Chapter 72 Can you recognize the shape of a drum?
Zhou Hai dragged a chair from the side and sat down, ready to discuss this matter with Xu Chuan.
That's right, it's communication, not instruction.
In his opinion, Xu Chuan's mathematical ability, which can study branch problems of weak Weyl-Berry conjecture, has reached a certain level.
"The origin of the Weyl-Berry conjecture comes from mathematician Mark Cuck in 1966. In a lecture that year, he raised a question that will go down in the history of science: 'Can anyone tell the shape of a drum from the sound? '"
"You can hear the shape of a drum through the sound? Is this possible?" A classmate next to Xu Chuan 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 not only impart knowledge during class, but also often chat with students.
After all, students are young, and sometimes their thinking on problems is very special and can bring unexpected surprises.
Moreover, it is far more useful to use some stories to prompt students to be curious about a certain field and get them into a learning state than to force knowledge onto them. This teaching method is also more suitable for universities.
"Mathematically speaking, stretching a membrane onto 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."
"Through these different sounds, you can really determine the shape of the drum."
"This involves the research of two mathematicians, Alan Connors and Walter van Suilekom."
"They extended the traditional framework of noncommutative geometry to handle spectral truncation of geometric space and tolerance relations that provide coarse-grained approximations of geometric space at finite resolution, and exploited the spectral truncation of circles to define a system of operators A propagation number that is proven to be an invariant under stable equivalence and can be used to compare approximations in the same space. ”
"Under this framework, we can describe the vibration of the 'drum' when it is struck through the wave equation. At the same time, because the edge of the 'drum head' is firmly attached to the rigid frame, we can think of the boundary of the wave equation The conditions are Dirichlet boundary conditions ”
"With these two pieces of data, and using diffusion equations and other methods, we can calculate the shape of the drum through the sound it makes, even if you have never seen it."
Zhou Hai explained with a smile, but directly said that the students who came to listen to the excitement were confused.
What is the spectral truncation of geometric space? What is the spectrum cutoff of a circle?
They all know what it means to distinguish the position by listening to the sound, but they have never heard of distinguishing the shape by listening to the sound.
Can mathematics really do this? It is not metaphysics!
You can tell what happened just by counting with your fingers. Isn’t this outrageous?
Xu Chuan, on the other hand, probably understood what Zhou Hai meant.
The so-called "distinguish the shape by listening to the drum" is actually the problem of the eigenvalue of the Laplacian operator in a region.
To ‘listen to drums and identify shapes’ through mathematics is related to another concept.
That is ‘diffused 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 with high concentration to places with low concentration. This phenomenon will occur whether it is so-called 'tangible' or 'intangible'.
For example, if you press a piece of copper and a piece of iron together, after a period of time, through instrument testing, 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 quite slow.
The same goes for sounds. As for the sound emitted by a drum, after clarifying the Dirichlet boundary conditions and initial vibration conditions, and then incorporating the time and diffusion equations, the shape and size of the drum can indeed be calculated.
Mathematics is so magical. Things that ordinary people find incredible or even metaphysical can be calculated for you step by step in mathematics.
Through Professor Zhou Hai's explanation, Xu Chuan roughly understood what the so-called spectral asymptotics of elliptic operators and the Weyl-Berry conjecture were all about.
To put it simply, you can turn the previous "listening to distinguish drum shapes" into a two-dimensional Weyl-Berry conjecture.
Mathematicians in the past have confirmed this, but not the Weyl-Berry conjecture in three dimensions or more complex conditions.
The current need 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.
This is the purpose.
As for what specific use this thing can be used 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 other things, there should be no practical applications for this conjecture at present.
But as for mathematics, to be honest, modern mathematics is actually very far away from the concept of "useful".
If a person does not have a strong, intrinsic interest in mathematics, it seems difficult to solve the question "Why should I study mathematics?"
When Richard Feynman, known as the ‘universal physicist’ in the last century, was young, he considered majoring in mathematics.
But when he went to the mathematics department for consultation, he asked, "What's 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, this big guy went to study physics.
It was he who proposed the unit of distance known as 'nanometer' as we all know it today.
Mathematics is a product of pure abstraction, and definitions and logic are the cornerstones of the mathematical system.
Mathematicians usually do not care about how mathematical concepts and derivation are related to the real world; mathematical conclusions may not be able to find prototypes in the real world.
However, with the development of technology and society, some results that were originally considered to have no practical significance will become meaningful.
For example, the "antimatter" he studied in his previous life has a certain connection with the negative roots of the quadratic equation, which seems useless today.
It's like you learned calculus but don't use it when buying groceries and think it's useless.
The famous historical figure Kangxi also asked the question what is the use of calculus.
Later, he probably felt that none of the tasks of "capturing Oboi, pacifying the San Francisco, conquering the WW, seizing the nine kings' direct descendants, regulating the Yellow River, writing eight-legged essays, and cultivating crops" required the use of calculus, so he felt that there was no need to promote it.
However, over time, the development and application of calculus has affected almost all areas of modern life.
Calculus is required for everything from modern missile flight calculations to taking a cold medicine.
Because through the decay pattern of drugs in the body, calculus can deduce the regular time for taking medication.
So don’t say that math is useless. If math is useless, you won’t even be able to take medicine at the right time.
(End of chapter)