite field.
In principle, it can be easily solved.
How to relate the solutions modulo p to integer solutions is an important problem in number theory.
The famous quadratic reciprocity law discovered by Gauss and Euler is the solution to this problem in the special case of quadratic polynomials.
Later, with the important discovery of class field theory in the early 20th century, this problem was solved for a larger class of univariate polynomial equations.
However, this type of equation is not limited by the degree of the polynomial, but depends on the intrinsic symmetry of the equation.
More precisely, it depends on its Galois group.
I have to say that the development of mathematics really depends on some great gods.
Not only Gauss, Euler and Riemann, Galois' revolutionary work in the early 19th century was the first introduction of group theory.
And use group theory to accurately measure the symmetries of polynomials.
As a result, for the first time, mathematicians were able to bypass tedious calculations and use deeper levels of abstract properties to deal with seemingly more concrete problems.
This also marked the beginning of modern algebra.
The complexity of a univariate polynomial lies in the complexity of the Galois group.
Class field theory deals with the case of commutative Galois groups.
As for the non-commutative case, it is much more complicated and has become an important goal of the modern Langlands program.
The Langlands Program was created by Professor Langlands, one of the three major reviewers of Chen Zhou's paper.
It can be said that, to a certain extent, L functions guided the development of modern algebra.
As a leading algebraist, the two difficult problems left by Professor Emil Artin can indeed be said to be two of the most crucial problems in the field of algebra.
But how much does this have to do with who I am now?
Chen Zhou said: "These are indeed two very important problems, but the solutions to these two problems are not so easy. If you are studying them, I wish you good luck."
Nott ignored Chen Zhou's words. She stared at Chen Zhou and said, "Don't you think solving such a difficult problem is very attractive?"
Chen Zhou frowned and looked at Nott. Was he trying to win him over?
Seeing that Chen Zhou didn't say anything, Noether continued: "We can even use this to solve a series of problems related to L functions! Including a series of problems including the Langlands Program!"
Chen Zhou grinned. This senior sister, I wonder if she hasn’t woken up yet?
Langlands program? BSD conjecture? Hodge conjecture? Riemann hypothesis?
This series of... questions?
Chen Zhou really wanted to ask her if she had ever solved a mathematical conjecture?
If not, he could tell her some experience.
Mathematical conjectures are not just mathematical fantasies that can easily solve a series of problems.
That is the wisdom of mathematicians and requires mathematical inspiration.
It is far from being as simple as just saying it.
"This..." Chen Zhou said hesitantly, "Just study it, don't count me."
Nott was stunned for a moment, then said, "Aren't you interested?"
Chen Zhou shook his head and said truthfully: "I am interested, but solving problems requires more than just interest."
After all, this series of questions really fascinated Chen Zhou.
To say I wasn't interested would be a lie.
I believe that no mathematician in the world is uninterested in the Riemann hypothesis, the BSD conjecture, and the Hodge conjecture.
After hearing what Chen Zhou said, Nott breathed a sigh of relief. This was the person he had his eyes on.
After a pause, Nott continued, "These two major problems were not only raised by Professor Emil Artin, nor were they just his research topics."
"These two major problems are also the research topics of Professor Emil Noether, Professor Richard Brauer and Professor Helmut Hasse."
"Especially Professor Emil Noether. As the queen of algebra, she had foresight in her research on these two problems!"
Nott's voice slowly changed from calm to excited again.
Especially when she talked about Emil Noether, the queen of algebra, her body seemed to tremble.
Chen Zhou noticed this and had his own answer in his mind.
It seems that my previous guess was correct.
The senior sister Noether in front of us has an extraordinary connection with the queen of algebra in the history of mathematics.
At the same time, Chen Zhou probably guessed Nott's intention in chatting with him for so long.
Sure enough, before Chen Zhou could ask, Nott calmed himself down: "Sorry, I lost my composure just now. You are probably wondering what the relationship is between me and Professor Emil Nott, right?"
"I'm indeed curious about the relationship between you two. As far as I know, Professor Emil Nott never married in his life?" Chen Zhou nodded, without hiding his thoughts.
Hearing this, Nott smiled slightly and explained, "Emile Nott is my great-grandmother."
Chen Zhou didn't react at first, but then he understood.
Professor Emil Noether has three younger brothers.
I guess, the senior sister Nott in front of me is the descendant of someone, right?
Chen Zhou didn't expect that his blind guess would actually be correct.
Is it so easy to meet a mathematical family in the United States?
Yes, my mentor, Professor Atin, and now the identity of this senior sister Nott has also been confirmed.
Chen Zhou thought for a moment and said, "So, this is why you want to study these issues?"
Nott nodded, her expression looking very heavy: "Since the death of my great-grandmother, although the Nott family has not produced another famous mathematician, no one in the Nott family has given up the glory in mathematics."
"My father told me since I was born that the children of the Noether family must regain the glory of mathematics in the past."
"So, our family's mission, or my mission, is to solve these mathematical problems."
"That's why I chose the field of algebra for research and study. My mentor, Professor Michel, and I have been trying to solve these difficult problems."
At this point, Noether's expression changed, and he said firmly: "I also believe that we can eventually solve these remaining mathematical problems, and I can also restore the mathematical glory of the Noether family!"
After listening to this, Chen Zhou looked at the exquisite girl in front of him and didn't know what to say.
At least, Chen Zhou still admires the courage to shoulder the family mission.
Putting aside other things, judging from the questions Noether asked himself last time, this girl’s mathematical talent is not bad.
Although I am not the best and cannot compare to myself, but with this strong inner strength, it is enough to achieve certain results in mathematics.
As for the difficult problem she mentioned, it is not just a matter of talent.
While Chen Zhou was thinking, Nott spoke again: "Student Chen Zhou, I solemnly invite you to join my and my advisor's research group and work with us to study these difficult problems. You don't have to reject me immediately. I hope you will seriously consider my invitation."
Nott's tone was sincere and his eyes were genuine.
The freshman dance was not the first time she met Chen Zhou. She had seen Chen Zhou's lecture before.
It was also from that lecture that Nott met Chen Zhou.
She was deeply impressed by the young student.
Hence, there was the consultation regarding the freshman dance issue.
It was a consultation on issues and also a test of strength.
After this period of time, Nott heard the professors' evaluation of Chen Zhou.
She finally made up her mind and invited Chen Zhou.
This is how we ended up with today's scene.
Chen Zhou asked in confusion: "Why me? I still think Professor Ating can help you better, right?"
Nott 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?”
Nott was silent for a while before he said, "Because of the relationship between Professor Emil Artin and Professor Emil Nott."
Chen Zhou was stunned for a moment, then said, "Sorry, I didn't mean to pry into your privacy."
Nott chuckled and said, "Besides, Professor Artin is an old scholar. You are definitely more interesting than him. Also, as people of the same age, it will definitely be easier for us to communicate."
Chen Zhou also smiled and said: "What's so interesting about academic exchanges? On the contrary, I think these mathematics professors are sometimes quite cute."
Nott was slightly stunned and looked up at Chen Zhou. Does this guy really not understand, or...
Chen Zhou looked at his watch again, then turned to Nott and said, "I probably can't accept your invitation. However, if you encounter any problems, need communication, or have suggestions, you can send me an email."
After saying that, Chen Zhou left Nott with a dazed look on his face and went straight back to his dormitory.
Looking at Chen Zhou's back, it took Nott a while to come back to his senses.
Although Chen Zhou rejected her, she had no intention of giving up.
Just like her, even though she knew that her talent in mathematics might not be outstanding.
But I still resolutely decided to take the path of mathematics.
Chen Zhou is someone she is very optimistic about.
Her intuition told her that Chen Zhou's mathematical talent was extremely terrifying!
