Volume 1: Student Life Chapter 453: Deligne's Lecture
In the lecture hall, Professor Deligne, wearing a blue shirt and with gray hair, was preparing materials for his lecture.
Looking at Deligne flipping through the materials, Chen Zhou sighed slightly.
Compared to his sometimes overconfident attitude, Deligne is a truly pure mathematician, confident and humble.
It is no exaggeration to say that even if Deligne had no preparation at all, his lecture would surely be very exciting and the hall would surely be full.
But now, what Chen Zhou saw was the other party's serious attitude.
In fact, Deligne is really a mathematical genius.
When he was in middle school, he learned the "Principles of Mathematics" of the French Bourbaki School from his mathematics teacher Nitsch.
The Bourbaki School's "Principles of Mathematics" is not an ordinary mathematics book.
This is a reinterpretation and understanding of modern mathematics. The content is very abstract and it is a very profound work.
It is basically a mathematics book at the university graduate level.
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 or even surpassed that of a mathematics undergraduate.
Later, when Deligne entered the Free University of Brussels to study mathematics, he became a student of the mathematician Tietz.
Professor Tietz is also a mathematics giant. He has won the Wolf Prize in Mathematics and the Abel Prize. He is a typical algebraist and is famous for his research in group theory.
Moreover, Titz and Deligne are old acquaintances.
When Deligne was still in high school, he often went to the university to audit Titz's classes and seminars, and was deeply appreciated by the teacher.
Chen Zhou remembered that he had read the story of Deligne and Tietz in a document.
It is said that one time, Deligne went on an outing with his classmates and would have missed a discussion class.
But when Titz found out about this, he simply postponed the seminar so that Deligne could attend the class smoothly.
It is precisely because of teachers like Titz that there is Deligne.
It was at Tietz's suggestion that Deligne went to Paris to study algebraic geometry and algebraic number theory, which were at their peak at the time.
It was also because of his trip to Paris that Deligne met the most important teacher in his life, who also had a great influence on him, the emperor of algebraic geometry, Grothendieck.
Paris at that time was home to many great masters, and it was the golden age of the French mathematical school.
Grothendieck and Searle, the youngest winner in history of the Fields Medal, happened to be holding a seminar in Paris to exchange and discuss the most cutting-edge issues in mathematics.
Grothendieck was responsible for algebraic geometry and Serre was responsible for algebraic number theory.
It was in such discussion groups that Deligne was further sublimated and quickly grasped the essence of the mathematical thoughts of these two masters.
Even Grothendieck, who many people thought was eccentric and difficult to get along with, was happy to lend his notes to Deligne so that he could organize and study them.
Grothendieck also bluntly stated that Deligne's mathematical level was already on par with his.
You have to know that Deligne was only in his twenties at that time.
In addition, Deligne obtained his doctorate from the Free University of Brussels at the age of 24 and was directly hired as a professor of mathematics at the university.
Later, when he was only 26 years old, Deligne became one of the four tenured professors at the French Institute of Advanced Studies in Science at that time with his strong mathematical ability.
The French mathematical community at that time was truly a gathering of stars.
In Chen Zhou's own words, this is the real life of cheating...
In fact, there are many geniuses like Deligne.
This is also one of the reasons why Chen Zhou has been pushing himself to move forward.
"Ahem..." Deligne on the stage coughed lightly and glanced at the people in the audience. "First of all, welcome everyone to my lecture today..."
"Many years ago, I proved the Weil conjecture in a very clever way, although the main idea was new and different."
"However, my proof avoided the question of whether the standard conjecture is correct or not, which left many people, including me, with great regret."
"That's why I never gave up on studying the standard conjecture for a long time afterwards. Especially two years ago, this regret accompanied me all day long..."
The words that Deligne used to open the game were unexpected to 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 the Weil conjecture is the greatest achievement in algebraic geometry in recent decades.
Throughout the 1960s, the Weil conjecture was a central research topic in algebraic geometry.
The main battlefield for research on the Weil conjecture is France.
In fact, Grothendieck's series of research and the mathematical ideas he proposed are basically centered around the Weil conjecture.
But even a great master of algebraic geometry like Grothendieck failed to solve this problem.
Of course, the reason why Grothendieck did not solve the Weil conjecture may not be a problem of his knowledge.
It's just because he didn't want to bypass the unsolved problem of the standard conjecture.
This is also what Deligne meant by what he just said.
Moreover, two years ago was the year when Grothendieck died.
Thinking of this, Chen Zhou suddenly felt that Deligne might be using this lecture to vent some emotions that had been in his heart for a long time.
Otherwise, no mathematician would use such an opening statement.
After Deligne finished saying this, he officially began his lecture without any pause.
The topic of the standard conjecture is the only topic he is currently working on.
It is also the only topic he is willing to spend time and energy to discuss in the future.
"If we use the homology theory defined by algebraic closed chains and then use the topological theory on categories, we can get a very good cohomology theory from this homology theory..."
"This cohomology theory can be called the dual of homology theory..."
Although Deligne's voice has been very flat from the beginning to now.
However, there was an inexplicable firmness in the voice.
The blueprint of modern mathematics that Chen Zhou had previously drawn up at Noether's invitation included the position of the standard conjecture.
At this moment, listening to Deligne's story.
Chen Zhou has a deeper understanding of this most important proposition in algebraic geometry.
The object of study of algebraic geometry is the algebraic variety defined by polynomial equations, or algebraic varieties.
It is roughly similar to the manifold defined by continuous functions in topology.
It’s just that manifold is a generalization of the concepts of curves and surfaces, and can have arbitrary dimensions.
An important property of polynomials is their globality.
But this does not prevent algebraic geometry and algebraic topology research from using the extremely powerful homology and cohomology theory as important tools.
Unlike the relatively clear theory of singular cohomology of manifolds in algebraic topology, the theory of cohomology in algebraic geometry is not so clear.
Just as there is a close connection between singular cohomology in algebraic topology and another class of groups now called topological K-theory, a great deal of information can be obtained about the topology of manifolds and other aspects.
Mathematicians naturally hope to have a similar theory in the homology theory of algebraic geometry.
Although algebraic K-theory was quickly constructed, the corresponding cohomology theory has only been constructed in a few very special cases.
This was seen 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 analytic structures other than the algebraic multiplicity itself to be defined.
For example, Betty cohomology and Hodge structures.
Moreover, the connections between various cohomology groups are not close.
Therefore, Grothendieck, who has always been committed to the study of cohomology theory in algebraic geometry, predicted the existence of a special type of mathematical objects formed by algebraic closed chains, that is, algebraic subvariants.
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 play the role of singular cohomology in algebraic topology.
In particular, there should be something like the Atiyah–Hertzbruch spectral sequence that links cohomology theory with algebraic K-theory.
And this special mathematical object is Grothendieck's Motive Theory, which is the standard conjecture.
What Deligne described is the discovery made in his study of the standard conjecture, which may be the long-sought "universal" cohomology.
"Here, we replace the closed interval [0, 1] in topological homotopy theory with an affine line..."
Deligne's words were clearly heard by Chen Zhou and stimulated Chen Zhou's sensitive mathematical nerves.
The research work that Deligne talked about in the lecture is actually an extremely abstract and formal work.
In particular, the establishment of cohomology theory involves the construction of a series of triangular categories and derived categories.
Abstract work in this category can easily fall into empty-handed metaphysical discussions.
The final long speech, but no practical results.
But Deligne has done a good job of both developing abstract concepts and using them to solve major practical problems.
I can only say that this is very much in the style of Grothendieck.
"The research on the standard conjecture is a long and arduous journey. I hope more mathematicians can participate in this grand proposition. Thank you all."
Deligne ended his lecture with encouragement.
The lecture was not too long, only about forty minutes.
But Chen Zhou believes that everyone who listens carefully will definitely gain a lot.
Deligne's research on the standard conjecture is probably the most insightful in the world today.
Many of the mathematical ideas contained therein have greatly inspired Chen Zhou.
So, after listening to this lecture, even though my brain was working very fast, I still felt a little tired.
But this harvest is not small.
Chen Zhou felt that if it weren't for his algebraic geometry, it would be relatively weak.
He will definitely have a deeper understanding.
But, none of this matters anymore.
The important thing is that 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 confusion: "What's wrong?"
Liu Maosheng hesitantly asked, "Well, did you understand everything Professor Deligne said? I saw that you were concentrating the whole time."
Chen Zhou nodded: "I can still keep up."
Liu Maosheng said "Oh" and stopped talking.
He was not surprised at all that Chen Zhou could keep up with Deligne's thinking.
Chen Zhou looked at the man strangely, then reacted immediately.
He glanced around again before asking, "Did you not understand?"
Liu Maosheng nodded somewhat embarrassedly.
Chen Zhou thought for a moment and said, "I'll organize the contents of the lecture and send it to you later."
When Liu Maosheng heard this, he suddenly looked up and looked at Chen Zhou in disbelief.
Then he nodded frantically and said, "Thank you, thank you junior brother, thank you boss..."
At this time, Zeng Zigu also came over silently and said, "Brother Junior, can you give me a copy as well?"
Chen Zhou nodded slightly, but also reminded: "I can give it to you, but you also have to read more literature and enrich yourself..."
Liu Maosheng and Zeng Zigu responded in unison, thinking that they had followed the right leader and that they would still have soup to drink.
Looking at the appearance of the two, Chen Zhou said no more.
There is no point in saying more, it all depends on the individual.
He also realized one thing at this moment, that it was a very rewarding lecture.
But that is based on people who can keep up with Deligne's mathematical thinking.
Most of the students here, like Liu Maosheng, may fall behind at the very beginning.
Apart from being confused and thinking that this was a book of heaven, the only things I could recognize were probably a few mathematical symbols.
But there is nothing we can do about it. The more difficult the mathematical problem is, the more likely it is that it belongs to only a few people.
After all, mathematics has never been a part of the lives of ordinary people.
After Chen Zhou expressed his willingness to give Liu Maosheng and Zeng Zigu the lecture content he had compiled.
There were a large number of people around him with burning eyes, staring at them blankly.
They are still very familiar with Chen Zhou, the new Kell Medal winner and the youngest winner in the history of the Kell Medal.
Therefore, they were very envious of Liu Maosheng and Zeng Zigu.
It's obvious at first glance that these two people are the type who are confused, but they have a big boss looking after them. They are so lucky!
They also want the lecture content compiled by Chen Zhou...
But they really don't have the nerve to say that.
But then they made their goal clear.
The way they looked at Liu Maosheng and Zeng Zigu became more and more passionate...
After the lecture, Chen Zhou originally planned to return to the hotel immediately with Liu Maosheng.
After all, the gains from this lecture still need to be sorted out by myself.
Stones from other mountains can be used to polish jade.
But you have to know how to use other people's knowledge and turn it into your own.
However, before Chen Zhou left the lecture hall, he was stopped by Deligne.
Deligne wanted to speak to him alone.
