Great Country Academician

After asking Professor Deligne for a week's vacation, Xu Chuan was sorting out the manuscript papers left by Professor Mirzakhani in his dormitory.

This sorting out is not a rough one.

Instead, learn the knowledge in these manuscripts in detail, and absorb and transform it into your own wisdom.

The legacy of a Fields Medal before death, even though it is only a part, is enough for an ordinary mathematician to study for several years or even half his life.

For Xu Chuan, the calculations in these leftover manuscripts are not precious things. They have a mathematical basis, and many people can calculate and deduce them.

But the ideas and mathematical methods and routes left in these formulas and handwriting are precious.

These things, even if they have not yet formed, are just some ideas, and they are also achievements that many mathematicians may not be able to make in their entire lives.

After all, among all the natural sciences, mathematics is undoubtedly the only one standing at the top of the pyramid if it depends on the degree of talent.

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

It can be said that there is no other subject that requires more talent than mathematics.

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

Math problems often require you to use some creativity to solve unfamiliar problems.

If the teacher's level is not enough, and you fail to find the correct method and direction by yourself, it is very likely that your hard work 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. There are many formulas in each knowledge category, but there are ingenious connections between these formulas; memory, calculation, argumentation, space, flexibility, transformation, all kinds of you Almost all of the skills that can be found in other subjects will be reflected in mathematics.

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

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

Although this is just a joke, mathematics is indeed a subject that cannot be learned well without talent.

Maybe you can get a full score in the college entrance examination by relying on various question-sea tactics and explanations from famous teachers before university, but after entering university or further study, you will soon lose the pace.

No matter how much time you spend and your best efforts, you won't necessarily be able to understand the meaning of certain mathematical topics, or learn to apply theorems and formulas that are more complicated than 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 the memory of 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 mathematics, and I don't know how many people still remember them.

I am afraid that one tenth of the people do not have it, let alone other mathematical formulas, theorems and data related to Pythagorean numbers.

If you are not talented in mathematics, learning mathematics may be quite painful.

It's not uncommon for a pen to be dropped in a class, and after picking it up, mathematics never keeps up with the rhythm.

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

"A fundamental result of algebraic geometry is that any algebraic variety can be decomposed into a union of irreducible algebraic varieties. This decomposition is called irreducible if any irreducible algebraic variety is not contained in any other algebraic variety."

"In constructive algebraic geometry, the above theorem can be realized constructively through the Ritt-Wu characteristic sequence method. Let S be the polynomial set of n variables with rational coefficients. We use Zero(S) to represent the polynomial in S over the complex field The set of common zeros of , that is, the algebraic variety.”

"."

"If the variable is renamed, it can be written as follows:

A(u, ···, uq, y)=Iyd+y's lower order term;

A(u, ···, uq, y, y2) = low order term of Iyd+y;

······

"Ap(u,..., Uq, y,..., yp)=IpYp+Yp's lower order term."

".Set AS ={A1···, Ap}, J is the product of the initial formula of Ai. For the above concepts, define SAT(AS)={P|There is 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 followed his two mentors, Deligne and Witten, and learned a lot.

Especially in the field of mathematics, group structure, differential equations, algebra, and algebraic geometry, it can be said that I have greatly enriched myself.

On the manuscript paper that Professor Mirzakhani left for him, there are some knowledge points related to differential algebraic varieties, which is what he is sorting out now.

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

In algebraic geometry, an algebraic variety is a set of common zero-point solutions of a set of polynomials. Historically, the Fundamental Theorem of Algebra establishes a link between algebra and geometry by showing that a univariate polynomial over the field of complex numbers is determined by its set of roots, which are intrinsic geometric objects.

Since the 20th century, the transcendental methods in algebraic geometry over the field of complex numbers have also made significant progress.

For example, De Ram's analytic cohomology theory, Hodge's application of harmonic integral theory, Kunihiko Kodaira and Spencer's deformation theory, etc.

This enables the study of algebraic geometry to apply theories such as partial differential equations, differential geometry, and topology.

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

But in algebraic varieties, there are still some important problems that have not been solved.

The two most critical ones are 'irreducible decomposition of differential algebraic varieties' and 'irreducible decomposition of differential algebraic varieties'.

Although mathematicians such as Ritt have already proved in the 1930s that any differential algebraic variety can be decomposed into a union of irreducible differential algebraic varieties.

But the constructive algorithm of this result has not been given.

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

Although it is a bit rough to say this, it is quite appropriate.

On Professor Mirzakhani's paper, Xu Chuan saw some of the female Fields Medalist's efforts in this regard.

Probably influenced by his previous exchange meeting in Princeton, Professor Mirzakhani is trying to determine whether SAT(AS1) includes SAT(AS2) given two irreducible differential series AS1 and AS2.

This is the core problem of 'Irreducible Decomposition of Differential Algebraic Varieties'.

Familiar with the entire manuscript, and following Professor Deligne's in-depth study in this area, he easily understood Professor Mirzakhani's thoughts.

In this central question, Professor Mirzakhani proposed a not-so-new but novel idea.

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

The inspiration and methods used to build these things come from his previous exchange meetings in Princeton and the proof papers of the Weyl-Berry conjecture.

"It's a very ingenious method. It may really be possible to extend algebraic varieties to algebraic differential equations. The process may be a little tortuous."

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

".In a broad sense, the problem of irreducible decomposition of differential algebraic varieties has been covered by the Ritt-Wu decomposition theorem."

"However, the Ritt-Wu decomposition theorem constructs the irreducible ascending series ASk in a finite step, and builds many decompositions, and in these decompositions, some branches are redundant. To remove these redundant branches, it is necessary to calculate SAT(AS )’s generation basis.”

".Because in the final analysis, it can be degraded into a Ritt problem. That is: A is an irreducible differential polynomial containing n variables, and it is determined whether (0,..., 0) belongs to Zero(SAT(A))."

"."

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

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

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

It's like writing an outline before writing a novel.

It can ensure that before you finish the book in your hand, the core plot will always be carried out around the main line; it will not be so outrageous that it was originally an urban entertainment, and you will go to immortality after writing.

Doing mathematics is slightly better than writing novels. Mathematics is not afraid of brain holes, but the fear is that you do not have enough basic knowledge and ideas.

In mathematics, the occasional inspiration and all kinds of whimsy are very important. An inspiration or an idea can sometimes solve a world problem.

Of course, there are quite a few people who have led their research to a dead end because of wrong ideas.

Putting it in the online literature circle, this is probably the kind of novel that has been written for a lifetime, and is still a rookie who has difficulty signing a contract, or has written countless books, and it is bound to skip books before a million words.

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

The things related to algebraic varieties are only part of the knowledge on the manuscript paper that Professor Mirzakhani left for him. 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 tickled his heart a bit, and he wished he could start researching it now, he still had to start and finish things.

After spending a few days, Xu Chuan properly sorted out all the papers that Professor Mirzakhani had left for him.

Thirty or forty pages of manuscript paper seemed like a lot, but after the actual sorting was completed, it took less than five pages to complete the records.

The real essence of ideas and knowledge points on the manuscript paper are actually not many, but more are some calculation data of Professor Mirzakhani's essays, and the useful subjects are basically derived from the methods used in the proof papers of the Weyl-Berry conjecture.

Of course, Professor Mirzakhani's knowledge is definitely more than this, but this is the intersection of the two.

Xu Chuan was very grateful that Professor Mirzakhani could bequeath these things to him.

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

According to these things, if the successor has a certain ability, there is a high probability that he can continue to make some achievements in it.

But Professor Mirzakhani had no selfish intentions, and instead gave these items to him, a 'stranger' who had only met once or twice.

This is probably the brilliance of the academic world.

After sorting out the useful things, Xu Chuan carefully stored the manuscript papers that Professor Mirzakhani had left for him, and put them in the bookcase specially for storing important materials.

It is not an exaggeration to treat these things with an attitude of respect, and he will definitely bring them back when he returns to China in the future.

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

There are still two days left for Professor Deligne's leave. Instead of going back early, it is better to use this time to try the problem of "incompressible decomposition of differential algebraic varieties".

This problem is really difficult, but the Ritt-Wu decomposition theorem has decomposed the corresponding differential algebraic variety into an irreducible differential algebraic variety, and the rest is to further obtain the irreducible decomposition.

If he didn't get Professor Mirzakhani's legacy, he probably wouldn't have thought of researching in this area.

His original goal was the automorphic form and automorphic L-function in the Langlands program, but now, it doesn't matter if the original goal is a little bit off.

Moreover, the field of "inreducible decomposition of differential algebraic varieties" is one of the mathematics fields he studied with Professor Deligne in the first half of this year.

Just use this question to test his learning results.

Thinking about it, Xu Chuan raised a confident smile at the corner of his mouth.

Using a world-class math problem as a test question for learning outcomes, such words will most likely be regarded as arrogant by others.

But he has such confidence.

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

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

"Introduction: Let k be a field, let k be algebraically closed, let G be a connected reduced algebraic group on k, let у be a collection of Borel subgroups of G, let B∈у, let T be a pole of B Large torus, let N be the normalizer of T in G, let W = N/T be the Weyl group."

"For any w ∈ W, let Gw = Bw˙B, where W ∈ N represents W"

"Let C ∈ W, let dC = min(l(W); w ∈ C) and let Cmin = { w ∈ C; l(w) = dC}"

".There exists a unique γ∈ G such that γ∩ Gw like

Whenever γJ∈G, γJ∩Gw, there is γγJ. Moreover, γ depends only on c"

PS: I don’t know what’s going on, it hasn’t been reviewed before, and it was reviewed again recently. It took a long time to revise and check at night before re-sending, and there is another chapter tonight.