The top student must be diligent.

Chapter 256 Geometric Langlands Program

Time passed quietly.

Now that I have decided to study the hail hypothesis, it is naturally not just talk.

So from that day on, Xiao Yi began to focus his main energy on the hail conjecture.

Of course, it had been a long time since he had studied pure mathematics problems, so at the beginning, Xiao Yi even felt a little uncomfortable.

After all, although mathematical tools are indispensable in the process of studying nuclear fusion, for nuclear fusion, the mathematical tools used are "useful", that is, these mathematical tools themselves are prepared for practical applications.

The mathematical tools used in the study of pure mathematics are basically "useless". Although they may be very useful for the study of pure mathematics, they are not very useful in terms of application.

So over the past few years, the mathematical tools Xiao Yi used were all useful, and now suddenly using these useless mathematical tools does make him feel a little uncomfortable.

"Tsk tsk, this is what they call unfamiliarity."

In the school office, Xiao Yi sighed.

However, it is not a problem for him.

As the relevant mathematical knowledge is recalled in the mind, the unfamiliarity will gradually fade away.

"However, this question does have a somewhat different feeling in it."

He simply used up a few sheets of draft paper and placed them on the table. Xiao Yi stroked his chin and raised his brows.

Well, it is indeed very different.

Other mathematical conjectures about prime numbers all involve analytic number theory or algebraic geometry, but the hailstone conjecture, which is also a conjecture in number theory, involves other aspects of mathematics.

For example, the power system.

Of course, this dynamical system refers to the dynamical system in mathematics, not any other dynamical system.

A system that uses functions to describe the changes in the state of a point in space over time is a dynamic system. Of course, there is no problem in applying such mathematical methods to dynamic systems in engineering.

All in all, it's probably an iterative process.

The calculation process of the hailstone conjecture is roughly equivalent to such an iterative process. Each step is transformed according to the parity of the current number, thus forming a whole sequence of digital changes.

In addition, according to the theory of fixed points and periodic orbits in dynamical systems, the hailstone conjecture is equivalent to the fact that for any starting positive integer, it will eventually enter a 4-2-1 cycle, that is, reach a fixed point or a periodic orbit.

So, if we can find such a fixed point or periodic orbit based on such a theory, it will provide very important help in proving the conjecture, or even directly complete the proof.

Of course, it's not just that.

This seemingly simple question may involve more areas.

For example, strange attractors in hybrid theory, topological entropy and complexity, etc.

All in all, this issue is definitely not simple.

Otherwise, some mathematicians would not call it "an extremely difficult problem that is completely beyond the scope of current mathematics" or even believe that "mathematics is not yet ready to deal with such a problem."

"Hmm... So, do we need new mathematical tools as well?"

Xiao Yi gradually fell into deep thought.

Mathematicians believe that this problem is beyond the scope of current mathematics, or that mathematics is not yet ready to deal with such a problem. Basically, this means that the current mathematical community does not have the mathematical tools to solve this problem well.

Therefore, it is necessary to invent new mathematical tools.

"Hmm... maybe the Langlands Program would be a good angle?"

Recalling the progress in mathematics in recent years, three years ago, in 2024, the geometric Langlands conjecture was solved.

Obviously, although he has become the main protagonist in the mathematical world over the years, other mathematicians have not fallen behind.

The geometric Langlands conjecture is a series of conjectures and conclusions obtained by stating the Langlands program in number theory on the function field of algebraic curves.

It is somewhat different from the Langlands Program. It can mainly connect disciplines such as algebraic geometry, representation theory, and even quantum mechanics. It can be said that it plays a very important role in mathematics.

When the paper was first published three years ago, Xiao Yi was invited to serve as a reviewer for this paper.

Unfortunately, he refused because he was working on other important projects at that time.

Now, can the geometric Langlands conjecture, which has been proven, help the study of the hail conjecture?

Although theoretically, the connection between the two is still a bit too weak. One involves highly abstract theories such as algebraic geometry and representation theory, and the other involves relatively specific problems such as number theory and dynamical systems. It seems that there is basically no connection between the two.

However, with mathematics, sometimes there doesn't seem to be much connection between the two, but that's just because you haven't tried.

As the saying goes, how do you know if you don’t try?
So, Xiao Yi started to try.

So putting the hail conjecture aside for the time being, Xiao Yi first found the paper that proved the geometric Langlands program.

If you want to find the relationship between the two, you must first figure them out.

This paper, which proves the geometric Langlands conjecture, is over 800 pages long and can be said to be quite complicated.

Of course, there are nine authors in total for this paper, including a mathematician from China named Chen Lin, who won the IMO gold medal when he was fifteen years old, and is considered a young mathematical genius.

"Well, I have something to do for the next week."

Xiao Yi stretched and then began to read the paper.

An 800-page paper might take a long time for others to read, but for him, it only took a few days.

Especially now that he has the lv7 BUFF, this is not difficult at all. Even for him, a week is a bit slow.

Mainly because he has other tasks.

As the general consultant and chief scientist of the National Energy Administration and the National Fusion Corporation, he has to shoulder the corresponding responsibilities. After all, his two positions are not empty titles. Especially now that they are just starting out, there are still many related things to do, especially with the dozen or so reactors that are being built at the same time.

He always gives relevant suggestions.

However, there are actually not many things that he needs to worry about.

This is all thanks to his original proposal.

After the Sun-Creating Project was completed, so many engineers had to be assigned somewhere; it was impossible for them all to continue staying at the Institute of Controlled Nuclear Fusion.

So, under the arrangement, these engineers went to the places where new nuclear fusion plants were being planned to be built to provide relevant construction experience.

They might be skilled engineers themselves, but they might not be familiar with nuclear fusion engineering or the new magnetic confinement nuclear fusion device such as UO structure at the beginning. But now, after so many years, they have all been brought up by Xiao Yi, and even those who are willing have become members of the National Fusion Company. Now there is also a special title for them, called nuclear fusion engineers.

When these new fusion reactors begin to be built, these engineers will be able to bring out new nuclear fusion engineers. In this way, China will sooner or later have sufficient nuclear fusion engineers. As for now, with these engineers, there will be fewer places where Xiao Yi is needed in the construction of new reactors, which makes his work much easier.

All in all, his main job now is to provide design advice and help new chief engineers check design drawings.

For example, Li Jiangan and Zhao Zhanyou have served as chief engineers for nuclear fusion project construction in the Yangtze River Delta and Beijing-Tianjin-Hebei economic circle respectively.

Zhao Zhanyou has now been promoted, and it has been basically confirmed that he will serve as the deputy general manager of the National Fusion Corporation. As for the related affairs of his original position as the director of the Plasma Research Institute, they have now been completely handed over to another former deputy director, and he is just waiting for the announcement at the meeting.

As for Li Jiangan, although he has reached the age to retire, as he had planned before, he still plans to make use of his remaining energy and try his best to see if he can make further progress before retiring completely, such as taking up some important positions in the National Energy Administration.

As a contributor to controlled nuclear fusion, everyone can be said to have gained something.

Of course, there were also many big moves internationally this week.

First, a number of Western countries, as well as Japan, South Korea, Australia and other countries jointly announced that they would establish a new joint nuclear fusion experimental organization, which would bring together all their relevant scientists to focus on the research of controlled nuclear fusion power generation technology. Some politicians directly stated in an interview: Since China can make it, then we can definitely make it too.

The planned investment amount starts at US$100 billion, and the total investment amount within five years will reach at least US$300 billion.

It looks very real, but I heard that the organization is currently arguing over the contribution ratio.

Then there is ITER, which has been inviting China to return to ITER. Of course, China has basically ignored this invitation and has not made any statement.

Of course, those who have good relations with China are trying to find out what conditions are needed to build a nuclear fusion reactor in their country.

However, for now, China has never indicated that it will build nuclear fusion reactors abroad.

For technology like this, we need to enjoy it more in China first, and then we can do anything about it after people abroad realize that they can’t develop it and start to get anxious.

In short, China is now sitting firmly on the Diaoyutai State, and it all depends on what kind of tricks other countries can come up with.

Just treat it as a show.

Just like that, a week passed.

……

Today, Xiao Yi did not go to school, but came to the Science Island Laboratory.

Anyway, now in Fei City, he can go to many places and have many offices. There are two of them, the school and the Science Island Laboratory, and there is also one each at the Plasma Institute and the Institute of Controlled Nuclear Fusion.

However, his favorite office is still the Science Island Laboratory.

After all, he designed it himself.

"Well, I finally finished watching it."

He put down the stack of papers in his hand, which was almost a hundred pages long.

He printed out the entire paper on the proof of the geometric Langlands conjecture in seven copies, reading one copy a day, and today he finished reading the last copy.

I closed my eyes and reviewed the various key proofs in my mind.

Xiao Yi fell into deep thought.
Although there are a lot of them, with his current memory, he can still recall them. Even if he only has some impressions, he can deduce them with his own abilities.

Until finally, his eyes lit up.

“It seems that we can really find the connection!”

"The geometric Langlands conjecture can realize the relationship between algebraic geometry and representation theory, and I can try to analyze the hailstone conjecture from the perspective of representation theory..."

A whole set of directions to try gradually formed in Xiao Yi's mind.

“For example, I could try to find a suitable group or algebraic structure, and then use the elements of this structure to represent the transformation rules in the hailstone conjecture.”

"Then we can start studying the representation of this group or algebraic structure in some vector space and see if we can find invariants or special vectors in the representation space that correspond to the hailstone sequence..."

"Then we use the classification of irreducible representations, Schur's lemma, etc. to analyze the properties of these invariants or special vectors..."

“At this point, perhaps we can try to combine the methods of the geometric Langlands program.”

"but……"

Xiao Yi quickly realized the problem with his idea.

"The transformation rules in the hailstone conjecture involve the parity of numbers, while classical group theory and representation theory mainly deal with algebraic operations. There is a compatibility issue between the two. How to reasonably combine the two may require some new ideas..."

Just this first step is already quite difficult.

At present, there is no experience in directly using representation theory tools to study the hail conjecture, which is considered a very new direction.

Even if you want to learn from others, you probably won’t be able to find anyone to learn from.

but……

Xiao Yi just smiled indifferently and then started to take action.

Difficulty is always relative.

When this question has an answer, the only thing left is the difference in IQ.

"The generation rule of this hailstone sequence can be temporarily regarded as a variation on the set of non-negative integers. Now, we define two types of generators: for any positive integer n, if n is an odd number, the generator is H_n=(3n+1, n); if n is an even number, the generator is H_n=(n/2, n). Now, we need to find out under what definition of multiplication operation these generators can form an infinite discrete group..."

In this way, Xiao Yi began his own deduction.

One piece of draft paper, two pieces of draft paper, three pieces of draft paper...

When thinking about the real problem, Xiao Yi finally experienced how powerful the lv7 buff was for him.

This seemingly difficult problem was solved by Xiao Yi as easy as cutting melons and vegetables.

"Done. This group, um... let's temporarily name it the Hail Group."

Xiao Yi made a simple decision.

"Next, we need to study the representation of hail groups in complex vector space. This is a bit troublesome..."

Xiao Yi raised his eyebrows slightly, but did not hesitate too much and continued his deduction.

Although it is still in its early stages, we have undoubtedly made a good start.

……

(End of this chapter)