Chapter 1087 Motive Theory
Inside a library activity room.
Lu Zhou was standing in front of a half-written whiteboard. He put down the marker in his hand, took two steps back, then spoke.
“… If we want to unify geometry and algebra, we have to change our view of numbers and shapes. We need to look for the similarities between their abstract concepts.”
Chen Yang was standing next to Lu Zhou. After contemplating for a second, he suddenly spoke.
“Like the Langlands program?”
Lu Zhou said in a serious manner, “It’s not just the Langlands program, but also motive theory. If we want to solve this problem, we have to find the relations.h.i.+p between different coh.o.m.ology theories.”
In fact, this was a common problem.
The connection between different coh.o.m.ology theories was divided into tens of thousands or even millions of unsolved conjectures and mathematical propositions.
The Hodge conjecture, which was an unsolved problem in the field of algebraic geometry, was one of the most famous examples.
However, interestingly enough, even though there were many difficult conjectures blocking the way, one could prove motive theory without having proven the other conjectures.
It was similar to Riemann’s hypothesis versus the generalized Riemann’s hypothesis on the Dirichlet function.
“… On the surface, it looks like we are researching a complex a.n.a.lysis problem, but in fact, it is also a problem concerning partial differential equations, algebraic geometry, and topology.”
Lu Zhou stared at the whiteboard and said, “A wise strategy would be to find an abstract factor that relates numbers and shapes. We can start with the relations.h.i.+p between a series of coh.o.m.ology theories, such as the Kunneth theorem and the Poincare duality. We can also apply this method to the L Manifold on the complex plane, the one I showed you earlier.”
Lu Zhou glanced at Chen Yang, who was standing next to him. He continued, “I need a theory that builds on top of the cla.s.sical theory of one-dimensional coh.o.m.ology, which is the Abel Jacobi theorem.
“Using this theory, we can study the direct-sum decomposition in motive theory and a.s.sociate H(v) with an irreducible motive.
“I planned on doing this myself, but there are other important things I have to do. I plan on finis.h.i.+ng the Grand Unified Theory by the end of the year, so you’ll be responsible for this part.”
Chen Yang went silent for a while before speaking, “Sounds interesting… If my interpretation is correct, if we find this theory, it will be able to help solve Hodge conjecture.”
Lu Zhou nodded and spoke.
“I’m not sure if it can solve Hodge conjecture or not, but it will inspire research on the Hodge conjecture.”
“I understand,” Chen Yang nodded and said, “I’ll give it a shot… I can’t guarantee I can solve this anytime soon.”
“It’s fine, this isn’t something that can be solved in a short amount of time. I’m not in a hurry anyway.” Lu Zhou smiled and then said, “But my advice is to give me an answer within two months. If you’re not confident, make sure to tell me in advance. I can do it myself.”
Chen Yang shook his head.
“It won’t take two months, two weeks should be enough.”
Chen Yang spoke confidently, like there was no doubt at all. The mathematical tools were already available, and Lu Zhou had even given him ideas on how to solve the problem.
This kind of work did not require out of the box thinking or creativity, it only required hard work.
And he had plenty of perseverance in him.
Lu Zhou looked at Chen Yang and nodded. He reached out and patted him on the shoulder.
“Okay, I believe in you!”
…
After Chen Yang left, Lu Zhou returned to the library and sat down in his chair. He flipped through the stack of theses on his table and continued to read while writing on a draft paper at the same time.
Looking at this from an overview perspective, the development of algebraic geometry could be split into two major directions. One was the Langlands program, the other was the motive theory.
The essence of the Langlands program was to establish links between seemingly unrelated areas in mathematics.
Motive theory, on the other hand, was less well-known compared to the Langlands program.
The paper Lu Zhou was reading about was written by the famous algebraic geometry expert Professor Voevodsky.
The Russian professor from the Princeton Inst.i.tute for Advanced Study proposed an interesting type of motive.
It was precisely what Lu Zhou needed.
“… The motive is the root of all numbers.”
Lu Zhou muttered to himself as he wrote on the draft paper, verifying the thesis calculations.
“For example, if we have a number n, n in base 10 is 100, n in base 2 is 1100100, n in base 8 is 144.
“Its expression only depends on whether we choose to count in base 2, base 8, or base 10. All of them correspond to the number n, just written in different forms of expression.
“N has a special meaning.
“It is not just an abstract number, but more of a mathematical concept.
“Motive theory is about a collection of uncountable n, named 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]…”
In fact, this was one of the core problems of algebraic geometry, which was the abstraction of numbers.
Different mathematical languages had been “translated” by humans through different notation systems. The abstract expression was the only true language of the universe.
People who used mathematics in their daily life might never realize this. Many religions and cultures that gave numbers special meaning did not actually understand what the “language of the universe” was.
People might ask what was the point of making calculations more complicated, but separating numbers from its representation could help people research the abstract meaning behind it.
In addition to laying the modern theoretical foundation of algebraic geometry, Grothendieck also proposed motive theory.
This theory was like a bridge that connected various coh.o.m.ology theories and algebra and geometry.
It was like the main melody of a symphony. Ever coh.o.m.ology theory could extract a theme from the main melody and modify it by changing the major, minor, or even tempo.
“… The coh.o.m.ology theories form a geometric object. This geometric object can be researched using his framework.”
“… I see.”
Lu Zhou had a flash of excitement in his eyes, and he suddenly stopped writing.
He had a feeling that he was close to the finish line.
This type of feeling came from the deepest part of his soul, and it was the best thing he had ever felt…
…