NOVEL NEXT Xueba starts with change
Chapter 453 Deligne's Lecture (2 in 14000 words)
Chapter 453 Deligne's Lecture (4000 in [-] [-] words)
In the lecture hall, Professor Deligne, who was wearing a blue shirt and had gray hair, was preparing materials for his lecture.
Seeing Deligne flipping through the materials, Chen Zhou sighed slightly.
Compared with his sometimes overconfident, Deligne is the kind of really very pure mathematician, confident and humble.
It is no exaggeration to say that even if Deligne did not have the slightest preparation, his lecture must be very exciting, and there must be a full house.
But now, what Chen Zhou saw was the other party's serious attitude.
In fact, Deligne is really a real mathematical genius.
When he was in middle school, he learned the French Bourbaki School's "Principles of Mathematics" from his mathematics teacher Nitz.
The "Principles of Mathematics" of the Bourbaki School is not an ordinary mathematics book.
This is a reinterpretation and cognition of modern mathematics, the content is very abstract, and it is a very broad and profound work.
Basically a mathematics book belonging to the graduate level of a university.
However, Deligne successfully read several of them and gained a lot of mathematical knowledge.
So much so that before Deligne entered university, his actual level had reached, even surpassed, the level of a mathematics undergraduate.
Later, when Deligne entered the Free University of Brussels to study mathematics, he became a student of the mathematician Titz.
This Professor Titz is also a mathematician. He has won the Wolf Prize in Mathematics and the Abel Prize. He is a typical algebraist and is famous for his research on group theory.
Moreover, Titz and Deligne are old acquaintances.
When Deligne was still in high school, he often went to the university to sit in on Titz's classes and seminars, and was appreciated by the teacher.
Chen Zhou remembered a document in which he saw the story of Deligne and Titz.
It is said that once Deligne went on an outing with his classmates and would have missed a seminar.
But after Titz found out, in order to allow Deligne to attend the class smoothly, he simply postponed the seminar.
It was also because of teachers like Titz that Deligne was born later.
Deligne also went to Paris to study algebraic geometry and algebraic number theory, which were in full swing at the time, at Titz's suggestion.
It was also because of going to Paris that Deligne met the most important teacher in his life and also the teacher who had a great influence on him, the emperor Grothendieck of algebraic geometry.
At that time, Paris, where masters gathered, was the golden age of French mathematics.
Grothendieck and Searle, the youngest winner in the history of the Fields Medal, happened to hold a seminar in Paris to exchange and discuss the most cutting-edge issues in the mathematics world.
Grothendieck was in charge of algebraic geometry, and Searle was in charge of algebraic number theory.
Deligne was sublimated again in such a seminar, and quickly mastered the essence of the mathematical thinking of these two masters.
Even Grothendieck, who many people think is eccentric and difficult to get along with, is more than happy to lend his notes to Deligne for him to organize and study.
And Grothendieck also bluntly said that Deligne's mathematics level is already on par with him.
You know, Deligne was only in his 20s at that time.
In addition, Deligne received a doctorate from the Free University of Brussels at the age of 24 and was directly employed as a professor of mathematics at the school.
Later, at the age of only 26, Deligne became one of the four tenured professors of the French Academy of Advanced Sciences at that time by virtue of his strong mathematical ability.
The French mathematics world at that time was really a gathering of stars.
In Chen Zhou's own words, this damn is the life that is really cheating...
In fact, there are still many geniuses like Deligne.
This is also one of the reasons why Chen Zhou has been urging himself to move forward.
"Ahem..." Deligne on the stage coughed lightly, and scanned the audience, "First of all, everyone is welcome to listen to my lecture today..."
"Many years ago, I proved the proposition of Weil's conjecture in a tricky way, although there are many new and different main ideas in it."
"However, my proof avoids the question of whether the standard conjecture is correct or not, which also makes many people, including me, leave a lot of regret."
"Because of this, I have not given up on the research of the standard conjecture for a long time since then, especially two years ago, this regret has been with me all day long..."
The words Deligne used to start the game were unexpected by many people.
Although it is certain that today's lecture is related to the standard conjecture, such an opening...
Chen Zhou took a deep look at Deligne on the stage.
It is no exaggeration to say that the proof of Weil's conjecture is the greatest achievement of algebraic geometry in recent decades.
Throughout the 20s, Weil's conjecture was a central research topic in algebraic geometry.
The main battlefield of Weil's conjecture research is France.
In fact, Grothendieck's series of studies and the mathematical ideas he proposed are basically based on Weil's conjecture.
But even Grothendieck, a great master of algebraic geometry, failed to solve this problem.
Of course, the reason why Grothendieck did not solve Weil's conjecture may not be his academic problem.
Just because he didn't want to bypass the unsolved problem of the standard conjecture.
This is also what Deligne said just now.
Moreover, Grothendieck died two years ago.
Thinking of this, Chen Zhou suddenly felt that Deligne might use this lecture to vent a certain emotion in his heart.
Otherwise, no mathematician would use such an opening statement.
After Deligne finished speaking, he officially started the content of his lecture without any pause.
The standard conjecture is the only topic he is currently working on.
It is also the only topic that he is willing to spend his mind on in the future.
"If you use the cohomology theory defined by the algebraic closed chain, and then use the topological theory on the category, you can get a good cohomology theory from this cohomology theory..."
"This cohomology theory can be called the dual of the cohomology theory..."
Although Deligne's voice, from the beginning to the present, is very flat.
However, there was an inexplicable firmness in the voice.
The blueprint of modern mathematics that Chen Zhou sorted out at Innott's invitation before has the position of the standard conjecture.
At this moment, listening to Deligne's narration.
Chen Zhou has a deeper understanding of the most important proposition in algebraic geometry.
The research object of algebraic geometry is an algebraic variety defined by polynomial equations, or an algebraic variety.
It is probably similar to the manifold defined by continuous functions in topology.
However, manifolds are an extension of the concepts of curves and surfaces, and can have any number of dimensions.
An important property of polynomials is their global nature.
However, this does not hinder the research of algebraic geometry and algebraic topology, which regard the extremely powerful theory of homology and cohomology as an important tool.
Unlike the singular cohomology theory of manifolds in algebraic topology, which is relatively clear, the cohomology theory in algebraic geometry is not so clear.
Like the close connection between singular cohomology in algebraic topology and another group now called topological K-theory, a great deal of information about the topology of manifolds etc. can be obtained.
Mathematicians naturally hope to have a similar theory in the homology theory of algebraic geometry.
Although the algebraic K-theory was quickly constructed, the corresponding cohomology theory was only constructed in a few very special cases.
And this has been regarded as a good progress in the research of algebraic geometry at that time.
On the other hand, the existing cohomology theory in algebraic geometry also has defects.
These cohomology theories often require topological and analytical structures other than the algebraic variety itself to define.
For example, Bettie cohomology and Hodge structure.
Moreover, the connection between various cohomology groups is not close.
Therefore, Grothendieck, who has always been committed to the study of cohomology theory in algebraic geometry, predicted the existence of a special mathematical object formed by algebraic closed chains, that is, algebraic subvariety.
Through these objects, a "universal" cohomology theory can be constructed, which has the common essence of all other good cohomology theories.
This "universal" cohomology theory should have the role of singular cohomology in algebraic topology.
In particular there should be similar Atiyah-Herzbruch spectral sequences linking cohomology theory and algebraic K-theory.
And this special mathematical object is Grothendieck's Motive theory, which is the standard conjecture.
What Deligne said is that in the study of the standard conjecture, the discovery of this possibility is the "universal" cohomology that has been sought for a long time.
"Here, we replace the closed interval [0, 1] in topological homotopy theory with an affine straight line..."
Deligne's words came into Chen Zhou's ears clearly, and stimulated Chen Zhou's sensitive math nerves.
The research work that Deligne mentioned in his lecture is actually an extremely abstract and formal work.
Especially for the establishment of cohomology theory, it involves the construction of a series of triangular categories and derived categories.
Abstract work in this category can easily fall into air-to-air metaphysical discussions.
The ultimate tirade with no real results.
But Deligne handled it well, both developing abstract concepts and using them to solve big practical problems.
It can only be said that this is very Grothendieck style.
"The research on the standard conjecture is difficult and long, and I hope that more mathematicians can participate in this grand proposition. Thank you."
Deligne ended his lecture with mutual encouragement.
The duration of this lecture, although not too long, is only about 10 minutes.
But Chen Zhou believes that everyone who listens carefully will definitely gain a lot.
Deligne's research on the standard conjecture should be regarded as the most insightful in the world.
Many mathematical ideas in it have greatly inspired Chen Zhou.
Therefore, after listening to this lecture, although the brain is running at high speed, I feel a little tired.
But this harvest is not insignificant.
Chen Zhou felt that if it wasn't for his algebraic geometry, it would be relatively weak.
He will definitely have a deeper experience.
But none of that matters anymore.
Importantly, he seems to have found some direction...
"Chen Zhou?"
The voice of Liu Maosheng beside him interrupted Chen Zhou's thoughts.
Chen Zhou turned his head in doubt: "What's wrong?"
Liu Maosheng faltered and asked: "Well, did you understand what Professor Deligne said? I see that you are engrossed in the whole process?"
Chen Zhou nodded: "I can still keep up."
Liu Mao let out an "oh" and stopped talking.
He was not surprised at all that Chen Zhou could keep up with Deligne's thinking.
Chen Zhou gave this person a strange look, and immediately reacted.
He glanced around again before asking, "Did you not understand?"
Liu Maosheng nodded embarrassingly.
Chen Zhou thought for a while and said, "I'll sort out the content of the lecture and send it to you later."
Hearing this, Liu Maosheng suddenly raised his head and looked at Chen Zhou in disbelief.
Then he nodded frantically and said, "Thank you, thank you junior, thank you boss..."
At this time, Zeng Zigu also silently came over and said, "Student brother, can you give me a share?"
Chen Zhou nodded slightly, but also reminded: "I can give it to you, but you also have to read more literature and materials yourself to enrich yourself..."
Liu Maosheng and Zeng Zigu agreed that it was true, but they felt that they were right with the boss, and they still had soup to drink.
Looking at the appearance of the two, Chen Zhou didn't say any more.
It's useless to say more, it's all up to the individual.
He also realized one thing at this moment, saying that it was a fruitful lecture.
But that's based on people who can keep up with Deligne's mathematical thinking.
Most of the students here, like Liu Maosheng and the others, may have fallen behind at the very beginning.
Except for the ignorant feeling that this is a heavenly book, there are probably only a few mathematical symbols that can be recognized.
But there is no way to do it. The more advanced the mathematical problem, the more it belongs to the field of only a few people.
After all, mathematics has never been the life of ordinary people.
After Chen Zhou said that he was willing to give Liu Maosheng and Zeng Zigu the content of the lecture he had organized.
Around him, there were a large number of people with hot eyes, staring at them blankly.
They are still very familiar with Chen Zhou, the new Cole Prize winner and the youngest winner in the history of the Cole Prize.
Therefore, they were very envious of Liu Maosheng and Zeng Zigu.
These two people seem to be the kind of dumbfounded at first glance, but they are so lucky to have a big boss take care of them!
They also want the content of the lecture organized by Chen Zhou...
But they really don't have the audacity to open their mouths.
But then, they made clear their goals.
The eyes they looked at Liu Maosheng and Zeng Zigu became more and more hot...
After the lecture, Chen Zhou originally planned to go back to the hotel with Liu Maosheng.
After all, the harvest of this lecture still needs to be sorted out by oneself.
Stones from other mountains can be used to attack jade.
But you have to be able to use other people's knowledge, you have to turn other people's knowledge into your own.
However, before Chen Zhou left the lecture hall, he was stopped by Deligne.
Deligne had something to say to him alone.
Thanks to book friend Kunlin Yuanshan for rewarding Chen Zhou's 100 starting coins!
(End of this chapter)
In the lecture hall, Professor Deligne, who was wearing a blue shirt and had gray hair, was preparing materials for his lecture.
Seeing Deligne flipping through the materials, Chen Zhou sighed slightly.
Compared with his sometimes overconfident, Deligne is the kind of really very pure mathematician, confident and humble.
It is no exaggeration to say that even if Deligne did not have the slightest preparation, his lecture must be very exciting, and there must be a full house.
But now, what Chen Zhou saw was the other party's serious attitude.
In fact, Deligne is really a real mathematical genius.
When he was in middle school, he learned the French Bourbaki School's "Principles of Mathematics" from his mathematics teacher Nitz.
The "Principles of Mathematics" of the Bourbaki School is not an ordinary mathematics book.
This is a reinterpretation and cognition of modern mathematics, the content is very abstract, and it is a very broad and profound work.
Basically a mathematics book belonging to the graduate level of a university.
However, Deligne successfully read several of them and gained a lot of mathematical knowledge.
So much so that before Deligne entered university, his actual level had reached, even surpassed, the level of a mathematics undergraduate.
Later, when Deligne entered the Free University of Brussels to study mathematics, he became a student of the mathematician Titz.
This Professor Titz is also a mathematician. He has won the Wolf Prize in Mathematics and the Abel Prize. He is a typical algebraist and is famous for his research on group theory.
Moreover, Titz and Deligne are old acquaintances.
When Deligne was still in high school, he often went to the university to sit in on Titz's classes and seminars, and was appreciated by the teacher.
Chen Zhou remembered a document in which he saw the story of Deligne and Titz.
It is said that once Deligne went on an outing with his classmates and would have missed a seminar.
But after Titz found out, in order to allow Deligne to attend the class smoothly, he simply postponed the seminar.
It was also because of teachers like Titz that Deligne was born later.
Deligne also went to Paris to study algebraic geometry and algebraic number theory, which were in full swing at the time, at Titz's suggestion.
It was also because of going to Paris that Deligne met the most important teacher in his life and also the teacher who had a great influence on him, the emperor Grothendieck of algebraic geometry.
At that time, Paris, where masters gathered, was the golden age of French mathematics.
Grothendieck and Searle, the youngest winner in the history of the Fields Medal, happened to hold a seminar in Paris to exchange and discuss the most cutting-edge issues in the mathematics world.
Grothendieck was in charge of algebraic geometry, and Searle was in charge of algebraic number theory.
Deligne was sublimated again in such a seminar, and quickly mastered the essence of the mathematical thinking of these two masters.
Even Grothendieck, who many people think is eccentric and difficult to get along with, is more than happy to lend his notes to Deligne for him to organize and study.
And Grothendieck also bluntly said that Deligne's mathematics level is already on par with him.
You know, Deligne was only in his 20s at that time.
In addition, Deligne received a doctorate from the Free University of Brussels at the age of 24 and was directly employed as a professor of mathematics at the school.
Later, at the age of only 26, Deligne became one of the four tenured professors of the French Academy of Advanced Sciences at that time by virtue of his strong mathematical ability.
The French mathematics world at that time was really a gathering of stars.
In Chen Zhou's own words, this damn is the life that is really cheating...
In fact, there are still many geniuses like Deligne.
This is also one of the reasons why Chen Zhou has been urging himself to move forward.
"Ahem..." Deligne on the stage coughed lightly, and scanned the audience, "First of all, everyone is welcome to listen to my lecture today..."
"Many years ago, I proved the proposition of Weil's conjecture in a tricky way, although there are many new and different main ideas in it."
"However, my proof avoids the question of whether the standard conjecture is correct or not, which also makes many people, including me, leave a lot of regret."
"Because of this, I have not given up on the research of the standard conjecture for a long time since then, especially two years ago, this regret has been with me all day long..."
The words Deligne used to start the game were unexpected by many people.
Although it is certain that today's lecture is related to the standard conjecture, such an opening...
Chen Zhou took a deep look at Deligne on the stage.
It is no exaggeration to say that the proof of Weil's conjecture is the greatest achievement of algebraic geometry in recent decades.
Throughout the 20s, Weil's conjecture was a central research topic in algebraic geometry.
The main battlefield of Weil's conjecture research is France.
In fact, Grothendieck's series of studies and the mathematical ideas he proposed are basically based on Weil's conjecture.
But even Grothendieck, a great master of algebraic geometry, failed to solve this problem.
Of course, the reason why Grothendieck did not solve Weil's conjecture may not be his academic problem.
Just because he didn't want to bypass the unsolved problem of the standard conjecture.
This is also what Deligne said just now.
Moreover, Grothendieck died two years ago.
Thinking of this, Chen Zhou suddenly felt that Deligne might use this lecture to vent a certain emotion in his heart.
Otherwise, no mathematician would use such an opening statement.
After Deligne finished speaking, he officially started the content of his lecture without any pause.
The standard conjecture is the only topic he is currently working on.
It is also the only topic that he is willing to spend his mind on in the future.
"If you use the cohomology theory defined by the algebraic closed chain, and then use the topological theory on the category, you can get a good cohomology theory from this cohomology theory..."
"This cohomology theory can be called the dual of the cohomology theory..."
Although Deligne's voice, from the beginning to the present, is very flat.
However, there was an inexplicable firmness in the voice.
The blueprint of modern mathematics that Chen Zhou sorted out at Innott's invitation before has the position of the standard conjecture.
At this moment, listening to Deligne's narration.
Chen Zhou has a deeper understanding of the most important proposition in algebraic geometry.
The research object of algebraic geometry is an algebraic variety defined by polynomial equations, or an algebraic variety.
It is probably similar to the manifold defined by continuous functions in topology.
However, manifolds are an extension of the concepts of curves and surfaces, and can have any number of dimensions.
An important property of polynomials is their global nature.
However, this does not hinder the research of algebraic geometry and algebraic topology, which regard the extremely powerful theory of homology and cohomology as an important tool.
Unlike the singular cohomology theory of manifolds in algebraic topology, which is relatively clear, the cohomology theory in algebraic geometry is not so clear.
Like the close connection between singular cohomology in algebraic topology and another group now called topological K-theory, a great deal of information about the topology of manifolds etc. can be obtained.
Mathematicians naturally hope to have a similar theory in the homology theory of algebraic geometry.
Although the algebraic K-theory was quickly constructed, the corresponding cohomology theory was only constructed in a few very special cases.
And this has been regarded as a good progress in the research of algebraic geometry at that time.
On the other hand, the existing cohomology theory in algebraic geometry also has defects.
These cohomology theories often require topological and analytical structures other than the algebraic variety itself to define.
For example, Bettie cohomology and Hodge structure.
Moreover, the connection between various cohomology groups is not close.
Therefore, Grothendieck, who has always been committed to the study of cohomology theory in algebraic geometry, predicted the existence of a special mathematical object formed by algebraic closed chains, that is, algebraic subvariety.
Through these objects, a "universal" cohomology theory can be constructed, which has the common essence of all other good cohomology theories.
This "universal" cohomology theory should have the role of singular cohomology in algebraic topology.
In particular there should be similar Atiyah-Herzbruch spectral sequences linking cohomology theory and algebraic K-theory.
And this special mathematical object is Grothendieck's Motive theory, which is the standard conjecture.
What Deligne said is that in the study of the standard conjecture, the discovery of this possibility is the "universal" cohomology that has been sought for a long time.
"Here, we replace the closed interval [0, 1] in topological homotopy theory with an affine straight line..."
Deligne's words came into Chen Zhou's ears clearly, and stimulated Chen Zhou's sensitive math nerves.
The research work that Deligne mentioned in his lecture is actually an extremely abstract and formal work.
Especially for the establishment of cohomology theory, it involves the construction of a series of triangular categories and derived categories.
Abstract work in this category can easily fall into air-to-air metaphysical discussions.
The ultimate tirade with no real results.
But Deligne handled it well, both developing abstract concepts and using them to solve big practical problems.
It can only be said that this is very Grothendieck style.
"The research on the standard conjecture is difficult and long, and I hope that more mathematicians can participate in this grand proposition. Thank you."
Deligne ended his lecture with mutual encouragement.
The duration of this lecture, although not too long, is only about 10 minutes.
But Chen Zhou believes that everyone who listens carefully will definitely gain a lot.
Deligne's research on the standard conjecture should be regarded as the most insightful in the world.
Many mathematical ideas in it have greatly inspired Chen Zhou.
Therefore, after listening to this lecture, although the brain is running at high speed, I feel a little tired.
But this harvest is not insignificant.
Chen Zhou felt that if it wasn't for his algebraic geometry, it would be relatively weak.
He will definitely have a deeper experience.
But none of that matters anymore.
Importantly, he seems to have found some direction...
"Chen Zhou?"
The voice of Liu Maosheng beside him interrupted Chen Zhou's thoughts.
Chen Zhou turned his head in doubt: "What's wrong?"
Liu Maosheng faltered and asked: "Well, did you understand what Professor Deligne said? I see that you are engrossed in the whole process?"
Chen Zhou nodded: "I can still keep up."
Liu Mao let out an "oh" and stopped talking.
He was not surprised at all that Chen Zhou could keep up with Deligne's thinking.
Chen Zhou gave this person a strange look, and immediately reacted.
He glanced around again before asking, "Did you not understand?"
Liu Maosheng nodded embarrassingly.
Chen Zhou thought for a while and said, "I'll sort out the content of the lecture and send it to you later."
Hearing this, Liu Maosheng suddenly raised his head and looked at Chen Zhou in disbelief.
Then he nodded frantically and said, "Thank you, thank you junior, thank you boss..."
At this time, Zeng Zigu also silently came over and said, "Student brother, can you give me a share?"
Chen Zhou nodded slightly, but also reminded: "I can give it to you, but you also have to read more literature and materials yourself to enrich yourself..."
Liu Maosheng and Zeng Zigu agreed that it was true, but they felt that they were right with the boss, and they still had soup to drink.
Looking at the appearance of the two, Chen Zhou didn't say any more.
It's useless to say more, it's all up to the individual.
He also realized one thing at this moment, saying that it was a fruitful lecture.
But that's based on people who can keep up with Deligne's mathematical thinking.
Most of the students here, like Liu Maosheng and the others, may have fallen behind at the very beginning.
Except for the ignorant feeling that this is a heavenly book, there are probably only a few mathematical symbols that can be recognized.
But there is no way to do it. The more advanced the mathematical problem, the more it belongs to the field of only a few people.
After all, mathematics has never been the life of ordinary people.
After Chen Zhou said that he was willing to give Liu Maosheng and Zeng Zigu the content of the lecture he had organized.
Around him, there were a large number of people with hot eyes, staring at them blankly.
They are still very familiar with Chen Zhou, the new Cole Prize winner and the youngest winner in the history of the Cole Prize.
Therefore, they were very envious of Liu Maosheng and Zeng Zigu.
These two people seem to be the kind of dumbfounded at first glance, but they are so lucky to have a big boss take care of them!
They also want the content of the lecture organized by Chen Zhou...
But they really don't have the audacity to open their mouths.
But then, they made clear their goals.
The eyes they looked at Liu Maosheng and Zeng Zigu became more and more hot...
After the lecture, Chen Zhou originally planned to go back to the hotel with Liu Maosheng.
After all, the harvest of this lecture still needs to be sorted out by oneself.
Stones from other mountains can be used to attack jade.
But you have to be able to use other people's knowledge, you have to turn other people's knowledge into your own.
However, before Chen Zhou left the lecture hall, he was stopped by Deligne.
Deligne had something to say to him alone.
Thanks to book friend Kunlin Yuanshan for rewarding Chen Zhou's 100 starting coins!
(End of this chapter)
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