Chapter 1087

The activity room of the library.

Facing the whiteboard that has written half, Lu Zhou takes back the marker in his hand and looks back at the whiteboard and says.

“…… If we want to solve the problem of the unity of algebra and geometry, we must separate "number" and "shape" from the general form of expression, and find the commonness between them in the abstract concept. "

Standing beside the boat, Chen Yang thought for a moment, then suddenly asked.

"The Langlands program?"

"It's not just the rananz program," Lu said seriously. "There's also the motive theory. To solve this problem, we have to find out the relationship between different cohomology theories."

In fact, this problem is a large category.

The problem of "the connection between different cohomology theories" can be subdivided into tens of thousands or even millions of unsolved conjectures, or mathematical propositions.

Hodge's conjecture, one of the outstanding problems in the field of algebraic geometry, is one of the most famous.

However, it is interesting to note that although there are so many extremely difficult conjectures in front of us, it is not necessary to solve all of them to prove the motive theory.

The relationship between them is just like the generalization of Riemann conjecture and Riemann conjecture on Dirichlet function.

“…… On the surface, we study a complex analysis problem, but in fact, it is also a problem of partial differential equation, algebraic geometry and topology. "

Looking at the whiteboard in front of us, Lu Zhou continued, "standing at the height of strategy, we need to find a factor that can relate the two in the abstract form of number and shape. In terms of tactics, we can start with a series of commonalities of cohomology theories, such as kunnetth formula, Poincare duality and so on, as well as the application method of l-manifold in complex plane that I showed you earlier. "

With that, Lu Zhou looks to Chen Yang, who is standing next to him.

"I need a theory that can carry forward the classic theory of one-dimensional cohomology, that is, the success of Jacobi and Abel cluster theories of curves, so as to facilitate the cohomology of all dimensions."

"Based on this theory, we can study the direct sum decomposition in the theory of motion, so that h (V) is related to irreducible motion."

"Originally, I planned to do it myself, but there are still some important parts worth me to complete. I intend to work out the grand unified theory within this year, and I'll give you this piece. "

Facing the request of Lu Zhou, Chen Yang thought for a while and said.

"That sounds interesting If I feel right, if I can find this theory, it will be the clue to solve Hodge's conjecture. "

The boat nodded and said.

"I don't know if I can solve Hodge's conjecture, but as the same kind of problem, its solution may be able to inspire the study of Hodge's conjecture."

"I see," Chen Yang nodded. "I will study it carefully when I go back But I can't guarantee to solve this problem in a short time. "

"It doesn't matter. This is not a task that can be completed in a short time, and I'm not particularly worried," continued Lu Zhou with a smile. "However, my suggestion is that you'd better give me a reply within two months. If you're not sure, you'd better tell me in advance. It's OK for me to make this one myself. "

Chen Yang shook his head.

"Two months is not enough, half a month It should be enough. "

It's not a self-confident speech, but a kind of affirmation that is almost declarative. The tools used are ready-made, and even the possible ways to solve the problem have been given by the land boat.

This kind of work doesn't need subversive thinking and creativity. It can be solved if you are willing to work hard.

And what he needs most is the perseverance to stick to one road.

Looking at the expressionless Chen Yang, Lu Zhou nodded and clapped his arm.

"Well, I'll give you this one!"

……

After Chen Yang left, Lu Zhou went back to the library, went to his previous position, sat down, opened the stack of unfinished documents on the table, continued the previous research, while using a pen to calculate on the draft paper.

From a macroscopic point of view, the development of algebraic geometry in modern times can be summed up in two major directions: one is the Langlands program, the other is the motive theory.

Among them, the spiritual core of the Langlands theory is to establish the essential connection of some seemingly unrelated contents in mathematics. Because many people have heard of it, they will not repeat it.

As for the motive theory, it is less famous than the Langlands program.

At this moment, the paper he is studying is written by Professor voevodsky, a famous algebraic geometer.

In this paper, the Russian professor from Princeton Institute of higher studies proposes a very interesting category of motive.And that's exactly what land boats need.

“…… The so-called motive is the root of all numbers. "

Read it in a low voice that only you can hear. While comparing the lines of calculation in the literature, Lu Zhou was working hard on the rough paper.

For example, if a number we call n can be expressed as 100 in decimal system, it can be 1100100 or 144 in fact.

The difference lies in whether we choose binary or octal to count it. In fact, no matter 1100100 or 144, they all correspond to the number of N, which is just a different form of elaboration of n.

Here, n is given a special meaning.

It is not only an abstract number, but also the essence of numbers.

What the motive theory studies is a set called capital n, which is composed of numerous n's.

As the root of all mathematical expressions, n can be mapped to any set of intervals, no matter [0, 1] or [0, 9], and all the mathematical methods of dynamic theory are equally applicable to it.

In fact, this has involved the core problem of algebraic geometry, that is, the abstract form of numbers.

Different from the language after all human beings "translate" through different base counting methods, this abstract expression method is the real language of the universe.

If we only use mathematics for our daily life, we may never realize this. In fact, many religions and cultures that give special meaning to numbers don't really understand "God's language"

some people may ask what else it can do besides making calculation more troublesome, but in fact, it is the opposite The number itself is separated from its expression form, which is more helpful for people to study the abstract meaning behind it.

In addition to laying the theoretical foundation of modern algebraic geometry, another great work of grotengdick is here.

He created a single theory, a bridge between algebraic geometry and various cohomology theories.

It is just like the main melody of a symphony. Each special theory of cohomology can extract its own theme material from it and perform according to its own keynote, major, minor or even original beat.

“…… All the cohomology theories together constitute a geometric object, which can be put into the framework of his research

“…… I see. "

The pupil gradually dyed a trace of excitement, and the tip of the pen in Lu Zhou's hand stopped.

A kind of premonition made him feel that he was close to the finish line.

This kind of excitement from the deep soul is more pleasant than the feeling of seeing the virtual reality for the first time

……

(for the part about the theory of motion, reference is Barry Mazur's famous "what is a dynamic", which is a scientific paper. It's really eye opening after reading.)