Chapter 1087: motive theory

Library's event room.
Facing the whiteboard half-written, Lu Zhou retracted the marker in his hand and looked back at the whiteboard in two steps.
"... If you want to solve the problem of the unity of algebra and geometry, you must strip‘ numbers ’and‘ shapes ’from general expressions, and look for similarities between them in abstract concepts.

Standing next to Lu Zhou, Chen Yang thought for a moment, then suddenly asked.
"Longlands Program?"
"It's not just the Langlands program," Lu Zhou said seriously, "and the tive theory. To solve this problem, we must figure out the connection between different cohomology theories."
In fact, this issue is a large category.
The problem of "connections between different cohomology theories" is continuously subdivided, and can even be split into tens or even millions of outstanding conjectures, or mathematical propositions.
One of the outstanding problems in the field of algebraic geometry, Hodge's conjecture, is one of the most famous.
However, it is interesting to note that although there are so many extremely difficult conjectures blocking the front, demonstrating tive theory does not need to solve all these conjectures.
The relationship between the two parties is as if the Riemann conjecture and the Riemann conjecture are generalized 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 white board in front of him, Lu Zhou continued, "At the height of strategy, we need to find a factor that can connect the two in the abstract form of numbers and shapes. Tactically, we can use formulas, poincare dualities, etc Wait for the commonality of a series of cohomology theories, and the application of the l-manifold on the complex plane that I showed you earlier. "
Then, Lu Zhou turned his attention to Chen Yang, who was 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 cluster theory of curves and the abel cluster theory to facilitate cohomology in all dimensions."
"Based on this theory, we can study the direct sum decomposition in tive theory to associate h (v) with irreducible tive."
"I originally planned to do this myself, but there are important parts worthy of me to complete. I plan to get the Grand Unified Theory within this year, and I will leave it to you."
Facing Lu Zhou's request, Chen Yang meditated for a while and said.
"It sounds interesting ... if I feel right, if this theory can be found, it should be a clue to solve Hodge's conjecture."
Lu Zhou nodded and said.
"I can't solve Hodge's conjecture, but as a problem of the same kind, its solution may inspire research on Hodge's conjecture."
"I see," Chen Yang nodded, "I will study it carefully when I go back ... but I can't guarantee that this problem 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 anxious," Lu Zhou continued with a smile, "however, my suggestion is that it is better to give it to me within two months A reply. If you are not sure, it is better to tell me in advance, I can do this by myself. "
Chen Yang shook his head.
"Not two months, half a month ... should be enough."
It's not a self-confident speech, but an affirmation in a near-declarative tone. The tools used are readily available, and even Lu Zhou has given possible ideas for solving the problem.
This kind of work that does not require subversive thinking and creativity can be solved with the utmost effort.
What he lacks most is the perseverance of a tendon on a road.
Looking at the expressionless Chen Yang, Lu Zhou nodded and reached out and patted his arm.
"Well, I'll leave this to you!"
...
After Chen Yang left, Lu Zhou returned to the library, sat down in his previous position, opened the stack of unread documents on the table, and continued his previous research while calculating with a pen on draft paper.
From a macro perspective, 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 tive theory.
Among them, the spirit of Langlands theory is to establish an essential connection between some seemingly irrelevant contents in mathematics. Since many people have heard of it, they will not repeat them.
As for the tive theory, it is less famous than the Langlands program.
At this moment, the paper he is studying is written by the well-known algebraic geometry professor voevodsky.
In the thesis, the Russian professor from the Institute of Advanced Studies in Princeton proposed a very interesting category of tive.
And this is exactly what Lu Zhou needs.
"... The so-called tive is the source of all numbers."
He whispered softly in a voice that only he could hear, while Lu Zhou sculpted on the draft paper while comparing the formulas in the literature.
As a popular example, if we call a number n, n can be expressed as 100 in decimal, then it can actually be 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 1100,100 or 144, they correspond to the number n, which is just a different elaboration form of n.
Here, n is given a special meaning.
It is both an abstract number and the nature of numbers.
What tive theory researches is a set called "uppercase n" consisting of numerous n.
As the source of all mathematical expressions, n can be mapped to a set of arbitrary intervals, whether it is [0, 1] or [0, 9], and all mathematical methods on the theory of tive are equally applicable to it.
In fact, this has involved the core issue of algebraic geometry, which is the abstract form of numbers.
It is different from all the languages ​​that humans have "translated" through different decimal notations ~ EbookFREE.me ~ This abstract expression is the language of the universe in the true sense.
And if we only use mathematics for our daily lives, we may not realize it for a lifetime. Many religions and cultures that give numbers special meanings do not actually understand the "God's language".
Some people may ask what else this can do besides making calculations more cumbersome, but in fact it is just the opposite. Separating the number itself from its representation will help people to study the abstract meaning behind it.
Apart from the theoretical foundations of modern algebraic geometry, Grothendieck lays another great work here.
He created a single theory that bridged the gap between algebraic geometry and various cohomology theories.
It is like the main theme of a symphony. Each special theory of homology can extract its own theme material from it, and perform it according to its own key, major, minor, or even original beat.
"... All the cohomology theories together form a geometric object, and this geometric object can be studied in the framework he opened up."
"... So it is."
A little bit of excitement gradually appeared in the pupils, and the pen tip in Lu Zhou's hand stopped.
A kind of deep premonition made him feel very close to the finish line.
This excitement from the soul is even more pleasing than the first time he saw the virtual reality world ...
...
(For the part about tive theory, refer to the famous "whatisative" by Barry Mazur, which is a science-based dissertation, which is really eye-opening after reading it.)