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∈, then φ(c) is the only idempotent class γ of g, such that γn 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 the Bruhat decomposition and the 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 it's like the one in the game, it's directly cheated and fed directly into the brain by the system.
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 ever 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 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 you come back."
It's not surprising that he asked, 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 phone again not long after.
"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 know the phone for 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 were kept 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 looking at it, turned around and immersed himself in research again.
"...Suppose gz=gl(n,c) is a general complex linear group, and b∈gz is an upper triangular subgroup. Then, gzbruhat is decomposed into a double-pertained set decomposition b\g1/b=nb is n *Linear isomorphism of n transformation matrices.”
".....a maximum ring t of the unitary group u(n):={diag(d, d2,...,dn): |dj|=1)....(1≤j≤n), diag Represents a diagonal matrix, d is a complex number. When u(n)nb=t, then the biperset decomposition of the subgroup g?u(n) is t\g1/t=nbwb."
"..."
The ballpoint pen in his hand re-wrote mathematical formulas one after another on the new manuscript paper. It was not until the last stroke was recorded on the new manuscript paper that Xu Chuan stopped the pen in his hand and stared at his results.
The manuscript papers spread out on the desk and even on the wooden floor record his achievements these days, which are not only the problem of 'irreducible decomposition of differential algebraic varieties', but also the other fields that this work has extended to.
In fact, he had already solved the problem of 'irreducible decomposition of differential algebraic varieties'.
On the way to solve the problem, he was inspired by algebraic varieties and polynomial equations, so he upgraded the original tool again and adapted it to the algebraic topology of projective algebraic varieties, non-singular complex algebraic varieties, and polynomial equations. That's why he's addicted to it.
Staring at the manuscript paper, Xu Chuan raised a smile at the corner of his mouth.
These days of busyness have paid off. He created a new mathematical tool on the problem of 'irreducible decomposition of differential algebraic varieties'.
With this tool, he may be able to attack the 'Hodge Conjecture', one of the seven millennium problems.
.......
