Library activity room.
Lu Zhou looked at the half-written whiteboard and put away the marker in his hand. He then took two steps back and looked at the whiteboard.
"… If we want to solve the problem of the unification of algebra and geometry, we must separate 'number' and 'shape' from the general expression and find their commonalities in abstract concepts."
Chen Yang was standing next to Lu Zhou. He thought for a bit and suddenly spoke.
"Langlands program?"
"Not just the Langlands program." Lu Zhou said in a serious manner, "There's also the theory of. If we want to solve this problem, we must figure out the connections between different cohomology theories."
In fact, this problem was a huge one.
The problem of "the connections between different cohomology theories" could be divided into tens of thousands or even millions of unsolved conjectures or mathematical propositions.
The unsolved Hodge conjecture in the field of algebraic geometry was one of them, and it was also the most famous one.
However, the interesting thing was that even though there were so many extremely difficult conjectures blocking the way, proving the theory didn't need to solve all of these conjectures.
The relationship between the two was like the generalization of the Riemann conjecture and the Riemann conjecture on the Dirichlet function.
"… On the surface, it looks like we are researching a complex analysis problem, but in fact, it is also a problem of partial differential equations, algebraic geometry, and topology."
Lu Zhou looked at the whiteboard in front of him and said, "From a strategic point of view, 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 common cohomology theories such as the kunh formula, ocare duality, and the application method of the L Manifold on the complex plane that I showed you earlier. "
Lu Zhou looked at Chen Yang, who was standing next to him.
"I need a theory that can carry forward the classical theory of one-dimensional cohomology, which is the success of the jabi cluster theory and abel cluster theory of curves, to facilitate cohomology of all dimensions.
"Based on this theory, we can study the direct sum decomposition in the theory of, so that h (v) is associated with irreducible.
"Originally, I planned to do this myself, but there's an important part that I need to complete. I plan to complete the Grand Unified Theory within this year. I'll leave this to you. "
Facing Lu Zhou's request, Chen Yang thought for a while and said.
"Sounds interesting … If I'm not wrong, if I can find this theory, it should be a clue to solving the Hodge conjecture."
Lu Zhou nodded and said.
"I'm not sure if it can solve the Hodge conjecture, but as a similar problem, it might be able to inspire research on the Hodge conjecture."
"I got it." 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 doesn't matter. This isn't a task that can be completed in a short period of time. Besides, I'm not particularly in a hurry." Lu Zhou smiled and continued, "However, my suggestion is to give me an answer within two months. If you're not sure, you'd better let me know in advance. I can do it myself. "
Chen Yang shook his head.
"It won't take two months, half a month … should be enough."
This was not a confident statement, but it was almost a statement. The tools were ready-made, and Lu Zhou had even given Chen Yang an idea to solve the problem.
This type of work didn't require creative thinking; it could be solved with hard work.
What he didn't lack the least was the willpower to stick to the same path with one tendon.
Lu Zhou looked at Chen Yang and nodded. He then patted Chen Yang's arm.
"Okay, I'll leave this to you!"
…
After Chen Yang left, Lu Zhou returned to the library and sat down in his original seat. He flipped through the stack of unfinished documents on the table and continued his previous research while he did some calculations on a piece of draft paper.
From a macro point of view, the development of algebraic geometry in recent times could be summed up in two major directions. One was the Langlands program, and the other was the theory of algebraic geometry.
The spiritual core of Langlands' theory was to establish connections between some seemingly unrelated mathematical concepts. Since many people have heard of it, I won't go into detail.
As for the theory of algebraic geometry, it wasn't as famous as the Langlands program.
The thesis he was reading was written by the famous algebraic geometry scholar Professor Voevodsky.
In the thesis, the Russian professor from the Princeton Institute for Advanced Study proposed a very interesting category of algebraic geometry.
This was exactly what Lu Zhou needed.
"… The so-called algebraic geometry is the root of all numbers."
Lu Zhou whispered in a voice that only he could hear. He compared the lines of calculations on the document while writing on the draft paper.
For example, if we call a number n, and n can be expressed as 100 in decimal, then it can be either 1100100 or 144.
The only difference in expression is whether we choose to use binary or octal to calculate it. In fact, whether it is 1100100 or 144, they both correspond to the number n. It's just that they have different representations of n.
Here, n was given a special meaning.
It was both an abstract number and the essence of numbers.
The theory of algebraic geometry was a set named capital n composed of countless n's.
As the root of all mathematical expressions, n could be mapped to any set of intervals, whether it was [0,1] or [0,9]. All of the mathematical methods of algebraic geometry were equally applicable to n.
In fact, this involved the core problem of algebraic geometry, which was the abstract form of numbers.
Different from the language that was "translated" by human beings through different arithmetic methods, this abstract method was the language of the universe in the true sense.
If we only used mathematics for our daily lives, we might never realize this. Many religions and cultures that gave special meaning to numbers didn't really understand the "language of God".
Some people might ask what was the use of this other than making calculations more troublesome, but in fact, it was the opposite. Separating the number itself from its expression was actually more helpful for people to study the abstract meaning behind it.
In addition to laying the theoretical foundation of modern algebraic geometry, Grothendieck's other great work was in this area.
He created a single theory that bridged the gap between algebraic geometry and various cohomology theories.
It was like the main melody of a symphony. Every special cohomology theory could extract its own theme material and play 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 put into the framework he created to study."
"… I see."
There was a hint of excitement in Lu Zhou's eyes, and the pen in his hand stopped.
He had a premonition that he was very close to the finish line.
This kind of excitement from the depths of his soul was even more pleasant than the first time he saw the virtual reality world …
…
(For the part about algebraic geometry, I refer to Barryazur's famous "whata" thesis.
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