A genius? I just love studying.

Chapter 106 Thesis Topic Selection
In Chengdu, at Happy Valley, Fang Wen, who had just gotten off the roller coaster, was still trembling and covered in sweat.

It was really exciting, but the body's instinctive reactions were simply too much to resist, and shouting on the roller coaster also left him somewhat exhausted.

"Senior brother, my legs are a little weak, could you help me up?"

The junior sister next to him took Fang Wen's arm and leaned half of her body against him.

Woohoo...

Suddenly, Fang Wen's phone vibrated twice.

"What's wrong? Can't we go back after we're done playing?"

The junior sister pouted, looking at Fang Wen who had taken out his phone with some dissatisfaction.

"I've set everyone else to 'Do Not Disturb,' and I'll only get notifications for messages from a few important people. Please wait a moment."

Fang Wen chuckled, having already opened the unread WeChat messages.

[If I want to write a paper, do you have any suggestions?]

"?"

Fang Wen looked at the names on the chat page and was momentarily confused.

Chen Hui?

What kind of academic paper do you, a high school student, need to write?
Is writing a paper something a high school student should be thinking about?

But considering what Chen Hui did, it doesn't seem so surprising.

"Hey, are you writing this paper to get extra points for independent admissions, or are you just trying to delve deeper into academic research?"

After a moment's thought, Fang Wen typed on his phone and replied.

Woo!
The message had barely been sent when the other side replied.

【Is there any difference? 】

"Of course there is a difference. If it's for extra points in independent enrollment, then you can just publish in some low-quality journals. The quality requirements are not high. Anyway, they only look at the impact factor of the journal that publishes your paper, and they won't pay much attention to your paper at all."

Fang Wen's hands traced afterimages across the phone's keyboard. As a seasoned keyboard warrior, even typing on a phone was no slower than typing on a real keyboard. "If you want to do academic research, you need to choose a direction carefully, then study it, and once you have results, you can naturally publish papers."

"Do academic research!"

Chen Hui replied without the slightest hesitation.

If the CMO can rank among the top 60 in the country, he won't need to worry about going to university at all. He now has some confidence in himself.

"Choosing a direction? Could you explain it to me in detail?"

Upon seeing Fang Wen's reply, Chen Hui was also invigorated. He had already searched for a lot of information online, but this point that Fang Wen mentioned was not mentioned online.

This is the drawback of the internet today. It seems to have abundant resources, but in some in-depth areas, without the guidance of experts, you simply cannot obtain useful information. Even if you do get useful information, you cannot distinguish between true and false information, because too much information also means too much noise.

Laymen simply cannot tell the difference.

That's a long story!
Fang Wen didn't reply immediately, but looked at his junior sister beside him, whose lips were already pouting, "Are you hungry? It's getting late, why don't we leave the valley first?"

"I remember there's a McDonald's just outside the valley, let's go grab something to eat!"

Upon hearing this, the junior sister, who was already somewhat displeased, immediately darkened her expression. "Who is it?"

Is replying to his messages more important than hanging out with me?

Fang Wen smiled wryly and handed over the phone.

"Chen Hui?"

The junior sister was astonished, "You know Chen Hui? Is it the Chen Hui we know?"

If it was the Chen Hui she knew, she couldn't figure out why her senior brother Fang Wen would be in contact with that person.

Fang Wen nodded without answering, but instead opened his chat history with Chen Hui.

"He wants to write a thesis?"

The junior sister raised her eyebrows, her face full of absurdity.

After hesitating for a moment, Fang Wen seemed to have made a decision. He reached out and swiped on the screen to the beginning of his chat history with Chen Hui, then released his hands, indicating to his junior sister to check it herself.

A dozen minutes later, the junior sister returned the phone to Fang Wen with a shocked expression.

"My God, is this even human?"

The junior sister muttered to herself, then looked at Fang Wen and took his hand. "Let's go. We can go to the amusement park anytime, but we don't get many opportunities to talk to big shots like this. Let's go eat McDonald's!"

She could understand what Chen Hui and Fang Wen were discussing because she also studied this field. Precisely because she could understand it, she understood its value even more.

She found it hard to imagine that this was something a high school student could understand.

Normally, shouldn't high school students have never even heard of these things?
At least, she had never heard of these terms before she went to college, or even before her junior year.

Even now, she still doesn't quite understand these things.

She's already 22 this year!
Fortunately, she had a good attitude. She went into McDonald's, ordered a bucket of chicken wings, and enjoyed them immensely. Her mood immediately improved.

Fried chicken is awesome!

Across from her, Fang Wen had no appetite for the cheese and beef stew in front of him, his thumbs frantically tapping on his phone.

"If you want to do academic research, there are many things to consider. For example, do you want to do pure theoretical mathematics or applied mathematics?"

Before Chen Hui could ask any questions, Fang Wen continued to explain, "If it's purely theoretical mathematics, then the two pearls are naturally for solving unsolved problems or developing new theories and tools."

Unsolved problems are easy to understand, such as the famous Millennium Problem, the Riemann Hypothesis, and Yang-Mills theory. However, these are all too advanced. There are many unsolved problems in mathematics that are not so difficult, such as Goldbach's Conjecture, which is very popular in China, and the Guagu Conjecture, which was recently solved by Chinese mathematicians.

Developing new theories and tools is easy to understand, such as Newton and Leibniz's invention of calculus, and Grothendieck's development of probability theory and category theory. "Of course, these are things researched by top scientists; the difficulty is too high. Even if I had five hundred years, I wouldn't find a breakthrough. But in these two directions, once you achieve results, they'll be at least at the top four levels, and you could even win numerous major awards. Having one such result could be enough to support you for a lifetime."

"However, snakes have their paths, rats have their ways, and we ordinary people also have our ways of survival."

We can extend existing theories to a wider range, or discover new perspectives on old conclusions, reinterpreting and generalizing them. For example, the extension of non-Euclidean geometry to Euclidean geometry, the extension from classical group theory to quantum groups, and the extension of real number analysis tools to the p-adjacent number field...

Of course, the examples I gave are all the achievements of experts. We can start with the simple ones, such as extending the concept of absolute value from real numbers to complex numbers...

In addition, it can be used to systematically classify certain types of mathematical objects or characterize their internal structure, such as the classification theorem for finite simple groups, and the representation classification of compact Lie groups...

We can also try optimization and simplification. The famous Terence Tao used harmonic analysis to simplify the prime number spacing problem after Zhang Yitang published the twin prime theorem.

The twin prime theorem proved for the first time that there are infinitely many pairs of prime numbers with a gap of less than 7000 million. However, Tao Shen used harmonic analysis to reduce this gap from 7000 million to 246!

This achievement is also at the fourth level!

Furthermore, it can reveal hidden mathematical structures or phenomena within known objects, such as discovering Mandelbrot sets in fractal geometry, or the connection between modular forms and elliptic curves...

It can even explore the foundations of mathematics and philosophy, all of which can yield published results, such as the impact of Gödel's incompleteness theorems on the foundations of mathematics.

Other examples include constructive proofs, which directly construct objects that satisfy the conditions without relying on existence theorems; and counterintuitive phenomena and paradoxes, which reveal mathematical phenomena that defy intuition and challenge traditional cognition, such as the Banach-Tarski paradox.

These are all areas where papers can be published, but of course, these are niche research areas. The mainstream of mathematical research remains the six directions mentioned earlier!

After typing so many words in one go, Fang Wen was exhausted. He put down his phone, picked up the iced cola on the table, and took a big gulp.

Seeing this, the junior sister smiled and handed Fang Wen a French fry as he put down the cola.

After taking a bite, Fang Wen felt refreshed and full of energy.

Then, suddenly remembering something, he slapped his forehead and picked up his phone again. "I forgot to mention, besides pure mathematical research, we can also try interdisciplinary research, applying mathematics to other disciplines such as physics and chemistry."

For example, Einstein's field equations are an application of differential geometry in general relativity, as are the applications of number theory in cryptography, and the recently popular condensed matter physics—all are closely related to mathematics.

Upon seeing the long string of text messages sent on WeChat, Chen Hui sincerely replied with a thank you and a fist-clasped emoji.

Having an expert to guide him makes all the difference; he suddenly felt the path ahead become clear.

Of course, the other party has already helped him a lot, and the choice he makes next is up to him.

Knowing yourself and your enemy is the key to winning every battle.

With the direction established, the next step is naturally to understand the specifics of each direction.

Chen Hui typed "Millennium Problem" into his browser, and soon the search results popped up on the page, including the P vs. NP problem, the Riemann Hypothesis, the Yang-Mills equations, and a brief introduction to these conjectures.

Chen Hui was quickly drawn to the description below: "Currently, only the Poincaré conjecture has been fully solved. Perelman's proof underwent rigorous peer review and was confirmed in 2006, but he declined the $100 million prize."

"He turned down a $100 million prize?"

"Where did the bonus come from?"

Chen Hui eagerly typed his question into the search box.

Soon, the truth came to light!
It turns out that these so-called Millennium Seven Problems were designated by the Clay Institute in the United States. They promised that anyone who solved any of the conjectures, as long as the solution was published in a mathematics journal and underwent a two-year verification period, would be awarded a one million dollar prize!

That's a million dollars!
That's equivalent to over seven million RMB!

This is not only enough for him to buy a house in Chengdu, but also enough for Rui Rui and Qing Shan to live a carefree life!
Proof is necessary!
Chen Hui felt his blood boiling, and a strong impulse surged in his heart!

He now has a clear vision of the future.

Chen Hui then searched for a large amount of information online.

But he still didn't quite believe the news on the internet, so he picked up his phone and sent Fang Wen another message.

Can you really win a million-dollar prize for solving the Millennium Puzzle?

Fang Wen, who had just finished eating the cheese and beef burger in front of him, looked at WeChat with a head full of question marks. Was he going to solve these conjectures that no one had been able to prove for hundreds of years?
Did he think the Millennium Problem was one plus one equals two?

For centuries, countless mathematical geniuses with extraordinary talent have failed and perished...

Even though Chen Hui has demonstrated remarkable mathematical talent, it is still far from enough to solve the remaining six Millennium Prize Problems.

Fang Wen had countless complaints in his mind, but in the end he only replied with two words - Yes!
"it is true!"

The moment he received a positive reply, Chen Hui felt full of energy.

He didn't know how others fared, but with his proficiency panel, he knew that if he wanted to, he could definitely succeed!
Open your browser and start searching for relevant knowledge. First, learn about these conjectures, then choose one that you think you can prove, and keep learning until you pluck that pearl!

After briefly learning about the details of the six conjectures, Chen Hui did not hesitate for long before choosing the Yang-Mills theory!
Because the person who proposed this theory was Chinese.

In fact, this description of the problem is inaccurate. A more accurate statement would be the existence of the general solution to the Yang-Mills equation and the mass gap assumption.

As a set of nonlinear partial differential equations, the classical form of the Yang-Mills equations has not yet found a globally rigorous solution. Although physicists have verified particle interactions through approximation methods such as perturbation theory and numerical simulations, the mathematical proofs of the existence, uniqueness, and convergence of the solution are still incomplete.

Although obtaining an approximate solution can make Yang-Mills theory work to some extent, finding an accurate solution is of great significance. To give a simple example, we all know that the closer the charges are, the stronger the electromagnetic force between them. So when the distance between the charges approaches zero, does the electromagnetic force become infinite?

Only by thoroughly solving the Yang-Mills equations can we accurately find the laws describing electromagnetic force, strong force, and weak force, just as Maxwell's equations accurately describe electromagnetic force and Newtonian mechanics describes gravity.

As for the mass gap, a more rigorous description is that for any compact gauge group, the quantum Yang-Mills field must have a positive mass gap.

This is because, in Yang-Mills theory, gauge bosons should be massless in order to achieve long-range effects. However, experiments have shown that the strong gluon and the weak W/Z bosons are short-range forces with mass. This contradiction tells us that although the current Yang-Mills theory can predict the laws of the three major forces well, it is incomplete.

Only by solving the existence problem of the Yang-Mills equation and the mass gap problem can the Yang-Mills theory be complete and truly unify the three major forces!

So this problem actually consists of two parts.

Proving the existence, uniqueness, and convergence of the general solution to the Yang-Mills equation is almost a purely mathematical problem, which made Chen Hui very satisfied, at least he was not completely without a foundation.

As for the issue of mass gaps, we can only study it after we have accumulated more experience. We could even consider improving our physics proficiency.

That's it, the Yang-Mills theory!
(End of this chapter)