Chapter 1088 Motive Theory

Library activity room.

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

"...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 look for the commonality between them in abstract concepts."

Standing next to Lu Zhou, Chen Yang thought about it for a moment, then suddenly asked.

"The Langlands Program?"

"It's not just the Langlands program," Lu Zhou said earnestly, "there is also the motive theory. To solve this problem, we must figure out the connection between different cohomology theories."

In fact, this question is a large category.

Continuously subdividing the problem of "the connection between different cohomology theories" can even be split into tens of thousands or even millions of unresolved conjectures, or mathematical propositions.

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

Interestingly, though there are so many extremely difficult conjectures standing in the way, it is not necessary to solve all of them to demonstrate the theory of motivation.

The relationship between the two parties is as close as the Riemann Hypothesis and the extension of the Riemann Hypothesis to the Dirichlet function.

"...on the surface it seems that what we are studying is a problem of complex analysis, 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, "From a strategic point of view, we need to find a factor that can relate the two in the abstract form of numbers and shapes. In terms of tactics, we can use the Kunneth formula, poincare dual Let’s start with the commonality of a series of cohomology theories, and the application method of L-manifolds on the complex plane that I showed you earlier.”

With that said, Lu Zhou turned his attention to Chen Yang who was standing next to him.

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

"Based on this theory, we can study direct sum decompositions in the theory of motives that relate H(v) to irreducible motives."

"Originally, I planned to do this part myself, but there are still important parts worthy of my completion. I plan to finish the grand unified theory within this year, and I will leave this part to you."

Facing Lu Zhou's request, Chen Yang pondered for a while before speaking.

"Sounds interesting... If my feeling is correct, if this theory can be found, it should be a clue to solve Hodge's conjecture."

Lu Zhou nodded and said.

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

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

"It's okay, this is not a task that can be completed in a short time, and I'm not particularly anxious," Lu Zhou smiled and continued, "However, my suggestion is that it is best to give me the task within two months. An answer. If you're not sure, you'd better let me know in advance, I can do this myself."

Chen Yang shook his head.

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

Not a confident statement, but an almost statemental affirmation. The tools used are ready-made, and even the possible ideas to solve the problem have been given by Lu Zhou.

This kind of work that does not require subversive thinking and creativity can be solved as long as you work hard.

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

Looking at the expressionless Chen Yang, Lu Zhou nodded and reached out to pat his arm.

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

...

After Chen Yang left, Lu Zhou went back to the library, went to his previous seat and sat down, opened the stack of unfinished documents on the table, continued the previous research, and calculated on the draft paper with a pen.

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

Among them, the spiritual core of Langlands theory is to establish an essential connection between some seemingly irrelevant contents in mathematics. Since many people have heard of it, I 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 the famous algebraic geometer Professor Voevodsky.

In the paper, the Russian professor from the Institute for Advanced Study in Princeton proposed a very interesting category of Motive.

And this is exactly what Lu Zhou needs.

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

Whispering in a voice that only he could hear, Lu Zhou compared the line-by-line calculation formulas in the document, and at the same time worked hard on the draft paper to calculate.

To give a popular example, if we call a number n, and n can be expressed as 100 in decimal, then in fact it can be either 1100100 or 144.

The way of expression is different, the difference is only in whether we choose binary or octal to count it. In fact, whether it is 1100100 or 144, they all correspond to the number n, which is just a different form of explanation of n.

Here, n is given a special meaning.

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

The study of motivation theory is a set called capital N composed of countless n.

As the root of all mathematical expressions, N can be mapped to any set of intervals, whether it is [0, 1] or [0, 9], and all mathematical methods about motivation theory are equally applicable to it.

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

Different from all human languages ​​"translated" by different base counting methods, this abstract method of expression is the real language of the universe.

And if we only use mathematics for our daily life, we may never realize this in our lifetime. Many religions and cultures that give numbers special meaning do not really understand the "language of God"

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

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

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

It is like the main theme of a symphony, from which each special cohomology theory can extract its own theme material, and play according to its own key, major or minor, or even original beats.

"...All cohomology theories together form a geometric object, and this geometric object can be studied under the framework developed by him."

"……I see."

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

A vague hunch made him feel that he was very close to the finish line.

This kind of excitement from the depths of the soul is even more pleasant than the feeling he experienced when witnessing the virtual reality world for the first time...

...

(For the part about motivation theory, refer to the famous "What is a Motive" by Barry Mazur, which can be regarded as a popular science paper. After reading it, it is really eye-opening.)