The top student must be diligent.

Chapter 258 Key Inspiration

This response from the Ministry of Foreign Affairs is certainly not because China is being kind or anything.

Because everyone knows that it is impossible for the United States to really invite China to help build it.

After all, the members of the JNEO organization are basically NATO and a number of countries that have relatively friendly relations with the United States. If they really start building research institutes and reactors, it would be equivalent to building a headquarters. If they invite China to help build it, wouldn’t it be a joke?

He will certainly be ridiculed by various media at that time. At least for the US president, the media under the opposition party will definitely laugh at him on this matter, saying that he obviously wanted to build a nuclear fusion reactor belonging to "democracy", but in the end he found the enemies of "democracy" to help build it.

Although he no longer has another term in office, such things will inevitably affect his party. After all, he still has to provide help for the party's next election.

Therefore, they will definitely only choose their own people to do this project.

But at this time, another question arises: if they are allowed to build the project themselves, can they complete it as soon as possible?

When it comes to strength in the field of infrastructure, no country in the world can surpass China.

That’s why the Sundial could be completed so quickly.

As for this JNEO organization...

That’s hard to judge.

Refer to the original ITER. The construction speed of ITER was slow, which was largely due to the fact that the construction speed was extremely slow. After a very long time of construction, not much was built.

If the United States wants to launch nuclear fusion experiments as soon as possible, it will probably have to spend a considerable amount of money on the construction costs.

Therefore, faced with the completely sarcastic response from the Ministry of Foreign Affairs, although many people in the US government were devastated, there was nothing they could do.

This is not World War II, and their workers are no longer the same workers who worked in Los Alamos.

……

Time flies and it is already mid-April.

HKUST, in Xiao Yi's office.

Xiao Yi was studying his own problems in his office as usual.

He doesn't need to worry about what happens in the international arena, and the United States' plans have no impact on him at all.

Over the past month, his research on the hail hypothesis has reached a fairly in-depth level.

"The representation of hail groups in complex vector space has now been completely figured out..."

Xiao Yi looked at the description of the hail group representation on the draft paper: [For each positive integer n, there exists a vector space V_n of dimension n, and defines the action of the hail group on V_n: if H_k is a generator, then it maps the kth basis vector in V_n to the 3k+1th (if k is odd) or k/2th (if k is even) basis vector, and the other basis vectors remain unchanged. ]

"Then, the characterization of the invariant vectors has been completed. In addition, in the representation space V_n, I just found a special type of vector that can remain unchanged under the action of the hail group, and proved that these invariant vectors correspond one-to-one to the periodic orbit of the hail sequence. In particular, this special vector directly corresponds to the invariant vector of the 4-2-1 cycle, which exists and is unique in every representation space."

Thinking of this, Xiao Yi smiled slightly.

Just getting to this point is already far ahead of the mathematics community.

This is because the mathematical community has not yet reached a very advanced level in research on this issue.

The most advanced achievement is probably that of Terence Tao in 2019.

He had used the method of logarithmic density to prove that in the sense of logarithmic density, almost all hailstone orbits would fall below the starting point for any given function, provided that the function diverged to infinity, no matter how slowly.

In a sense, his proof is equivalent to saying that almost all natural numbers conform to the Collatz conjecture.

However, this proof may be sufficient in physics, but for mathematics, almost all is never "all".

Just like in mathematics, infinity is not equal to everything.

However, despite this, Tao's achievement is still considered one of the most important achievements in the field of Collatz conjecture.

Now, although the law found by Xiao Yi has not obtained "almost all" the results like Terence Tao, it can be regarded as another path or direction leading to the final proof.

When proving mathematical problems, direction is very important.

Just like a tree, there may be quite a lot of branches, but there is only one fork that can lead to the highest point.

Of course, there may be more than one "highest point" in mathematics, but by the same token, once you go in the wrong direction, you may not reach the final answer in the end.

"But now the last question is..."

“Is there a unified method to generate invariant vectors in every representation space?”

Xiao Yi fell into deep thought.

This is the most difficult problem in the following research.

He has been thinking about this issue for almost two weeks.

The other problems mentioned above only hindered him slightly, but this problem alone made it difficult for him to find a good idea.

“Perhaps, we should try other angles?”

Xiao Yi thought about it for a moment.

However, at this moment, there was a knock on the office door.

"Please come in."

He looked at the door, and then Liang Qiushi opened the door and walked in.

"Teacher! I can't take it anymore! Please help!"

As soon as he came in, Liang Qiushi shouted hurriedly.

Xiao Yi asked with a puzzled look: "What's wrong?"

Liang Qiushi said helplessly: "I have encountered the last problem in my thesis and I still don't know how to solve it!"

"I can only ask the teacher for help!"

Xiao Yi burst out laughing and said, "Wow, you only came to me now. There's only one and a half months left before the deadline."

Liang Qiushi clasped his hands together, leaned forward and said, "I really have no other choice, teacher, this question is too difficult!"

Xiao Yi shook his head and said, "Okay, okay, I told you when you chose the topic that your question was difficult. To be honest, the fact that you are stumped now is beyond my expectation."

Hearing what the teacher said, Liang Qiushi chuckled and seemed a little proud.

Xiao Yi didn't care about him and said, "Okay, show me where you are stuck now."

"Ok."

Liang Qiushi then took out a stack of draft paper from his bag and began to show it to Xiao Yi.

"That's the problem..."

"I still can't figure out the proof that the friendly measure μ on the function field F^n must be a non-uniform strong extremum, and the proof of the extension to the non-homogeneous Baker-Sprindzuk conjecture."

He looked quite embarrassed, and it was obvious that this problem had stumped this mathematical genius.

Xiao Yi took it and carefully read Liang Qiushi's proof. Liang Qiushi's thesis topic involves generalizing the method of non-homogeneous Diophantine approximation in the real number field to the function field.

Diophantine approximation theory has always been a very difficult topic in the field of number theory. It studies the approximation of rational numbers or algebraic numbers to real numbers, as well as the related measurement theory and counting theory. It has a very long research history, which can be traced back to the research work of Diophantus in ancient Greece. Similarly, it has always been one of the topics that the mathematical community has studied over and over again.

Liang Qiushi's topic is of course quite difficult.

It may be quite difficult for an average doctoral student to understand this paper.

So when Liang Qiushi chose this topic, Xiao Yi tried to persuade him.

What I didn't expect was that he was able to research this topic to such an extent, which was quite surprising.

Moreover, Liang Qiushi had actually completed a result before. He gave the equivalent properties between extreme value measure and non-uniform extreme value measure, which can be regarded as the transfer principle in the non-homogeneous case.

And this achievement is already sufficient to be completed as a master's thesis. Publishing it in the first district is a piece of cake. Even being rated as an outstanding thesis is completely casual.

Liang Qiushi completed this achievement last year, so he was not satisfied with that result and continued to conduct in-depth research, eventually reaching this level.

Xiao Yi took a quick look and then said, "I remember that you seemed to have proved before that the friendly measure is strongly contracted, right?"

"Yes." Liang Qiushi asked doubtfully, "But what does this have to do with the current problem?"

Xiao Yi smiled and said, "Since you have completed this step, what should you do next? Actually, you just need to think about it carefully."

"For example, think more about your previous results, the equivalence between extreme value measures and non-uniform extreme value measures. In other words, your method can also prove that strong extreme values ​​and non-uniform strong extreme values ​​are also equivalent."

After hearing Xiao Yi's reminder, Liang Qiushi's eyes flashed with a thoughtful gleam, and then he entered a state of thinking.

As time passed, he suddenly realized something and said in surprise, "Transfer principle!"

"Yes, that's it." Xiao Yi snapped his fingers. "Now, we already know that the friendly measure μ is strongly extremal. This is the result of Ghosh's proof, which you have already cited in your previous results. On the other hand, μ is also strongly contractive. So, you know what to do next."

Liang Qiushi nodded repeatedly: "I understand."

But then a puzzled expression appeared on his face again: "So how do we solve the non-homogeneous Baker-Sprindzuk conjecture next?"

Xiao Yi smiled slightly and said, "For a conjecture, we must first figure out what kind of problem this conjecture is asking us to discuss."

"For Y∈F^(m×n) and θ∈F^m, we define the inhomogeneous Diophantine index ω(Y,θ) to be the supremum of the real number ω that allows the system of inequalities to be solved under arbitrarily large T."

"Here, the system of inequalities describes the approximation of Yq+p+θ to q. Similarly, we can define a multiplicative version of the inhomogeneous exponential ω×(Y,θ)."

"This conjecture shows that the transfer principle from homogeneous approximation in the almost everywhere sense to non-homogeneous approximation for arbitrary translation θ is valid under certain conditions, which reflects the pursuit of the mathematical community for the extension of Diophantine approximation theory from homogeneous to non-homogeneous."

"Now that we have proved that the friendly measure is non-uniformly strongly extremal over the function domain, the key is to establish the connection between the natural measure and the friendly measure on the analytic non-degenerate manifold."

"But actually, by now, this problem is already very clear. Here I recommend a paper to you. If you go back and read it, you will probably be able to figure it out."

Then Xiao Yi turned on the computer, searched for a while, and finally found a paper.

“Yes, it’s this paper, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, from 2007.”

Xiao Yi sent the link of the paper to Liang Qiushi on WeChat.

"Please read this essay carefully later. If you still don't understand it after reading it, come back to me."

Liang Qiushi's eyes suddenly widened. He didn't expect that his teacher solved his problem almost effortlessly and even directly recommended a paper to him.

And it’s from 2007?

Is this the strength of the youngest mathematical master in history?
He expressed his gratitude sincerely: "I understand. Thank you, teacher."

Xiao Yi nodded, then said: "Okay, do you have any other questions?"

"No more, no more. If you have any more questions, I will come to you again." Liang Qiushi shook his head repeatedly.

"Well, remember to finish your paper early. Don't forget that the final draft must be submitted before June."

"Sure, sure!" Liang Qiushi gestured OK: "Don't worry, I've already written down the previous results. If the content of this expansion is not completed, I will send the previous results to you at that time. It doesn't matter."

Xiao Yi rolled his eyes: "Then you might as well publish your previous results as soon as possible."

Liang Qiushi waved his hand, said nothing, said goodbye and left the office directly.

Only Xiao Yi was left again.

However, at this moment, Xiao Yi recalled the process of instructing Liang Qiushi just now, and a thoughtful look appeared on his face.

"Is Diophantus approaching?"

"What if we use the Diophantine approximation to study the generation of invariant vectors?"

He pondered a little in his mind.

"Convert the problem into a problem of solving a system of Diophantine equations?"

The inspiration in my mind began to jump up, and various thoughts began to collide with each other.

Eventually, his intuition told him that this idea had potential!

"If we fix a set of bases, represent the linear transformation ρ(g) by the matrix A_g, and represent the vector v by the coordinate vector (x_1, ..., x_n), then the invariance condition ρ(g)v=v becomes a set of linear equations."

He picked up the draft paper next to him and started writing on it.

[A_g (x_1,…,x_n)^T =(x_1,…,x_n)^T, g∈G]

"If G is finitely generated, say by g_1, ..., g_m, then the above condition is equivalent to a system of Diophantine equations..."

At this moment, Xiao Yi's eyes gradually lit up.

“It seems that it really works!”

As long as it is converted into a system of Diophantine equations, he can use the Diophantine approximation method to directly give a constructive solution!

And now, he is very close to this goal.

"Liang Qiushi, Liang Qiushi, I didn't expect you could give me some inspiration."

Xiao Yi sighed.

Although it was not a direct inspiration, it can only be said that when he was thinking about the theory of Diophantine equations just now, he inadvertently triggered this inspiration.

Now, it has become a little easier for him to get such inspiration.

Even...it can no longer be called inspiration, but rather...common feeling?
"Okay, now it's time to tackle this step."

Xiao Yi raised his eyebrows slightly, and then continued with his research.

He could foresee that once this step was resolved, the remaining obstacles would become quite simple.

……

(End of this chapter)