From failed candidate to chief scientist

Chapter 73 Do you have any special interest in people named Gu?
"Restrictive Problems of Fourier Transform".

This was the topic Lin Mo chose for his next research.

The reason why Lin Mo chose to study this problem first is because the NS equation is a partial differential equation. Studying the Fourier transform will help Lin Mo have a deeper understanding of the solution of partial differential equations, which is helpful for the solution of the NS equation.

This is also the reason why Lin Mo chose to solve the Fourier transform restriction problem as the first research problem.

However, after deeply understanding the limitations of Fourier transform, Lin Mo felt that his system was trapped.

In 1970, American mathematician Charles Fefferman used Fourier transform to calculate the unit sphere S
Some restrictive results on n1 lead to a general result on the Bochner-Ries average problem.This triggered research on the limiting properties of Fourier transform.

Of course, this is the name of the 10000 scientific problems. In fact, this problem has another name, and its name is the Kakeya conjecture.

In 1917, the Japanese mathematician Soichi Kaketani proposed the famous Kaketani problem in mathematics. Its mathematical expression is: a line segment with a length of 1 moves as a rigid body on a plane in any way, whether it is rotation or translation. In short No matter what means are used, just to turn 180 degrees and make a U-turn, let me ask: What is the minimum area to be swept?

While raising this question, Kakeya Soichi also gave his own guess, which is the so far unsolved Kakeya conjecture: the area of ​​the smallest simply connected domain may tend to zero!

Of course, it is not accurate to completely equate the study of the limitations of Fourier transform with Kakeya's conjecture.

The results achieved by studying Kakeya's conjecture can promote the progress of research on limiting problems of Fourier transform.However, it does not mean that completely solving Kakeya’s conjecture will completely solve the limitations of Fourier transform.

So in a way, 10000 scientific problems are harder problems than those unsolved conjectures.

This made Lin Mo feel EMO for a moment.

But people, there always needs to be challenges to have fun, right?

Lin Mo was not doing it for some mission reward.

……

"Restrictive problem of Fourier transform?"

Tian Fang nodded.

"Okay, since you have made your choice, I will repay..."

Before Tian Fang finished speaking, he seemed to remember something and suddenly froze.

"Kake...Kakeya guesses?"

Tian Fang opened his mouth.

I asked you to choose an easy one, but it's a good idea for you to just choose Kakeya's conjecture.

First came the Kakutani conjecture (Kratts conjecture also known as the Kakutani conjecture), and then came the Kaketani conjecture.

Do you have any special interest in people named Gu?
Is Kakeya’s conjecture so easy to prove?

The popular point of Kakeya's conjecture is to achieve a three-point U-turn in zero space.

How can this be?

In 1971, Henry Cunningham created a simply connected Kakeya set with a very small area within the unit circle, which solved the problems of simple connectivity and boundedness.Reduced the area of ​​Kakeyaji to π/108=
0.029.

But there is still a long way to go to completely prove the 0 given by Kakeya’s conjecture.

Even, according to Cunningham's method, the proof approaching 0 cannot be achieved.

Now Lin Mo said he wants to study this...

Tian Fangyi felt that this was not even half as simple as studying NS equations.

"What's wrong? Director Tian, ​​do you have any questions?" "It's just a short-term study. If you feel it's not suitable, wait until I finish the study and you can choose the next topic. How about it?"

Tian Fang's mouth twitched, short term?A little research?

All right, just be happy.

"No, you can study at ease, and I will arrange the establishment of the project."

Tian Fang admitted that he was scared.

It's going to rain, my mother is going to get married, let him go.

Our old Tian can't keep up with the genius' thinking.

Tian Fang said goodbye to Lin Mo with a sad expression and turned to leave.

After sending Tian Fangyi away, Lin Mo concentrated on his research. He found some information on Kakeya's conjecture and started reading it.

Why did Soichi Kakeya propose the Kakeya Conjecture?

This is inseparable from the national conditions of China.

The prototype of this problem originally raised by Soichi Kaketani was: a samurai was attacked by the enemy while going to the toilet, and was hit with a hail of bullets, and he only had a short stick. In order to block the shots, he needed to rotate the stick 360°. (The pivot point can vary).But the toilet is very small, so the area swept by the stick should be as small as possible.How small can the area be?
If the hero Jin Yong had been present at that time, he would probably have told him that the Taoist priests on Wudang Mountain in the Xia Kingdom could give him the answer because they were good at a circle-drawing sword technique.

Changing the sword with you makes it turn as you wish; moving without moving is endless, this is Tai Chi.

Therefore, the highest state of swordsmanship is the answer to Kakeya's conjecture, and the minimum area approaches zero.

So what Lin Mo wants to study now is the highest level of this sword technique.

It's a bit far-fetched, but it's a rough statement. This question seems simple, but to truly prove it is like cultivating swordsmanship to the highest level, which is extremely difficult.

Soichi Kaketani and many mathematicians devoted themselves to this problem.

Soichi Kaketani thought of using the tricuspid hypocycloid. In this case, the area swept by the line segment is π/8.

In 1928, the former Soviet mathematician Besikovich used a constructive proof method-the Peron tree.

Rotate three Peyron trees by 3, 0°, and 120° respectively and stack them together. The final figure has line segments with side length ≥ 240 at each corner, forming a Besikovich set with an arbitrarily small area. .

This seems to solve the Kakeya conjecture problem, but there are still problems, because the Peilong tree is a complex structure and is not single-connected.

It's as if what a samurai needs to dance is not a short stick, but a shield composed of countless short sticks. Of course, if the samurai is fast enough, he can use the short stick to dance the shield instantly, which is also considered It can satisfy Kakeya's conjecture.

However, this is obviously unrealistic, so Besikovich's proof is not perfect.

Until 1971, Cunningham used the finite star method to reduce the minimum value to π/108.

Afterwards, no one can make a more effective proof on this basis.

After reading this information, Lin Mo started thinking.

Cunningham's method can no longer be used when π/108 is reached. Obviously this method does not work.

Besikovich's method can approach 0 infinitely, but how to solve the problem of single connection?
Lin Mo thought for a while and suddenly thought of topology.

Besikovich's Peyron Tree is, to put it bluntly, a topological structure. However, this topological structure is not perfect enough to meet the requirements of Kakeya's problem. So, can you build a topological structure yourself?What to use to solve this problem?
Just do it when you think of it. Fortunately, when solving Kratz's conjecture, Lin Mo had a lot of dealing with topology, so he was very proficient. He just picked up the pen and started to write and draw.

(End of this chapter)