Volume 1: Student Life Chapter 454: Mathematicians' Feelings
Chen Zhou greeted Liu Maosheng and Zeng Zigu, then followed Deligne to a nearby cafe.
In the cafe.
When the waiter asked what he needed, Chen Zhou just ordered a glass of boiled water.
To be honest, Chen Zhou doesn’t like drinking coffee very much. He really can’t get used to it.
In comparison, boiled water is the real popular drink.
Looking at the young man opposite, Deligne asked softly: "Do you know why I am looking for you?"
Chen Zhou thought for a moment and said uncertainly, "On the topic of the standard conjecture?"
Deligne nodded: "It's related to the teacher's topic."
After getting a positive answer, Chen Zhou became even more confused.
It seems that I can't help him solve the standard conjecture?
Moreover, I have no experience in researching standard conjectures, so I can't provide much help.
Could it be that Professor Deligne thinks highly of me and thinks that he can definitely handle the topic he is researching, so he is here to persuade me to study the standard conjecture?
I'm afraid you're making fun of yourself, right?
Chen Zhou asked himself, he is definitely no match for the old man in front of him.
The standard conjecture is a difficult problem not only now but also in the future.
At least for now, he is far from being able to solve this problem.
Deligne had been observing Chen Zhou. After noticing Chen Zhou's puzzled expression, he did not explain immediately.
Instead, he picked up the coffee, took a sip, and then told a story slowly.
"I remember that it was in 1959 when Dwork used the p-adic method to prove the first Weil conjecture, that is, the zeta function of a finite body solution is a rational function."
"Later, in 1964, my teacher Grothendieck made a further proof of the Weil conjecture, which was more general, and introduced his 'formal system of six operations'."
"Later, the teacher proved the second Weil conjecture, which is that the zeta function of the solution shape satisfies a certain functional equation."
"It was also during that period in the 1960s that finding a way to prove the final Weil conjecture became the main source of inspiration for the teacher's research and also became his obsession."
"The 'standard conjecture' was constructed and proposed during this period. The teacher always believed that if these standard conjectures could be proved, the entire Weil conjecture could be proved. And he was deeply involved in this..."
Listening to Deligne's story, the puzzled expression on Chen Zhou's face gradually disappeared.
He vaguely guessed the reason for Deligne's unique opening remarks in the lecture.
As Deligne said, the standard conjecture was Grothendieck's last regret.
It was also Deligne's regret that he proved the Weil conjecture by bypassing the standard conjecture.
And Deligne was telling Chen Zhou the reason for this regret.
"Because I was deeply influenced by my teacher, when I started to study the Weil conjecture, I also focused on the standard conjecture."
"It was not until later that I saw Professor Lan Jin's paper, which discussed the theory of classical modular forms, which the teacher did not know."
"And this key concept successfully helped me bypass the standard conjecture and prove Weil's final conjecture."
“That’s why some mathematicians later said that in order to prove the final Weil conjecture, more classical materials were needed, and this was exactly Grothendieck’s blind spot.”
"But in fact, this is the biggest misunderstanding about teachers!"
There was a strange emotion in Deligne's voice, so much so that it began to tremble slightly.
At this moment, Chen Zhou was obviously a qualified listener.
At the same time, he was trying to understand Deligne's emotions.
Deligne's words also made Chen Zhou confirm his guess.
In today's lecture, Deligne made such an opening statement indeed because of Grothendieck.
It was also two years after Grothendieck's death that Deligne used such an occasion to vent certain emotions.
But what Chen Zhou didn’t understand was why it was two years later?
Also, why is the other party saying these words to me?
Deligne paused for a moment, calmed down a little, and then continued: "In the teacher's mind, the Weil conjecture is important because it is the tip of the iceberg, reflecting the basic mathematical structure he wants to discover and develop."
"If we can prove the standard conjecture, we can reveal this structure more deeply. In other words, the Weil conjecture should serve the standard conjecture."
"In the process of proving the Weil conjecture, we should not bypass the standard conjecture!"
"If I use Motive Theory, the professor will definitely be very interested because it means that the standard conjecture has a new development."
"But I have overlooked this tip of the iceberg and completely deviated from the standard conjecture..."
At this point, Deligne sighed softly: "That's why the teacher is not very interested in my flattering proof."
Chen Zhou didn't know how to comfort this master of mathematics.
But Chen Zhou heard the cause and effect clearly.
Chen Zhou felt that Deligne’s lecture today was probably the first step to make up for Grothendieck’s regrets and his own regrets.
After a moment's silence, Deligne picked up the coffee and took another sip.
Looking at the young man opposite him, Deline suddenly said, "Do you have any thoughts on today's lecture?"
Chen Zhou thought for a moment and said, "Let's talk about Milnor's conjecture. I think these new research results should be able to promote the solution of Milnor's conjecture and provide the key to solving many propositions."
The Milnor conjecture is a proposition that connects cohomology theory with algebraic K-theory. It also involves the theory of quadratic forms and is one of the most important problems in algebraic K-theory in the past two or three decades.
More importantly, if the Milnor conjecture is solved.
Then, this breakthrough in mathematical theory will be an extremely critical first step in solving the Beilinson-Lichdenbaum conjecture and the Bloch-Kato conjecture.
These mathematical conjectures are the basic "results" that have been obtained by Beilinson, Bloch, Lichtenbaum and others over the years when they studied standard conjectures and are valid for special algebraic varieties.
This result is put in double quotes because these basic propositions are just guesses.
But now, Deligne's research seems to make it possible for these conjectures to become theorems.
It also makes it possible for Grothendieck's long-cherished standard conjecture to develop further.
Although Chen Zhou's answer was not so clear.
If an ordinary person were to listen, they might not have any idea what Chen Zhou was talking about.
However, when Deligne heard Chen Zhou's answer, his eyes suddenly lit up.
The look in his eyes when he looked at Chen Zhou also became a little subtle.
"I originally thought that your research field has always been limited to analytic number theory, and your understanding of algebraic geometry is not deep enough. But your answer exceeded my expectations."
Deligne praised Chen Zhou without reservation.
Chen Zhou explained: "Probably no mathematician would not be interested in a difficult problem like the standard conjecture. Moreover, my advisor is Professor Artin of algebraic geometry."
Deligne smiled and nodded: "I almost forgot, your graduate advisor is Michael Artin."
After a pause, Deligne asked Chen Zhou: "Since you also said that you are interested in the standard conjecture, have you ever thought about turning to study this topic?"
Chen Zhou was stunned by the question. Is this the ultimate purpose of finding himself?
But before Chen Zhou could answer, Deligne said, "I am very optimistic about your research in this area, and I also believe that you can achieve the same results as in analytic number theory."
"Also, I wonder if you are interested in my research group and would like to come and study with me?"
Chen Zhou looked at Deligne with a strange expression. What exactly did this old man mean?
He first told me a story, then asked me what I thought about listening to the lecture, and now he directly invited me to join his research group.
Is this the legendary technique of throwing a brick to attract jade and attracting people to join the group?
"Um... Professor Deligne..." Chen Zhou said hesitantly, "I think I may not be suitable for conducting research on standard conjectures at the moment... So, thank you for your kindness..."
Unexpectedly, after hearing this, Deligne just smiled and said, "It doesn't matter. The solution to the standard conjecture will not take another one or two years. When you are ready, we can study it together."
Chen Zhou was speechless. This was too...
But then, Deligne continued, "This invitation does not mean that I am the main one, or that you come here to serve my research group. The relationship between the two of us will be an equal partnership."
Chen Zhou looked up at Deligne. The meaning of these words, or the weight of these words, was a bit too heavy.
Although Chen Zhou has just won the Cole Prize in Number Theory, the highest honor in the field of number theory.
But in front of Deligne, he was just a toddler who had just started in the world of mathematics.
In any case, he was not equal to Deligne in becoming a collaborator in the research of the standard conjecture.
It’s not that Chen Zhou’s abilities are totally inadequate.
Just because the opponent is Deligne.
Sensing Chen Zhou's change, Deligne said softly, "My teacher encountered great technical difficulties when studying the standard conjecture and developing the Motive theory."
"I believe you have learned from my lecture. The most serious problem in the study of standard conjectures is that in order to achieve the teacher's idea of Motive theory, we must construct enough algebraic closed chains."
"But unfortunately, even though the teacher has spent a lot of energy and wisdom, he still cannot find the Motive Theory in his heart."
"And since then, no one has succeeded..."
Chen Zhou didn't react for a moment. He didn't understand why Deligne said such a thing again.
Chen Zhou hesitated and said, "Professor Deligne, your research is getting closer to this theory, isn't it?"
Deligne shook his head slightly. “Although the mathematical community has not yet confirmed my research results, I know that this is just the beginning, and it can’t even be called close to the theory.”
After hearing what Deligne said, Chen Zhou remained silent.
He felt that Deligne was a little pessimistic.
In fact, the importance of this research result is gradually becoming apparent.
Just as Chen Zhou had just thought, the series of propositions foreshadowed by this achievement are undoubtedly important.
Deligne glanced at the silent Chen Zhou, picked up the cold coffee, and put it down again.
"I have always felt that it was because of his obsession with Motive theory and the study of the standard conjecture that he was unable to solve the Weil conjecture."
"But I solved the Weil conjecture in a clever way..."
Chen Zhou clearly heard a hint of sadness in Deligne's voice.
Pursing his lips, Chen Zhou said, "But it is undeniable that the proof of the Weil conjecture is one of the greatest achievements in algebraic geometry in recent decades."
Deligne shook his head: "If I hadn't bypassed the standard conjecture, I might have been able to accept this statement calmly. But now, all I have left is regret..."
After a pause, Deligne looked at Chen Zhou and finally expressed his true thoughts.
"Chen Zhou, I know it may be presumptuous to say this, but I have to give a reasonable explanation for what I just said."
Chen Zhou nodded slightly, and Deligne continued: "The study of the standard conjecture should require mathematicians to continue to try and eventually complete it."
"Just like Fermat's Last Theorem, after hundreds of years of attempts, it was finally proved by Professor Wiles. The same should be true for the standard conjecture."
"As the originator of the standard conjecture, my teacher Grothendieck was the first mathematician to study the standard conjecture. Unfortunately, he was not able to ultimately solve this difficult problem, which may belong to the future."
When talking about the future, there was a look of expectation in Deligne's eyes.
"I should have been the one who was closest to the teacher's ideas in the study of the standard conjecture, but unfortunately..." Deligne continued, "In order to make up for this regret, in recent years, I have always been persistent in the study of the standard conjecture and have achieved the results you know."
"However, in my research on the standard conjecture, I know that the solution to this conjecture will not be mine. I am just a guide for future generations."
"So, I hope you will not give up the research on the standard conjecture after me..."
After Deligne finished speaking, Chen Zhou finally understood.
Deligne is looking for the "successor" of the standard conjecture.
Chen Zhou felt that this was probably what the inheritor meant.
From Grothendieck to Deligne, this is a continuous trend.
But after Deligne, he himself had no suitable "successor".
So, he found Chen Zhou.
Of course, Chen Zhou felt that Deligne’s obsession with the standard conjecture mostly came from his desire to make up for his regrets.
And the emotions contained in this heart are all the teacher-student relationship between him and Grothendieck.
From this conversation, Chen Zhou finally understood the feelings of mathematicians.
It starts with mathematics and ends with mathematics.
To be honest, Deligne's request was not that difficult for Chen Zhou.
On the road of mathematics, Chen Zhou, who is "determined" to build his own mathematical edifice, cannot avoid the standard conjecture, the most important problem in the field of algebraic geometry.
However, Chen Zhou had to figure out one thing.
That is, why him?
In contrast, Schultz, a super genius in algebraic geometry.
Should n't it be more appropriate?
