Chapter 71 Weyl-Berry Conjecture
On the podium, Zhou Hai stopped the chalk calculation on the blackboard, frowned at the white writing, turned around, picked up the blackboard eraser, and erased all the mathematical formulas and calculations on it.
He calculated for more than ten minutes and finally confirmed that there was a dead end road.
For the first time since teaching, he was stumped by a student's question. It was incredible.
However, after carefully thinking and studying the topic in the notebook, he felt relieved.
It's normal that he couldn't solve this problem in a short time.
Although the title was handwritten by Xu Chuan, it was actually part of a weakened form proof of a world-class mathematical conjecture.
Even if he devotes himself to research, he may not be able to solve this kind of problem. If he wants to find an idea or directly prove it in a test, it will be as difficult as climbing to the sky.
"Where did you get this question?"
After confirming that he could not solve the questions on the notebook in a short time, Zhou Hai returned to Xu Chuan's seat, returned the notebook in his hand to him, and asked curiously.
As an undergraduate, you have started to come into contact with such things as world conjecture?
Or is it a question he found specially, on purpose?
He has been teaching for more than 20 years and has seen students one after another.
I have seen talented young people and mediocre talents, I have seen people who study hard and move forward step by step, and I have also seen people who are opportunistic and want their tutors to help them write their graduation thesis.
There are many people who ask teachers for super-syllabus questions and difficult questions in order to gain a good impression in front of the teacher and make the teacher feel that they are studying seriously.
Of course, he didn't think Xu Chuan was this kind of person.
But no matter what, when a freshman begins to come into contact with world-class conjectures, it will always make people feel surprised and doubtful.
And it would be okay if he found the questions himself. After all, if many college students are interested in a certain subject, they will look for some questions on the Internet or in the library to try to solve.
But if this topic was conceived by himself during the study process, it would be so shocking and incredible.
Not to mention a freshman, even the graduate students, doctoral students he teaches, and even a university professor may not have such ability.
Study old knowledge, integrate it, and then expand to new boundaries and new problems based on this.
This is something that can only be done by top mathematicians or mathematicians who have studied extremely deeply in a certain field and have almost reached the end.
How can a freshman come this far?
impossible!
Absolutely impossible!
Zhou Hai didn't believe that a freshman could do this, so he asked Xu Chuan where he got this question.
After hearing Zhou Hai's inquiry, Xu Chuan took out "Factorization of Linear Operators and Geometric Properties of Banach Space" from his schoolbag again, turned to the last three chapters, and handed it to him.
"This book contains some descriptions of spectral asymptotic methods and problems on connected regions with fractal boundaries."
"If we remember Nn(r)=#{(Q1,...,Qn)∈"|qi+...+q
"Starting from Theorem 3.1, when combining the second term asymptotically of the Dirichlet spectral counting function to expand the adjacent connected branches of the special non-connected region, we encountered this problem in the notebook."
Xu Chuan briefly explained the source of the problem in the notebook, which attracted a shocked and surprised look from Professor Zhou Hai.
"Is this question really developed by your own research?"
Zhou Hai opened his lips slightly, feeling that his mouth was dry. After swallowing hard, he asked in disbelief.
"What's wrong? Is there any problem?" Xu Chuan raised his head and asked with some confusion.
"Then do you know what the extension of this problem will be?" Zhou Hai asked eagerly.
Xu Chuan shook his head. He really didn't know this. These questions in his notebook were all recorded by him while he was reading and studying. Regarding the area of spectral asymptotic methods and problems on connected regions with fractal boundaries, he had never studied it in his previous life, and he was not sure what these problems would correspond to when expanded.
"It's the Weyl-Berry conjecture!"
Zhou Hai lowered his voice, but there was a hint of trembling and excitement in his tone.
"Weyl-Berry conjecture? Is it a problem in the field of functional analysis?"
Xu Chuan asked doubtfully. He had never heard of this conjecture.
After all, the field of mathematics is too vast. Even if it is a world-class conjecture, there are a lot of conjectures and problems. He was not a math major in his previous life, so it is normal for him not to know some mathematical conjectures.
Zhou Hai dragged a chair over from the side, sat down and continued: "The Weyl-Berry conjecture, the full name is the spectral asymptotic of the elliptic operator and the Weyl-Berry conjecture."
"It mainly studies the spectral asymptotics of elliptical operators, the inverse spectral problem and the construction of spectral fractal regions such as fractal drum theory and the conjectures of nonlinear analytic Gevrey-like microlocal analysis, which are world-class conjectures."
"Of course, it's normal that you don't know this conjecture. It's not as famous as Fermat's conjecture, the four-color conjecture and Goldbach's conjecture, and it's not as difficult as it is."
"If we divide it according to the difficulty of solving the conjecture, it should belong to the level of conjecture between T2 and T3."
"Teacher, can you tell me more about this conjecture?" Xu Chuan asked with interest.
He did know a little bit about the classification of levels mentioned by Zhou Haikou.
Any problem is difficult to solve, and mathematics is no exception.
In the world of mathematics, there are many conjectures and problems.
The most famous and common ones are the seven major millennium mathematical problems such as the 'Riemann Hypothesis', 'Yang-Mills Gauge Field Existence and Mass Interval Hypothesis' and 'P=NP Problem'. These problems are basically T0 level.
There are currently about ten mathematical conjectures and problems at the T0 level.
If you solve any problem, you can get the Fields Medal, and you can go to any university in the world to be a professor or even the director or dean of the mathematics department.
From the T0 level down, the T1 level includes Goldbach's conjecture, the four-color problem, Langlands' reciprocity conjecture, and some of Hilbert's twenty-three questions.
Here I would like to mention Goldbach’s conjecture, which Minke focuses on research. Its difficulty is actually worthy of the T0 level.
But the year before last, in 2013, Harold Hoefgot, a researcher at the Ecole Normale Supérieure in Paris, published two papers, announcing that he had completely proved the weak Goldbach conjecture.
The 'weak Goldbach's conjecture' has been proven, which caused Goldbach's conjecture to lose the relative integrity of the conjecture, so it dropped from T0 to T1 level.
However, this does not mean that the difficulty of solving it has been reduced. In fact, if you purely look at the difficulty of solving it, its difficulty is still at the T0 level.
By the way, most civilians study Goldbach's conjecture because they can only understand it. For other conjectures, even at the T2T3 level, they can't even understand what the title means.
If you solve a T1-level conjecture, you can also win the Fields Medal, and you can also become a professor or even the director or dean of the mathematics department in any university in the world.
Further down, there are mathematical conjectures and difficult questions at T2 and T3 levels.
There are many conjectures about this kind of staircase, and Xu Chuan cannot name every one.
If you insist on talking about it, Model's conjecture derived from Poincaré's conjecture, weak Goldbach's conjecture derived from Goldbach's conjecture, twin prime conjecture, Hilbert's twenty-three questions, etc. can all be put into in this staircase.
As for the Weyl-Berry conjecture mentioned by Zhou Hai, he indeed did not know and had not studied it.
(End of chapter)