The library's activity room.

Facing the half-written whiteboard, Lu Zhou withdrew the marker pen in his hand, took two steps back and looked at the whiteboard and said.

“…… In order to solve the problem of the unity of algebra and geometry, it is necessary to separate 'number' and 'shape' from the general form of expression, and find the commonality between them in the abstract concept. ”

Standing next to Lu Zhou, Chen Yang thought for a moment before suddenly asking.

"The Langlands Program?"

"It's not just the Langlands program," Lu Zhou said earnestly, "but also otive theory, and to solve this problem, we have to figure out how different cohomology theories relate to each other." ”

In fact, the question is a big one.

The question of the relationship between the theories of cohomology on different sides is constantly subdivided, and it can even be broken into tens or even millions of unsolved conjectures, or mathematical propositions.

The Hodge conjecture, an unsolved problem in the field of algebraic geometry, is one of them, and one of the most famous.

Interestingly, however, while there are so many extremely difficult conjectures in the way, it is not necessary to solve all of them in order to demonstrate the otive theory.

The relationship between the two sides is like the generalization of the Riemann conjecture and the Riemann conjecture on the Dirichlet function.

“…… On the surface, we are studying a complex analysis problem, but in fact it is also a problem of partial differential equations, algebraic geometry, and topology. ”

Looking at the whiteboard in front of him, Lu Zhou continued, "Standing at a strategic level, we need to find a factor in the abstract form of number and shape that can correlate the two. Tactically, we can start with the commonalities of cohomology theory such as the Kunh formula, the OCARE duality, and so on, as well as the application of the L manifold to the complex plane that I showed you earlier. ”

As he spoke, Lu Zhou turned his gaze to Chen Yang, who was standing next to him.

"I need a theory that builds on the success of the classical theory of cohomology in one dimension, the Jabi cluster theory and the Abel cluster theory of curves, so that all dimensions can be cohomology."

"Based on this theory, we can study the straight-sum decomposition in the otive theory, so that h(v) is associated with irreducible otive."

"Originally, I was going to do this on my own, but there are still important parts that I need to complete. I'm going to finish the Great Unification Theory within this year, and I'll leave this piece to you. ”

Facing Lu Zhou's request, Chen Yang pondered for a while and spoke.

"Sounds interesting...... If I'm right, if I can find this theory, it should be a clue to the solution of Hodge's conjecture. ”

Lu Zhou nodded and said.

"I don't know if it can solve the Hodge conjecture, but as a problem of the same kind, its solution may inspire research into the Hodge conjecture."

"I see," Chen Yang nodded, "I'll study it carefully when I get back...... But I can't guarantee that I'll be able to solve this problem anytime soon. ”

"It's okay, this is not a task that can be completed in a short period of time, not to mention that I am not in a particularly hurry," Lu Zhou smiled and continued, "However, my suggestion is that it is best to give me an answer within two months." If you're not sure, it's best to let me know in advance, and it's okay for me to do this myself. ”

Chen Yang shook his head.

"Not in two months, half a month...... It should be enough. ”

Not a statement of confidence, but an affirmation that borders on a declarative tone. The tools used are ready-made, and even the possible ideas for solving the problem have been given by Lu Zhou.

This kind of work that does not require subversive thinking and creativity can be solved with the same effort.

And what he lacks most is the perseverance to stick on a road.

Looking at Chen Yang, who was expressionless, Lu Zhou nodded, reached out and patted his arm.

"Well, I'll leave this piece to you!"

……

After Chen Yang left, Lu Zhou returned to the library, walked to his previous position and sat down, opened the stack of unfinished documents on the table, and continued his previous research while calculating on the scratch paper with a pen.

From a macro point of view, the development of algebraic geometry in modern times can be summarized into two major directions, one is the Langlands program, and the other is the otive theory.

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

As for the otive theory, it is less well-known than the Langlands Program.

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

In his dissertation, the Russian professor from the Institute for Advanced Study in Princeton proposes a very interesting category of otive.

And this is exactly what Lu Zhou needs.

“…… The so-called otive is the root of all numbers. ”

Whispering in a voice that only he could hear, Lu Zhou calculated the scratch paper while comparing the lines of calculations on the literature.

To give a layman's example, if a number we call n, n can be represented as 100 in decimal, then it can actually be either 1100100 or 144.

The way it is expressed is different, the difference is only whether we choose binary or octal to count it. In fact, whether it is 1100100 or 144, they all correspond to the number n, but they are different forms of elaboration of n.

Here, n is given a special meaning.

It is both an abstract number and the essence of a number.

The otive theory is a collection of countless N's called uppercase N.

As the root of all mathematical expressions, n can be mapped to a set of arbitrary intervals, whether 0,1 or 0,9, and all mathematical methods of otive theory are equally applicable to it.

In fact, this already touches on the core problem of algebraic geometry, which is the abstract form of numbers.

Unlike all human beings who have "translated" the language through different base notations, this abstract expression is the language of the universe in the true sense.

And if we only use mathematics for everyday life, we may not realize it for the rest of our lives, and many religions and cultures that give numbers special meaning do not actually understand the "language of God"

One might ask what this does other than make the calculations more cumbersome, but in fact it is the opposite, and decoupling the number itself from its representation is more helpful for people to study the abstract meaning behind it.

In addition to laying the theoretical foundations of modern algebraic geometry, Grothendieck's other great work lies in this.

He created a single theory that bridged the gap between algebraic geometry and a wide variety of cohomology theories.

It is like the main theme of a symphony, from which each particular theory of upper cohomology can draw its own thematic material and play it in its own key, major, or minor, or even in an original tempo.

“…… All the theories of cohomology together form a geometric object, and this geometric object can be studied within the framework he has opened. ”

“…… I see. ”

A hint of excitement gradually appeared in his pupils, and the tip of the pen in Lu Zhou's hand stopped.

A vague premonition made him feel like he was close to the finish line.

This excitement from the depths of his soul was even more enjoyable than the feeling he felt when he first witnessed the world of virtual reality......

……

(The part about the theory of otive, referring to Barryazur's famous "Whata Otive", is a popular science paper, and it is really eye-opening after reading it.) ）