Chapter 180 Use world-class math problems to test your learning
After asking Professor Deligne for a week's leave, Xu Chuan went to his dormitory to sort out the manuscript papers left for him by Professor Mirzakhani.
This time I am not going through a rough review.
Instead, study the knowledge in these manuscripts in detail and absorb it into your own wisdom.
The legacy of a Fields Medal before his death, even if it is only a part, is enough for an ordinary mathematician to study for several years or even half a lifetime.
For Xu Chuan, the calculations in these leftover manuscripts are not precious things. They have mathematical foundations and many people can calculate and deduce them.
However, the thoughts, mathematical methods and routes left behind in these formulas and handwriting are extremely precious.
These things, even if they have not yet been formed, are just some ideas, and they are also achievements that many mathematicians may not be able to achieve in their lifetime.
After all, among all the natural sciences, if we talk about the degree of reliance on talent, mathematics is undoubtedly the only one that stands at the top of the pyramid.
Even physics and chemistry are slightly less dependent on talent than mathematics.
It can be said that no other subject requires more talent than mathematics.
This is a subject that requires strong logical thinking to 'really' learn well.
Mathematics problems often require you to use a certain amount of creativity to solve unfamiliar problems.
If the teacher's level is not good enough and you fail to find the correct method and direction on your own, your efforts may be in vain and the more you learn, the more you will collapse.
It is necessary to have not only forward thinking but also reverse thinking. There are many formulas in each knowledge category, and there are ingenious connections between these formulas; memory, calculation, argumentation, space, flexibility, transformation, all kinds of you
Almost all of the techniques that can be found in other subjects will be found in mathematics.
Many netizens said that the fear of being dominated by mathematics has nothing to do with age. When I was a child, I was afraid of learning by myself, but when I grew up, I was still afraid of tutoring children.
Some netizens also said that people can do anything if they are pushed hard, 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 full marks in the college entrance examination by relying on various question-test tactics and explanations from famous teachers before college, but after entering college or studying in more depth, you will soon be unable to keep up.
No matter how much time you spend and your best efforts, you may not be able to understand the meaning of certain mathematical topics, nor can you learn to apply theorems and formulas that are more complex than those in high school.
For example, the Pythagorean Theorem is something you will learn when you enter junior high school.
Hook three strands, four strings and five.
This is the memory of many people.
However, many people just remember this sentence, which is the most common Pythagorean number.
These are the most basic mathematics, and I don’t know how many people still remember them.
I am afraid that one in ten people does not have it, let alone other mathematical formulas, theorems and data related to Pythagorean.
If you are not talented in mathematics, learning mathematics may be quite painful.
It's not a weird thing to drop a pen in class, and after picking it up, you can't keep up with the rhythm in mathematics.
.......
In the dormitory, Xu Chuan was sorting out the manuscript papers left for him by Professor Mirzakhani, and at the same time sorting out some of the knowledge he had learned in the past six months.
"A basic result of algebraic geometry is that any algebraic variety can be decomposed into the union of irreducible algebraic varieties. This decomposition is called irreducible if any irreducible algebraic variety is not included in other algebraic varieties."
"In constructive algebraic geometry, the above theorem can be constructively realized through the Ritt-Wuite series method. Let s be a polynomial set of n variables with rational coefficients. We use zero(s) to represent the polynomial in s in the complex field.
The set of common zeros is an algebraic variety."
"...."
"If you rename it through a variable, it can be written in the following form:
a?(u?,···, uq, y?)=i?y??d? The lower order term of y?;
a?(u?,···, uq, y?, y2)= i?y??d? The lower order term of y?;
······
"ap(u?,···, uq, y?,···, yp)= ip?yp the lower order term of yp."
"...Suppose as ={a1···, ap}, j is the product of the initial formula of ai. For the above concept, 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 learned a lot from two mentors, Deligne and Witten.
Especially in the field of mathematics, group structures, differential equations, algebra, and algebraic geometry can be said to have greatly enriched myself.
The manuscript paper that Professor Mirzakhani left for him contains some knowledge points related to differential algebraic varieties. It is this knowledge that he is currently sorting out.
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 polynomial set. Historically, the fundamental theorem of algebra established a connection between algebra and geometry, which shows that a single-variable polynomial in the complex number field is determined by its
Determined by the set of roots, which are intrinsic geometric objects.
Since the 20th century, significant progress has been made in transcendental methods in algebraic geometry over the complex number field.
For example, De Lamm's analytic cohomology theory, the application of Hodge's harmonic integral theory, Kodaira Kunihiko and Spencer's deformation theory, etc.
This allows the study of algebraic geometry to apply theories such as partial differential equations, differential geometry, and topology.
This chapter is not finished yet, please click on the next page to continue reading the exciting content! Among them, the core algebraic varieties of algebraic geometry have also been applied to other fields. Today's algebraic varieties have been extended to algebraic differential equations in parallel.
Partial differential equations and other fields.
However, there are still some important problems that remain unresolved in algebraic varieties.
The two most critical ones are 'irreducible decomposition of differential algebraic varieties' and 'irreducible decomposition of differential algebraic varieties'.
Although mathematicians such as Ritt had proved as early as the 1930s that any difference algebraic variety can be decomposed into the union of irreducible difference algebraic varieties.
However, the constructive algorithm for this result has not been given.
To put it simply, mathematicians already know that the result is correct, but cannot find a way to verify the result.
Although this is a bit crude, it is quite appropriate.
On Professor Mirzakhani’s manuscript, Xu Chuan saw some of the female Fields Medalist’s efforts in this regard.
Probably influenced by his previous exchange meeting at Princeton, Professor Mirzakhani is trying to determine whether sat(as1) contains sat(as2) given two irreducible differential ascending sequences as1 and as2.
This is the core problem of ‘irreducible decomposition of differential algebraic varieties’.
Being familiar with the entire manuscript and having studied in depth with Professor Deligne, he easily understood Professor Mirzakhani's ideas.
In this core issue, Professor Mirzakhani proposed an idea that is not entirely new but is novel.
She tried to push it further by constructing an algebraic group, subgroup and torus.
The inspiration and methods used to build these things came from his previous exchange meetings at Princeton and the proof paper of the Weyl-berry conjecture.
...
"A very clever method. It may be possible to generalize 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 trace 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 actually been included by the Ritt-Wu decomposition theorem."
"But the Ritt-Wu decomposition theorem constructs an irreducible ascending sequence ask within a finite step, and constructs many decompositions, and among these decompositions, some branches are redundant. To remove these redundant branches, you need to calculate sat(as
) is the generative basis of."
"...Because in the final analysis, it can ultimately be reduced to a ritt problem. That is: a is an irreducible differential polynomial containing n variables. Determine whether (0,···, 0) belongs to zero(sat(a
)).”
"..."
With the ballpoint pen in his hand, he laid down his thoughts on the printing paper word by word.
This is the basic work before starting to solve the problem. Many mathematics professors or scientific 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, then go through them in detail and sort them out.
It's like writing an outline before writing.
It can ensure that before you finish the book in your hand, the core plot will always revolve around the main line; it will not be so outrageous that it was originally an urban entertainment novel and you will become a fairy as you write it.
Doing mathematics is slightly better than writing. Mathematics is not afraid of imagination. What is afraid is that you do not have enough basic knowledge and ideas.
When it comes to mathematical problems, occasional inspiration and various whimsical ideas are very important. An inspiration or an idea can sometimes solve a world problem.
Of course, there are many people who end up in dead ends in their research because of wrong ideas.
In the online literary world, this means that after a lifetime of writing, you are still a newbie who has a hard time signing a contract, or that you have written countless books and will definitely end up reading before you reach a million words.
.....
After sorting out the thoughts in his mind, Xu Chuan temporarily put down the ballpoint pen in his hand.
Things related to algebraic varieties are only part of the knowledge that Professor Mirzakhani left for him on the manuscript paper. What he has to do now is to sort out all these dozens of manuscript papers, rather than plunge into the research of new problems.
Although this problem tickled his heart a little and he wanted to start studying it now, he still had to start and finish everything.
After spending a few days, Xu Chuan properly sorted out all the manuscript papers that Professor Mirzakhani left for him.
Thirty or forty pages of manuscript paper seemed like a lot, but after the actual arrangement was completed, it took less than five pages to complete the record.
In fact, there are not many real essential ideas and knowledge points on the manuscript paper. What is more is some calculation data from Professor Mirzakhani's essays. The useful subjects basically come from the method used in the paper to prove the Weyl-berry conjecture.
Of course, Professor Mirzakhani’s knowledge is definitely more than this, but this is the point where the two of them meet.
Xu Chuan was very grateful that Professor Mirzakhani could leave these things to him.
Because of these manuscript papers, she can leave them to her students or future generations.
According to these things, if the successor has certain abilities, there is a high probability that he can continue to make some achievements in this.
But Professor Mirzakhani had no selfish motives and instead gave these things to him, a 'stranger' whom he had only met once or twice.
This is probably the glory of academia.
.......
After sorting out the useful things, Xu Chuan carefully collected the manuscript papers that Professor Mirzakhani left for him and put them in a bookcase specially designed to store important information.
These things cannot be treated with too much respect, and he will definitely take them back when he returns to China in the future.
After dealing with this, 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 "irreducible decomposition of differential algebraic varieties".
This problem is indeed difficult, but the Ritt-Wu decomposition theorem has decomposed the corresponding differential algebraic variety into an irreducible differential algebraic variety. The rest is to further obtain the irreducible decomposition.
If he had not received the inheritance from Professor Mirzakhani, he probably would not have had the idea to research in this area.
Originally his goal was the automorphic form and the automorphic l-function in the Langlands program, but now, it doesn't matter if the original goal is slightly relaxed.
Moreover, the field of ‘irreducible decomposition of differential algebraic varieties’ is one of the fields of mathematics that 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 on his lips.
Using a world-class mathematical problem as a test question for learning achievement will most likely be regarded as arrogance by others.
But he has such confidence.
This is not brought about by studying mathematics in this life, but developed by climbing to the top in the previous life.
...
Taking a stack of manuscript paper from the table, Xu Chuan read through the ideas he had compiled before, then pondered for a moment and turned the ballpoint pen in his hand.
"Introduction: Let k be a field, let k be algebraically closed, let g be the connected reduced algebra group on k, let y be the variety of the borel subgroup of g, let b∈y, let t be the pole of b
Large torus, let n be the regularizer of t in g, let w = n/t be the weyl group..."
"For any ˙ b, where w∈n represents w..."
“Suppose c∈ w, let d(l(w); w∈
={ w∈c; l(w)= dc}....."
“...there exists a unique γ∈ g such that γn gw? and so on
Whenever γj∈ g, γjn gw?, there is γ?γ j. And, γ only depends on c..."
.......
ps: I don’t know what happened. It has not been reviewed before, but it has been reviewed again recently. It took a long time to revise and check it at night before re-releasing it. There is another chapter tonight.