Chapter 182 Moving Towards Hodge's Conjecture

In the dormitory, Xu Chuan was immersed in the derivation and construction of a new tool.

In the manuscript left to him by Professor Mirzakhani, he did put forward some ideas in the field of ‘irreducible decomposition of differential algebraic varieties’.

But it's just an idea, and even this idea is not that perfect.

At least, in Xu Chuan's eyes, these basic ideas still have many loopholes. What he is doing now is to further expand on the work of Professor Mirzakhani.

“Based on this, a mapping φ: W——G, C’ → γC can be obtained.”

“Let W be the set of conjugate classes of ellipses (consisting of elements that have no eigenvalue 1 in the reflected representation of W). Here are some properties of φ.

(b) φ is surjective;

(c)φ| Wel:Wel—→G is internally mapped.

(d) If C∈Wel and w∈Cmin, then φ(C) is the only idempotent class γ of G, such that γ∩ Gw is the conjugate action of B on Gw

(e) If C ∈ W — Wel, then φ(C) has a simple description, which is similar to φ of the Levi subgroup of the true parabolic subgroup of g.

Following Professor Mirzakhani's ideas, Xu Chuan tinkered along the way and reintroduced Bruhat decomposition and Weyl group.

On this basis, let’s go further to domain theory.

After a long time, Xu Chuan finally stopped what he was doing. He turned on his computer and started searching for relevant information.

Mathematics is such a huge subject. Now in the 21st century, there are dozens, if not hundreds, of various fields derived from basic mathematics.

It is impossible to read all the knowledge in these fields and memorize it by heart. It is impossible for human beings to complete such a task, and it will take a lifetime to accomplish it. Unless, like in the novel, you cheat directly and the system directly feeds it into your brain.

But unfortunately, although he was reborn, he did not have a so-called system.

Although his memory is not weak, he still needs to learn and search for all new knowledge and things he needs to use bit by bit.

Especially when entering a frontier field that no one has entered before, the knowledge and literature required are even greater.

In Deligne's office at the Institute for Advanced Study in Princeton, Edward Witten opened the door and walked in.

"Huh? Isn't Xu Chuan here with you?"

Seeing that Deligne was the only one in the empty office, Witten asked in surprise.

At this point, his student should study in Deligne's office as usual.

Deligne raised his head, glanced at Witten, and said calmly: "He took a leave and hasn't come to study for a while."

Hearing this, Witten asked: "Where did you go to receive the award? When will you come back."

It's not surprising that he asked this, after all, in his memory, this student basically went out to receive awards when he took leave.

"That's not true. He is still at Princeton. His request for leave seems to be related to Professor Mirzakhani who passed away some time ago. I heard from Xu Chuan that Professor Mirzakhani left a manuscript for him. He should be still sorting out the manuscript. As for what When he comes back, you have to call him and ask him.”

With that said, Deligne looked at the calendar and added: "But it's been a long time since he took leave this time. If I remember correctly, he seemed to have taken leave last month. It's been more than a month." "

"Let me call and ask. I have something to do with him."

Witten shrugged, took out his mobile phone from his pocket, found Xu Chuan's number and dialed it, but he put down the mobile phone again before long.

"What's wrong?" Deligne asked casually.

"Maybe something happened. I got through, but was hung up quickly."

Witten put the phone back into his pocket and did not call again. He knew that Xu Chuan would not hang up his call if it was not urgent.

"Then just wait. If he sees it, he should call you. Do you want a cup of coffee?" Deligne stood up and made a cup of coffee for Witten.

Both he and Witten were relieved about Xu Chuan. Even if the student extended his vacation from one week to one month, he didn't bother much.

In the dormitory, Xu Chuan yawned and got up from the bed.

Getting up, washing up, holding a bag of yogurt in his mouth and a piece of bread in his hand, he walked towards the desk while eating.

After finishing the breakfast in his hands in a few mouthfuls, Xu Chuan quickly passed through the manuscript paper on the desk, then picked up the ballpoint pen beside him and sank into it.

Not long after, there was a half-written piece of white manuscript paper. The black handwriting on it spread out and gradually covered the entire white paper.

But then, the ballpoint pen that was writing stopped, as if he was dissatisfied with it, and seemed to have made a new discovery. In short, a hand with clear joints took a brand new manuscript paper from the side and began to write again. stand up.

The manuscript paper, which had been filled with nearly two-thirds of the writing, was swept aside.

Accompanying it are other manuscript papers also filled with mathematical formulas and symbols, piled messily on the long and narrow desk.

Not only that, there were also a lot of manuscript papers scattered on the wooden floor under the desk, some of which were crumpled into a ball, while others remained flat.

I don’t know how much time passed, but the phone on the table suddenly vibrated.

The harsh ringtone broke the silence. Xu Chuan touched his phone impatiently, pressed the call without even looking at it, turned around and immersed himself in research again.

“.Let Gz=GL(n,C) be a general complex linear group, and B∈Gz be an upper triangular subgroup, then GzBruhat decomposition is a bi-bunch decomposition B\G1/B=∏BωB. Weyl group W is a linear isomorphism of N*N transformation matrices.”

“.A maximal ring T:={diag(d,d2,…,dn):|dj|=1)(1≤j≤n) of unitary group U(n), Diag represents a diagonal matrix, d is a complex number. When U(n)∩B=T, the bi-bunch decomposition of subgroup GU(n) is T\G1/T=∏BωB.”

“.”

He rewrote mathematical formulas on the new manuscript paper with his ballpoint pen, and only stopped writing the last stroke on the new manuscript paper, and stared at his achievements.

The manuscripts scattered on the desk and even on the wooden floor recorded his achievements in the past few days, not only the problem of "irreducible decomposition of differential algebraic clusters", but also other fields extended by this work.

In fact, he had already solved the problem of "irreducible decomposition of differential algebraic clusters".

On the way to solve it, inspired by algebraic clusters and polynomial equations, he upgraded the original tools again and adapted them to projective algebraic clusters, algebraic topology of non-singular complex algebraic clusters and polynomial equations, which is why he is addicted to it.

Staring at the manuscript, Xu Chuan raised a smile.

These days of busyness have been rewarding. He has created a new mathematical tool for the problem of "irreducible decomposition of differential algebraic clusters".

With this tool, he may be able to launch an attack on the "Hodge conjecture", one of the seven millennium problems.