Scholar’s Advanced Technological System

The activity room of the library.

Faced with half of the whiteboard, Lu Zhou took back the marker in his hand and said two steps back to the whiteboard.

"...To solve the problem of unity of algebra and geometry, we must separate the 'number' and 'form' from the general expression, and look for the commonality between them in the abstract concept."

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

"Langland program?"

“It’s not just the Langlands program,” Lu Zhou said seriously. “There is also motive theory. To solve this problem, we must figure out the relationship between different cohomology theories.”

In fact, this issue is a big category.

The problem of "the relationship between different cohomology theories" is broken down and can even be split into tens of thousands or even millions of unresolved conjectures, or mathematical propositions.

The Hodge conjecture in the field of algebraic geometry is one of the most famous ones.

Interestingly, however, although there are so many extremely difficult conjectures to block, the argument for motive does not need to solve all of these conjectures.

The relationship between the two sides is as if the Riemann conjecture and the Riemann conjecture are as popular as 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 the height of strategy, we need to find a factor that can relate the two to the abstract form of numbers and shapes. In tactics, we can use the kunh formula, poinare duality Waiting for a commonality in a series of cohomology theories, and the way I used to show you the l-shaped form on the complex plane."

Said, Lu Zhou will look at Chen Yang standing next to him.

"I need a theory that can carry forward the classic theory of one-dimensional cohomology, that is, the success of the jaobi cluster theory of the curve and the abel cluster theory, in order to cope with the homology of all dimensions."

"Based on this theory, we can study the direct sum decomposition in the motive theory, making h(v) associated with irreducible motive."

"I originally planned to do this myself, but there are important parts that are worth completing. I plan to get the big unified theory within this year, and this piece will be handed over to you."

In the face of Lu Zhou’s request, Chen Yang pondered for a while and said.

"It sounds a bit interesting... If I feel right, if I can find this theory, it should be a clue to the Hodge conjecture."

Lu Zhou nodded and said.

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

"I know," Chen Yang nodded. "I will study it carefully after I go back... but I can't guarantee that it will be solved in a short time."

"It doesn't matter, this is not a task that can be completed in a short time, let alone I am not particularly worried," Lu Zhou said with a smile. "However, my suggestion is that it is best to give it to me within two months." A reply. If you are not sure, it is best to let me know in advance, I can do it myself."

Chen Yang shook his head.

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

It is not a statement of self-confidence, but an affirmation of a near-statement. The tools used are readily available, and even the possible ideas for solving the problem, Lu Zhou has already given.

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

What he lacks most is the perseverance of a shackle on a road.

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

"Well, this piece will be handed over to you!"

......

After Chen Yang left, Lu Zhou returned to the library and walked to his previous position to sit down and flipped through the pile of unfinished documents on the table. While continuing the previous research, he calculated it on the draft paper with a pen.

From a macroscopic 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, Langland's theory, the spiritual core is to establish an essential connection between some seemingly irrelevant content in mathematics. Since many people have heard of it, they will not repeat them.

As for the motive theory, it is not so famous relative to the Langlands program.

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

In the paper, the Russian professor from the Princeton Institute of Advanced Studies presented a very interesting motive category.

And this is exactly what Lu Zhou needs.

"...the so-called motive is the root of everything."

With a voice that only he could hear, he whispered softly, and Lu Zhou made a calculation on the draft paper against the line of calculations in the literature.

As a general example, if a number we call n, n can be expressed as 100 in decimal, then it can be either 1100100 or 144.

The way of expression is different, the only difference is whether we choose binary or octal to count it. In fact, whether it is 1100100 or 144, they 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 motive theory studies a set of uppercase n consisting of an infinite number of n.

As the root of all forms of mathematical expression, n can be mapped to a set of arbitrary intervals, whether it is [0, 1] or [0, 9], and all mathematical methods of motive theory are equally applicable to it.

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

Different from all the language that humans "translate" through different hexadecimal notation, this abstract expression method is the language of the universe in the true sense.

And if we use mathematics only for everyday life, we may not realize this for a lifetime. Many religions and cultures that give special meaning to numbers do not actually understand the language of God.

Some people may ask what it can do besides making calculations more cumbersome, but in fact the opposite is true. Separating the numbers themselves from their expressions is more conducive to people studying the abstract meaning behind them.

In addition to laying the theoretical foundations of modern algebraic geometry, Grottendick is another great work.

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

It is like the main theme of a symphony. Each special cohomology theory can extract its own theme material and play it according to its own tone, major, or minor tune or even the original tempo.

"...all upper homology theories together form a geometric object, and this geometric object can be put into the framework he has developed."

"...It turned out to be."

Gradually, there was a fascinating look in the pupil, and the tip of the hand in Lu Zhou stopped.

A premonition in the dark makes him feel that he is very close to the finish line.

This excitement from the depths of the soul is more enjoyable than the first time he witnessed the feelings of the virtual reality world...

......

(About the motive theory, the reference to the famous "whatisamotive" of barry·mazur is a scientific paper. It is really eye-opening after reading it.)