Great Country Academician

Zhou Hai dragged a chair from the side and sat down, ready to discuss this with Xu Chuan.

That's right, it's communication, not pointers.

In his opinion, Xu Chuan's mathematical ability, who can study the weak Weyl-Berry conjecture branch problem, has reached a certain level.

"The source of the Weyl-Berry conjecture comes from the mathematician Mark Carker in 1966. In a lecture that year, he raised a question that left a name in the history of science: 'Can anyone hear the shape of a drum from the sound? '"

"Hear the shape of the drum through the sound? Can this also be done?" Beside Xu Chuan, a student who came over to listen asked curiously.

Zhou Hai smiled, and didn't mind the students interrupting him. University and junior high school are two completely different learning environments.

In universities, some teachers often chat with students in addition to imparting knowledge in class.

After all, students are young, and sometimes their thinking on problems is very special, which will bring unexpected surprises.

Moreover, it is far more useful to make students curious about a certain field through some stories and let them enter the learning state than to force knowledge to them. This teaching method is more in line with universities.

"Mathematically, stretching a membrane over a rigid support creates a two-dimensional drum."

"Drums of different shapes produce sound waves of different frequencies when struck, and therefore produce different sounds."

"With these different sounds, it's really possible to determine the shape of the drum."

"It involved the work of two mathematicians, Alan Conners and Walter van Suylecombe."

"They extend the traditional framework of non-commutative geometry to deal with spectral truncation of geometric spaces and provide tolerance relations for coarse-grained approximations of geometric spaces at finite resolution. A propagation number that is shown to be an invariant under stable equivalence and can be used to compare approximations of the same space."

"In this framework, we can describe the vibration of the 'drum' when it is struck by the wave equation, and because the edge of the 'drum head' is firmly attached to the rigid frame, we can think that the boundary of the wave equation The condition is the Dirichlet boundary condition."

"With the data of these two pieces, and through the diffusion equation and other methods, we can calculate its shape from the sound of the drum, even if you have not seen it."

Zhou Hai explained with a smile, but directly said that he was stunned and came to listen to the lively students.

What is spectral truncation in geometric space? What is the spectrum truncation of the circle?

They all know what it means to recognize the position by listening to the sound, but they have never heard of recognizing the shape by listening to the sound.

Can mathematics really do this? It is not metaphysics!

You can know what happened by pinching your fingers, which is too outrageous, right?

Xu Chuan, on the other hand, probably understood what Zhou Hai meant.

The so-called "distinguishing shape by listening to the drum" is actually the problem of the eigenvalue of the Laplacian operator in a region.

To "listen to the drum and distinguish the shape" through mathematics is related to another concept.

That is 'diffusion imagination'.

We all know that if you drop a drop of ink into clear water, the ink will spread over time.

This is the phenomenon of diffusion.

As time goes by, substances will spontaneously diffuse from places of high concentration to places of low concentration, whether it is so-called 'tangible' or 'intangible', this phenomenon will occur.

For example, you press a piece of copper and a piece of iron together. After a period of time, you will find that there is copper on the surface of the iron and iron on the surface of the copper. This is also diffusion, but the process is very slow.

The same goes for the sound.

As for the sound from a drum, after the Dirichlet boundary conditions and initial vibration conditions are clarified, and then brought into the time and diffusion equations, the shape and size of the drum can indeed be calculated.

Mathematics is so miraculous that ordinary people find it incredible or even metaphysical, but in mathematics, it can be calculated for you step by step.

Through Professor Zhou Hai's explanation, Xu Chuan roughly understood what the spectral asymptotics of the so-called elliptic operators and the Weyl-Berry conjecture are all about.

To put it simply, you can see the two-dimensional Weyl-Berry (Weyl-Berry) conjecture from the previous "listening to the drum shape".

Mathematicians in the past have proven this, but not the Weyl-Berry conjecture in three dimensions or more complex conditions.

The current demand is whether mathematicians can find a fractal framework, so that the three-dimensional or more complex Weyl-Berry conjecture can be established under this fractal framework, and can make Ω measurable under this fractal framework.

That's the purpose.

As for the specific use of this thing after it is confirmed?

Probably it can be used to study the shape of stars in the universe and the size of the universe. As for others, there should be no one that can be used for this conjecture at present.

But mathematics, to be honest, modern mathematics is actually very far away from the concept of "useful".

It seems difficult to answer the question "why should I study mathematics" without one's own strong, intrinsic interest in mathematics.

Richard Feynman, known as the "universal physicist" in the last century, considered choosing a major in mathematics when he was young.

But when he went to the Department of Mathematics for consultation, he asked, "What is the use of learning Mathematics?".

Then the old professor of the mathematics department told him that since you asked this question, then you do not belong here, you do not belong to the mathematics department.

Then, the big guy went off to study physics.

The distance unit known to all of us as 'nanometer' was proposed by him.

Mathematics is a product of pure abstraction, and definitions and logic are the cornerstones of a mathematical system.

Mathematicians usually don't care about how mathematical concepts and derivations are related to the real world; mathematical conclusions may not necessarily find their prototypes in the real world.

However, with the development of technology and society, some results that were considered to be meaningless will become meaningful.

For example, the "antimatter" he studied in his previous life has a certain connection with the negative root of the quadratic equation that seems useless today.

It's like you learned calculus, but you don't use it at all when you buy groceries and think it's useless.

Kangxi, a famous historical figure, also asked the question of what is the use of calculus.

Later, he probably felt that there was no need to use calculus for "capturing Oboi himself, flattening the San Francisco, taking ww, nine kings seizing the heir, governing the Yellow River, writing stereotyped essays, and cultivating crops", so he felt that there was no need to promote it.

Over time, however, the development and application of calculus has influenced almost all areas of modern life.

Calculus is needed for everything from modern missile flight calculations to taking cold medicine.

Because through the decay law of the drug in the body, calculus can deduce the regular time of taking the drug.

So don't say that mathematics is useless. If mathematics is useless, you can't even take the medicine correctly.