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

In the manuscript left to him by Professor Mirzahani, there were indeed some ideas for the field of 'unreducible decomposition of differential algebraic clusters'.

But it's just an idea, and even this idea isn't so perfect.

At least, in Xu Chuan's eyes, there are still a lot of loopholes in these basic ideas. What he is doing now is to build on Professor Mirzahani.

“.... On this basis, a mapping φ can be obtained: w-g, c'→γc. ”

"Let w be a set of conjugate classes of ellipses (consisting of elements that do not have an eigenvalue of 1 in the reflective representation of w). Here are some of the properties of φ.

(b) the φ is full-shot;

(c)φ| wel:wel—→g is internally mapped.

(d) If C∈ wel and w∈ then φ (c) is the only idempotent class γ of g, such that γn gw is the conjugation of b on gw......

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

.......

Following Professor Mirzahani's line of thought, Xu Chuan tinkered along the way and reintroduced the Bruhat decomposition and the Weyl group.

On this basis, we will further work on domain theory.

After a long time, Xu Chuan finally stopped what he was doing, he turned on the computer, and began to search for relevant information.

The discipline of mathematics is so huge that in the 21st century, there are hundreds of fields derived from basic mathematics, and there are dozens of fields.

It is impossible to read all the knowledge in these fields and memorize it by heart, it is impossible for human beings to accomplish such a task, and it is impossible to do it in a lifetime. Unless, like in the middle, it is directly opened and hung up, and the system is poured directly into the brain.

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

Although his memory is not weak, all the new knowledge and things he needs to use still need to be studied and searched little by little.

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

.......

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

"Huh? Isn't Xu Chuan here with you? ”

Seeing that Deligne was alone in the empty office, Witten asked in some surprise.

At this point, his student should normally be studying in Deligne's office.

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

Hearing this, Witten asked, "Where did you go to receive the award?" When will I be back. ”

It's no wonder that he asked, after all, in his memory, this student basically went out to receive the award when he asked for leave.

"That's not, he's still in Princeton, and the leave seems to have something to do with Professor Mirzahani who died some time ago, and I heard Xu Chuan say that Professor Mirzahani left a manuscript for him, and he should still be sorting out the manuscript. As for when you'll be back, you'll have to call him and ask him. ”

As he spoke, Deligne glanced at the calendar and added: "But he has indeed been on leave for a long time this time, if I remember correctly, it seems that he took a leave last month, and it has been more than a month." ”

"I'll give him a call and ask, I'm looking for something with him."

Witten shrugged, took out his phone from his pocket, found Xu Chuan's phone and dialed it, but he didn't wait long for him to put down his phone again.

"What's wrong?" Deligne asked casually.

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

Witten stuffed the phone back into his pocket and didn't call it a second time, he knew that it was impossible for Xu Chuan to hang up his phone if there was nothing important.

"Then wait, if he sees it, he should give you a call, would you like a coffee?" Deligne got up and made a cup of coffee for Witten.

For Xu Chuan, he and Witten were relieved, even if this student directly extended the vacation from a week to a month, he didn't ask much.

........

In the dormitory, Xu Chuan yawned and got out of bed.

Getting up, washing, holding a bag of yogurt and a piece of bread in his hand, he walked to the desk as he ate.

After sipping out the breakfast in his hand, Xu Chuan quickly went through the manuscript paper on the desk, and then picked up the ballpoint pen on the side and sank into it.

After a while, a half-written white manuscript paper, with black handwriting on it, stretched out and gradually covered the entire white paper.

But then, the ballpoint pen that was writing stopped, as if dissatisfied with it, and as if there was something new, in short, a bony hand directly took a new piece of manuscript paper from the side, and began to write again.

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

Beside it were other manuscript papers, also filled with mathematical formulas and symbols, piled up in a messy manner on a long, narrow desk.

Not only that, but the wooden floor under the desk was also littered with a lot of manuscript paper, some of which was crumpled and some of which remained flat.

I don't know how long it has been, but the mobile phone on the table suddenly vibrates.

The shrill ringtone broke the silence, Xu Chuan touched the phone a little impatiently, and without looking at it, he directly pressed the call, turned around and plunged into the study again.

“..... Let gz=gl(n,c) be the general complex linear group, and b∈gz is an upper trigonometric subgroup, then gzbruhat decomposition into double culture set decomposition b\g1/b=nb is a linear isomorphism of the n*n transform matrix. ”

“..... One of the largest rings of the unitary group u(n) t:={diag(d,d2,...,dn):|dj|=1).... (1≤j≤n), diag denotes a diagonal matrix, and d is a plural. When u(n)nb=t, the double culture set of the subgroup g?u(n) is decomposed into t\g1/t=nbwb. ”

“......”

The ballpoint pen in his hand rewrote the mathematical formulas one by one on the new manuscript paper, and it was not until the last stroke was recorded on the new manuscript paper that Xu Chuan stopped the pen in his hand and looked at his results.

The mess of manuscript paper on his desk, or even on the wooden floor, records his work these days, not just the problem of the 'unretractable decomposition of differential algebraic clusters,' but also extends to other areas as a result of this work.

In fact, he had already solved the problem of 'unreducible decomposition of differential algebraic clusters'.

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

Staring at the manuscript paper, a smile rose at the corner of Xu Chuan's mouth.

These days have been rewarding, and he has created a new mathematical tool for the problem of 'unreducible decomposition of differential algebraic clusters'.

And with this tool, he may be able to attack the 'Hodge conjecture', one of the seven millennial puzzles.

.......