Zhou Hai pulls a chair from the side and sits down, ready to exchange this with Xu Chuan.
Yes, it is communication, not guidance.
In his opinion, Xu Chuan's mathematical ability to study the bifurcation problem of weak Weyl Berry conjecture has reached a certain level.
"The source of Weyl Berry's conjecture comes from Mark Kark, a mathematician in 1966. In a lecture that year, he raised the question of the history of science: 'Can anyone hear the shape of a drum from the sound?'"
"Listen to the shape of the drum through the sound? Can this also be done?" asked a curious classmate near Xu Chuan.
Zhou Hai smiled, and did not mind students interrupting his speech. College 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 their thinking about problems can sometimes be very special, which will bring unexpected surprises.
Moreover, it is more useful to stimulate students' curiosity in a certain field through some stories than to force them to learn knowledge. This teaching method is also more suitable for universities.
"From a mathematical point of view, a membrane is stretched and sleeved on a rigid support, thus forming a two-dimensional drum."
"Drums of different shapes will produce sound waves of different frequencies when they are struck, so they will 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 Suelekom."
"They extended the traditional framework of noncommutative geometry to deal with the spectral truncation of geometric space and the tolerance relationship of providing coarse-grained approximation of geometric space under finite resolution.... They also defined a propagation number for the operator system by using the spectral truncation of a circle, and proved that it is an invariant under stable equivalence, and can be used to compare approximations in the same space."
"Under this framework, through the wave equation, we can describe the vibration of the 'drum' when it is struck. At the same time, because the edge of the 'drum surface' is firmly attached to the rigid shelf, we can think that the boundary condition of the wave equation is the Dirichlet boundary condition."
"With these two pieces of data, through diffusion equation and other methods, we can calculate its shape through the sound of the drum, even if you haven't seen it."
Zhou Hai explained with a smile, but he said that he was stunned to listen to the lively students.
What is spectral truncation in geometric space? What is the spectral truncation of a circle?
They all know what it means to listen to sound and identify position, but they have never heard of how to listen to sound and identify shape.
Can mathematics really do this? It's not metaphysics!
You can tell what happened by pinching your fingers. It's a bit far away, isn't it?
It was Xu Chuan who understood the meaning of Zhou Hai.
The so-called "listening to drums and distinguishing shapes" is actually the eigenvalue problem of Laplacian operators in a region.
Another concept is related to "listening to drums and distinguishing shapes" through mathematics.
That is' diffusion imagination '.
We all know that if a drop of ink is dropped into water, the ink will spread over time.
This is the diffusion phenomenon.
As time goes by, substances will spontaneously diffuse from places with high concentrations to places with low concentrations, whether 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, you will find that there is copper on the surface of iron and iron on the surface of copper. This is also diffusion, but the process is very slow.
The sound is the same.
The sound from a drum, after the Dirichlet boundary conditions and initial vibration conditions are clear, can be calculated by the time and diffusion equation.
Maths is so magical that ordinary people think that things that are inconceivable or even metaphysical can be calculated for you step by step in mathematics.
.......
Through professor Zhou Hai's explanation, Xu Chuan basically understood what is the so-called spectral asymptotic of elliptic operators and Weyl Berry conjecture.
To put it simply, you can see the two-dimensional Weyl Berry conjecture from the previous "listening to sound and identifying drum shape".
Mathematicians in the past have confirmed this, but have not confirmed the Weyl Berry conjecture under three-dimensional or more complex conditions.
The current demand is whether mathematicians can find a fractal framework to make the three-dimensional or more complex Weyl Berry conjecture hold true under this fractal framework, and to make partial Ω measurable under this fractal framework.
This is the purpose.
As for what's the specific use of this thing after it is confirmed?
It may be useful to study the shape and size of stars in the universe. As for others, the conjecture that can be used in practice should be gone at present.
But mathematics. To be honest, modern mathematics is very far away from the concept of "usefulness".
If a person does not have a strong internal interest in mathematics, it seems difficult to solve the problem of "why should I study mathematics".
Richard Feynman, who was known as an 'all-round physicist' in the last century, once considered taking mathematics as a major when he was young.
But when he went to the math department for consultation, he asked, "What's the use of learning math?".
Then the old professor of the mathematics department told him that since you asked this question, you don't belong here, you don't belong to the mathematics department.
Then the big man went to learn physics.
Now we all know that the distance unit of 'nanometer' is proposed by him.
Mathematics is the product of pure abstraction, and definition and logic are the cornerstone of mathematical system.
Mathematicians usually do not care about the relationship between mathematical concepts and derivation and the real world; Mathematical conclusions may not be able to find prototypes in the real world.
However, with the development of science, technology and society, some results that were previously considered to be meaningless will become meaningful.
For example, the "antimatter" he studied in his last life has a certain connection with the negative root of the quadratic equation that seems useless today.
It's just like you learned calculus, but you don't use it when you buy vegetables.
The historical celebrity Kangxi also asked the question of what is the use of calculus.
Later, he probably felt that none of the 'catching and worshiping oneself, leveling San Francisco, closing the WW, nine kings seizing the throne, governing the Yellow River, writing eight part essays, and cultivating crops' needed to be used in calculus, so he didn't think it was necessary to promote them.
However, with the passage of time, the development and application of calculus has affected almost all fields of modern life.
From modern missile flight calculation to taking a cold medicine, calculus is needed.
Because through the decline law of drugs in the body, calculus can deduce the regular time of taking drugs.
So don't say math is useless. If math is useless, you can't even take medicine.
......