Chapter 181: Using World-Class Math Problems to Test Your Learning

After asking Professor Deligne for a week's vacation, Xu Chuanqian sorted out the manuscripts left by Professor Mirzakhani in his dormitory.

This time, he did not just go through them roughly.

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

The legacy of a Fields Medalist before his death, even if it was only a part, was enough for an ordinary mathematician to study for several years or even half a lifetime.

For Xu Chuan, the calculations in these manuscripts were not precious things. With a mathematical foundation, many people could calculate and deduce them.

But the ideas, mathematical methods and routes left in these formulas and handwriting were extremely precious.

These things, even if they have not yet taken shape, are just some ideas, and they are the results that many mathematicians may not be able to produce in their entire lives.

After all, among all natural sciences, if we talk about the degree of reliance on talent, mathematics is undoubtedly the only one standing at the top of the pyramid.

Even physics and chemistry are slightly inferior to mathematics in terms of the degree of reliance on talent.

It can be said that no other subject relies more on talent than mathematics.

This is a subject that requires strong logical thinking to be "really" learned well.

Mathematical problems often require you to use a certain degree of creativity to solve unfamiliar problems.

If the teacher's level is not enough, and you can't find the right method and direction yourself, it is very likely that your efforts will be in vain, and the more you learn, the more frustrated you will be.

Not only should you have positive thinking, but you should also have reverse thinking. There are many formulas in each knowledge category, and there are clever connections between these formulas; memory, calculation, argumentation, space, flexibility, transformation, and various techniques you can find in other subjects are almost all reflected in mathematics.

Many netizens said that the fear of being dominated by mathematics has nothing to do with age. I was afraid of studying by myself when I was a child, and I was still afraid of tutoring my children when I grew up.

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

Although these are just some jokes, mathematics is indeed a subject that you cannot learn well without talent.

Maybe you can get full marks in the college entrance examination before college by relying on various tactics of the sea of ​​questions and the explanations of famous teachers, but after entering college or more in-depth study, you will soon be unable to keep up with the pace.

No matter how much time you spend and how hard you try, you may not understand the meaning of some math topics, nor learn to apply theorems and formulas that are more complicated than those in high school.

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

The hypotenuse is three times the square of the hypotenuse, the square of the hypotenuse is four times the square of the hypotenuse.

This is the memory of many people.

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

But what about the rest?

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

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

I’m afraid that one in ten people don’t have them, let alone other math formulas, theorems and data related to Pythagorean numbers.

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

It’s not uncommon for someone to drop a pen in a class, pick it up, and never keep up with math again.

In the dormitory, Xu Chuan sorted out the manuscript paper Professor Mirzakhani left for him, while also sorting out some of the knowledge he had learned in the past six months.

"A basic result in algebraic geometry is that any algebraic cluster can be decomposed into the union of irreducible algebraic clusters. This decomposition is called irreducible if any irreducible algebraic cluster is not contained in any other algebraic cluster."

"In constructive algebraic geometry, the above theorem can be constructed by the Ritt-Wu characteristic sequence method. Let S be a set of polynomials with rational coefficients and n variables. We use Zero(S) to represent the set of common zeros of polynomials in S over the complex field, i.e., algebraic clusters."

"."

"If we rename the variables, we can write it in the following form:

A(u, ···, uq, y)=low-order terms of Iyd+y;

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

······

"Ap(u, ···, Uq, y, ···, yp)=low-order terms of IpYp+Yp. ”

“. Let AS ={A1···， Ap}, J be the product of Ai’s initial formula. For the above concepts, define SAT(AS)={P|There exists a positive integer n such that J nP∈(AS)}”

On the manuscript paper, Xu Chuan rewrote some of the knowledge points in his mind with a ballpoint pen.

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

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

And on the manuscript paper left to him by Professor Mirzakhani, there are some knowledge points related to differential algebraic clusters, and he is now sorting out this knowledge.

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

In algebraic geometry, algebraic clusters are the sets of common zero solutions of polynomial sets. Historically, the fundamental theorem of algebra established a connection between algebra and geometry, which shows that a polynomial of a single variable over a complex field is determined by its root set, and the root set is an intrinsic geometric object.

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

For example, de Rham's analytic cohomology theory, Hodge's application of harmonic integral theory, Kodaira and Spencer's deformation theory, etc.

This allows the study of algebraic geometry to apply partial differential equations, differential geometry, topology and other theories.

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

However, there are still some important problems in algebraic clusters that have not been solved.

The two most critical ones are "irreducible decomposition of differential algebraic clusters" and "irreducible decomposition of difference algebraic clusters".

Although mathematicians such as Ritt have proved as early as the 1930s that any difference algebraic cluster can be decomposed into the union of irreducible difference algebraic clusters.

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

Simply put, mathematicians already know that the result is correct, but they can't find a way to verify this result.

Although this statement is a bit crude, it is quite appropriate.

In Professor Mirzakhani's manuscript, Xu Chuan saw some of the experience of this female Fields Medal winner in this direction.

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

This is the core problem of "irreducible decomposition of differential algebraic clusters".

Familiar with the entire manuscript and having studied in depth with Professor Deligne in this regard, he easily understood Professor Mirzakhani's ideas.

In this core problem, Professor Mirzakhani proposed an idea that is not completely new but also novel.

She tried to make further progress by constructing an algebraic group, subgroup and torus.

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

"It's a very clever method. Maybe it can really extend algebraic clusters to algebraic differential equations. The process may be a little tortuous."

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

". In a broad sense, the problem of irreducible decomposition of differential algebraic clusters has actually been included in the Ritt-Wu decomposition theorem."

"But the Ritt-Wu decomposition theorem constructs an irreducible ascending sequence ASk in a finite number of steps and constructs many decompositions. In these decompositions, some branches are redundant. To remove these redundant branches, you need to calculate the generating basis of SAT(AS)."

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

"."

The ballpoint pen in his hand laid out his thoughts 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 unique to Xu Chuan.

Write down the problem and your own ideas and thoughts clearly with pen and paper, then go through them in detail and organize them.

This is like writing an outline before writing a novel.

It can ensure that the core plot is always centered around the main line before you finish the book in your hand; it will not be so outrageous that the original urban entertainment article will become a fairy while writing.

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

In mathematical problems, occasional inspiration and various strange ideas are very important. An inspiration or an idea may sometimes solve a world problem.

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

In the online literature circle, this is probably a rookie who has written novels for a lifetime and is still difficult to sign a contract after a lifetime of struggle, or a rookie who has written countless books and must jump to the book before a million words.

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

The algebraic clusters were only part of the knowledge on the manuscripts that Professor Mirzakhani left him. What he had to do now was to sort out all the dozens of manuscripts, rather than diving into the research of new problems.

Although this problem made him itch and he wanted to start researching it now, he still had to finish what he had started.

It took Xu Chuan several days to properly sort out all the manuscripts that Professor Mirzakhani left him.

Thirty or forty pages of manuscripts seemed a lot, but after the actual sorting, it took less than five pages to record them all.

There were not many truly essential ideas and knowledge points on the manuscripts, but there were more calculation data in Professor Mirzakhani's essays, and the useful subjects basically came from the methods used in the proof paper of the Weyl-Berry conjecture.

Of course, Professor Mirzakhani's knowledge was definitely more than this, but this was the only intersection between the two.

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

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

According to these things, if the successor has certain abilities, there is a high probability that he can continue to make some achievements on them.

But Professor Mirzakhani did not have any selfish intentions, but 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 original manuscripts left by Professor Mirzakhani and put them in the bookcase specially used to store important materials.

These things should not be treated with too much respect, and he will definitely take them back when he returns to China in the future.

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

Professor Deligne still has two days of leave. Instead of going back early, it is better to use this time to try the problem of "irreducible decomposition of differential algebraic clusters".

This problem is indeed difficult, but the Ritt-Wu decomposition theorem has decomposed the corresponding differential algebraic cluster into irreducible differential algebraic clusters. The rest is to further obtain irreducible decomposition.

If he had not received Professor Mirzakhani's legacy, he probably would not have thought of researching in this direction.

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 slightly put aside.

Moreover, the field of 'irreducible decomposition of differential algebraic clusters' is one of the mathematical fields he studied with Professor Deligne in the first half of this year.

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

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

Using a world-class mathematical problem as a test question for learning results, such words will most likely be regarded as arrogance by others.

But he has such confidence.

This is not brought by studying mathematics in this life, but cultivated by climbing to the top in the previous life.

Taking a stack of manuscripts from the table, Xu Chuan read the ideas he had sorted out before again, and then pondered for a while and turned the ballpoint pen in his hand.

"Introduction: Let k be a field, assume k is algebraically closed, let G be a connected reduced algebraic group on k, let у be a cluster of Borel subgroups of G, let B∈у, let T be a maximal torus of B, let N be a normalizer of T in G, let W = N/T be a 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

Whenever γJ∈ G, γJ∩ Gw, there is γγ J. And, γ depends only on c"

PS: I don't know what happened. It was not reviewed before, and it was reviewed again recently. I revised and checked it for a long time before reposting it. There is another chapter tonight.