After asking Professor Deligne for a week's vacation, Xu Chuanqian sorted out the manuscript paper left for him by Professor Mirzahani in his dormitory.

This time, it's not a rough one.

Instead, I studied the knowledge in these manuscripts in detail and absorbed them into my own wisdom.

The dying legacy of a Fields Medal, though only a part, is enough for an average mathematician to study for several years or even half a lifetime.

For Xu Chuan, the calculations in these leftover manuscript papers are not a precious thing, there is a mathematical basis, and many people can calculate and derive them.

However, the ideas and mathematical methods and lines left behind in these formulas and handwriting are precious.

These things, even if they have not yet taken shape, are just some ideas, and they are the results of many mathematicians who may not be able to make them in their lifetime.

After all, among all the natural sciences, mathematics is undoubtedly the only one at the top of the pyramid if it is to be said to be dependent on the degree of talent.

Even physics and chemistry are slightly inferior to mathematics in terms of dependence on talent.

It's fair to say that there is no other subject with more talent than mathematics.

It is a subject that requires strong logical thinking to 'really' learn well.

Math problems often require a certain amount of creativity on your part to solve unfamiliar problems.

If the teacher's level is not enough, and you fail to find the right method and direction on your own, it is very likely that your efforts will be in vain, and the more you learn, the more you will collapse.

There must be not only positive thinking but also reverse thinking, and there are many formulas in each knowledge category, and there are ingenious connections between these formulas; Memorization, calculations, argumentation, spatial, flexibility, transformation, and all the kinds of skills you can find in other subjects are almost all mathematically represented.

Many netizens said that the fear of being dominated by mathematics has nothing to do with age, and they are afraid of learning by themselves since they were children, and they are still afraid of tutoring children when they grow up.

Some netizens also said that people can do anything when they are forced to do anything, except for math problems.

Even though it's just a joke, it's true that mathematics is a subject that you don't have talent for and can't learn well.

Maybe you can get a full score in the college entrance examination before college, relying on various tactics and explanations from famous teachers, but after entering college or more in-depth study, you will soon be unable to keep up with the rhythm.

No matter how much time you spend and try your best, you may not be able to understand the meaning of certain math topics and learn to apply theorems and formulas that are more complex than in high school.

For example, the Pythagorean theorem, which is something that you will learn when you enter junior high school.

Hook three strands, four strings, five.

This is a memory for many people.

However, many people also remember this sentence, which is the most common Pythagorean number.

But what about the back?

（5，12，13）（7，24，25）（9，40，41，）...... 2n+1，2n^2+2n，2n^2+2n+1.......

These are the most basic maths, and I don't know how many people still remember them.

I'm afraid that one in ten people don't, let alone other mathematical formulas, theorems and data associated with the Pythagorean number.

If you don't have a talent for math, learning math will be quite painful.

It's not a strange thing that you lose a pen in class and pick it up and never keep up with the rhythm of math.

.......

In the dormitory, Xu Chuan was sorting out the manuscript paper left to him by Professor Mirzahani, and at the same time, he was also sorting out some of the knowledge he had learned in the past six months.

"A fundamental consequence of algebraic geometry is that any algebraic cluster can be decomposed into unions of irreducible algebraic clusters. This decomposition is called irreducible if any one irreducible algebraic cluster is not included in the other algebraic clusters. ”

"In constructive algebraic geometry, the above theorem can be constructively realized by the ritt-Wu eigencolumn method, where s is a polynomial set of n variables with rational coefficients, and we use zero(s) to represent the set of common zeros of the polynomials in the complex field in s, i.e., the algebraic cluster."

“.......”

If renamed by a variable, it can be written like this:

a? (u?，···， uq， y?) =i?y?? the lower order term of d?+y?;

a? (u?，···， uq， y?， y2)= i?y?? the lower order term of d?+y?;

······

"ap(u?,···, uq, y?,···, yp) = ip?yp+yp."

“...... Let as ={a1···, ap}, j be the product of the initial expression of ai. For the above concept, define sat(as)={p|the existence of a positive integer n such that j np∈(as)}........"

On the manuscript paper, Xu Chuan used a ballpoint pen to rewrite some knowledge points in his mind.

In the first half of this year, he learned quite a lot from the two mentors he followed, Deligne and Witten.

Especially in the field of mathematics, group construction, differential equations, algebra, and algebraic geometry can be said to have greatly enriched themselves.

Professor Mirzahani left him a piece of paper related to differential algebraic clusters, which he is now compiling.

As we all know, algebraic clusters are the most basic research objects in algebraic geometry.

Whereas, in algebraic geometry, an algebraic cluster is a set of common zero-point solutions of a polynomial set. Historically, the fundamental theorem of algebra established a connection between algebra and geometry, which states that a univariate polynomial on a field of complex numbers is determined by its set of roots, which are intrinsic geometric objects.

Since the 20th century, there have also been significant advances in transcendental methods in algebraic geometry in complex domains.

For example, de Lam's analytically cohomology theory, Hodge's application of harmonic integral theory, Kunihiko Kodaira and Spencer's deformation theory, and so on.

This makes it possible to apply the theories of partial differential equations, differential geometry, topology, etc., to the study of algebraic geometry.

Among them, the core algebraic clusters of algebraic geometry have also been applied to other fields, and today's algebraic clusters have been extended to the fields of algebraic differential equations and partial differential equations in parallel.

But in the algebraic cluster, there are still some important problems that remain unsolved.

The two most critical of these are 'irreducible decomposition of differential algebraic clusters' and 'irreducible decomposition of differential algebraic clusters'.

Although ritt and other mathematicians have proved as early as the thirties of the twentieth century that any differential algebraic cluster can be decomposed into an irreducible union of differential algebraic clusters.

However, the constructive algorithm for this result has not been given.

To put it simply, mathematicians know that the result is right, but they can't find a way to calculate the result.

It's a bit crude, but it's appropriate.

On Professor Mirzahani's manuscript, Xu Chuan saw some of the experiences of the female Fields Medal winner in this regard.

It is probably influenced by his previous exchange in Princeton, where Professor Mirzahani tried to determine whether the SAT(AS1) contains the SAT(AS2) given two irreducible differential liters of AS1 and AS2.

This is the central problem of the 'unreducible decomposition of differential algebraic clusters'.

Having familiarized himself with the whole manuscript and having studied it in depth with Professor Deligne, it was easy for him to understand Professor Mirzahani's thoughts.

In this central question, Professor Mirzahani proposed an idea that was not entirely new, but was also novel.

She tries to take it a step further by constructing an algebraic group, subgroup, and torus.

The inspiration and methods used to build these things came from his previous exchange in Princeton and his paper proving the Weyl-Berry conjecture.

......

"It's a clever way to really generalize algebraic clusters to algebraic differential equations, maybe the process will be a little more tortuous......"

Staring at the handwriting on the manuscript paper, Xu Chuan's eyes showed a hint of interest, and he pulled a piece of printing paper from the table, and the ballpoint pen in his hand recorded it on it.

“..... Broadly speaking, the problem of unreducable decomposition of differential algebraic clusters is already covered by the ritt-Wu decomposition theorem. ”

But the ritt-Wu decomposition theorem constructs irreducible ascending column ask in finite steps, and constructs many decompositions, some of which are redundant. To get rid of these redundant branches, you need to compute the generation base of sat(as). ”

“...... Because at the end of the day, it can eventually degrade into a ritt problem. That is, a is an irreducible differential polynomial with n variables, and determines whether (0, ···, 0) belongs to zero(sat(a)). ”

“......”

With a ballpoint pen in his hand, he laid the thoughts in his heart on the printing paper word by word.

This is the basic work before starting to solve the problem, and many mathematics professors or researchers have this habit, and it is not unique to Xu Chuan.

Write down the questions and your thoughts and ideas clearly with pen and paper, and then go through them in detail and sort them out.

It's like writing an outline before writing.

It ensures that the core plot revolves around the main line until you finish the book in your hand; It's not so outrageous that it was originally an urban entertainment article, and it was written and written to cultivate immortals.

Doing math is slightly better than writing, math is not afraid of brains, what I am afraid of is that you do not have enough basic knowledge and ideas.

When it comes to mathematical problems, occasional inspiration and whimsical ideas are very important, and an idea or an idea can sometimes solve a world's problem.

Of course, there are many people who have fallen into a dead end in their research because of wrong ideas.

Put it in the online literature circle, this is probably a rookie who has written for a lifetime, and it is still difficult to sign a contract after a lifetime, or he has written countless books, and he must jump into a book before a million words.

.....

After sorting out the thoughts in his mind, Xu Chuan temporarily put down the ballpoint pen in his hand.

Algebraic clusters are just some of the knowledge that Professor Mirzahani left him on paper. What he needs to do now is to sort out all these dozens of manuscript papers, instead of diving headlong into new problem research.

Although this question tickles his heart, and he can't wait to start studying it now, he still has to start and finish his work.

It took Xu Chuan a few days to properly sort out all the manuscript papers that Professor Mirzahani had left for him.

Thirty or forty pages of manuscript paper, which seems like a lot, and after the real sorting is completed, it takes less than five pages to complete the record.

In fact, there are not many ideas and knowledge points on the original manuscript paper, but more of them are some calculation data from Professor Mirzahani's essays, and the useful subjects are basically derived from the methods used in the proof paper of the Weyl-Berry conjecture.

Of course, Professor Mirzahani certainly has more than that, but that's how the two meet.

Professor Mirzahani was able to leave these things to him, and Xu Chuan was very grateful.

Because of these manuscripts, she can leave them to her students or future generations.

According to these things, if the inheritor has a certain ability, there is a high probability that he will be able to continue to make some achievements in this.

But Professor Mirzahani was not selfish, and instead gave these things to him, a 'stranger' who had only met him once or twice.

This is probably the glory of the academic world.

.......

After sorting out the useful things, Xu Chuan carefully put away the manuscript paper left by Professor Mirzahani and put it in a bookcase dedicated to storing important materials.

These things cannot be treated with any respect, and when he returns to China in the future, he will definitely bring them back.

After dealing with this, Xu Chuan sat back at the table.

For example, Professor Deligne has two days to go on leave, so instead of going back early, it is better to use this time to try the problem of 'unreducible decomposition of differential algebraic clusters'.

This is a difficult question, but the ritt-Wu decomposition theorem has decomposed the corresponding differential algebraic clusters into irreducible differential algebraic clusters, and the rest is to further obtain the unreducible decomposition.

If he had not been left behind by Professor Mirzahani, he probably would not have had the idea of studying in this area.

Originally, his goal was the self-defense form and the self-defense L function of the Langlands program, but now, it doesn't matter if the original goal is lowered a little.

And the field of 'unreducible decomposition of differential algebraic clusters' was one of the areas of mathematics that he studied with Professor Deligne in the first half of this year.

Let's use this question to test his learning results.

Thinking about it, a confident smile rose at the corner of Xu Chuan's mouth.

Using a world-class math problem as a test of learning outcomes is likely to be seen as arrogant by others.

But he has such self-confidence.

This is not brought about by studying mathematics in this life, but by climbing all the way to the peak in the previous life.

......

Taking a stack of manuscript paper from the table, Xu Chuan looked at the ideas he had sorted out before, then groaned for a moment and turned the ballpoint pen in his hand.

"Introduction: Let k be a domain, assume that k is algebraically closed, let g be a connected reduced algebraic group on k, let y be a cluster of the borel subgroup of g, let b ∈y, let t be the maximum torus of b, let n be the normalizer of t in g, let w = n/t be the weyl group......"

"For any ̇ b, where w∈n represents w......."

"Let c∈ w, let d(l(w); w∈ ={ w∈c； l(w)= dc}.....”

“...... There is a unique γ∈ g, such that γn gw?

Whenever γj∈ g, γjn gw?, there is γ?γ j. And, γ depends only on c......"

.......

ps:I don't know what's going on.,I haven't been reviewed before.,Recently, it was reviewed again in a row.,It took a long time to revise and check it at night before it was re-sent.,There's another chapter tonight.。