The Science Fiction World of Xueba

(5,000-word chapter, the second half has a lot of mathematics history science content, I originally wanted to skip it directly, after all, the nearly 2,000-word content may affect everyone’s reading experience, but think about it carefully, let’s keep it , everyone finds it interesting, then take a closer look, if you find it boring, you can skip it directly)

Of course, Pang Xuelin understood why Liu Tingbo was so excited. In the domestic mathematics field, Peking University was the champion, while Fudan and Tsinghua University were slightly behind.

In recent years, Jiang University's Department of Mathematics has developed moderately in China, and few particularly influential mathematicians have been produced. Pang Xuelin is already one of the best.

If Pang Xuelin proves the BSD conjecture this time, his academic achievements will be enough to form a school of his own in the domestic mathematics circle.

At that time, as long as Pang Xuelin is still in Jiangda, it will be a towering tree that can compete with Peking University Mathematics Institute. Do you know how many outstanding young scholars have been produced in Shuimu's Yau Mathematics Center in the past few years since Qiu Chengtong came to Shuimu? .

Although there is still a certain gap between Shuimu's mathematics department and Peking University, it is no longer inferior to Fudan, an old and powerful mathematics school in China.

The existence of Pang Xuelin will play a similar role to that of Qiu Chengtong.

And this BSD conjecture report meeting will be the first shot fired by Jiang Da Mathematics Institute in the international mathematics community!

In the next few days, there was a dark tide in the mathematics world, and Pang Xuelin's paper on arXiv entered the field of vision of most mathematicians who paid attention to this field.

Even in some domestic university forums, there have gradually been discussions about Pang Xuelin's proof of the BSD conjecture.

"Have you heard? The BSD conjecture was proved by a professor named Pang Xuelin of Jiang University!"

"Who is Pang Xuelin, I've never heard of it!"

"Didn't you read the news? A talented mathematician who has emerged in China in recent years. He graduated with a doctorate at the age of 22 and published a top journal paper. He studied under Professor Tao Zhexuan. He returned to China this year and went directly to Jiangda University as a professor and doctoral supervisor."

"Looking at the resume is quite awesome, but the BSD conjecture is one of the seven major problems of the millennium, why does it feel unreliable!"

"I heard from my professor that the paper is still in the peer review stage, and it is said that it has been approved by many big names."

"Who are you teaching?"

"Aksayi Venkatesh! The winner of the Fields Medal in 2018. His main research fields are analytic number theory, topology, and representation theory."

"Fuck, it turned out to be Stanford's senior brother, senior brother, please be respected by my younger brother!"

...

Discussions like this are endless, and there are even similar rumors on the campus of Jiang University.

On the morning of September 11, 2019, today is Pang Xuelin's first class since joining Jiangda University.

At seven o'clock in the morning, he arrived at the auditorium early.

Class time is eight o'clock. When Pang Xuelin arrived, the classroom was empty. Pang Xuelin found a seat in the first row and sat down. Then he turned on the computer and continued to write the PPT for the report meeting.

About twenty minutes later, Pang Xuelin suddenly felt a scent of fragrance coming from his nose. When he raised his head, he saw a tall, tall, beautiful girl with a high nose, deep eyes, and delicate face, who looked a bit like a mixed race sitting in the first row of the classroom. Down.

The classroom they were in was a large lecture hall. The two sat in the first row across the aisle, Pang Xuelin sat on the left side of the aisle, and the girl sat on the other side across the aisle.

The two looked at each other, and the girl smiled slightly at Pang Xuelin and said good morning.

Pang Xuelin nodded to her, but did not speak.

After a while, the girl saw Pang Xuelin typing on the keyboard, got up and walked to Pang Xuelin's side, and asked curiously: "Student, what are you doing?"

"Make a PPT!"

"Are you going to make a report? By the way,

Why haven't I seen you before, and you're here for class too? Today is Pang Shen's first class, and many students from other majors come to worship the God of Learning. "

Pang Xuelin was slightly taken aback, then raised his head and said, "What about you?"

The girl said: "I am from the Department of Mathematics, and "Abstract Algebra" is a basic course, so of course there is no need to skip the class."

Pang Xuelin couldn't help being taken aback. The ratio of male to female students in the mathematics department has always been disparate. Such a beautiful girl chooses to study mathematics. The boys in his class are blessed.

At this time, the girl said again: "But today I will be able to meet the idol, and it makes people excited just thinking about it. Hee hee, have you heard a piece of news, now the forum is saying that Pang Shen proved the BSD conjecture, then But it is one of the seven major problems of the millennium, and it is said that the Clay Institute has also offered a prize of one million dollars."

"I don't think Professor Pang did his research for that money!"

The girl gave him a blank look, and said: "Of course not, but don't you think Pang Shen is very powerful? He is only 23 years old, and he is the same age as us, so he is already a full professor in our school. Compared with others, I feel that I have lived for nothing in the past twenty years."

Pang Xuelin nodded and said, "That's why you should work harder!"

The girl raised her little hand and said, "Yes, I must work harder. My goal is to be admitted to Pang Shen's graduate school in two years, hehe!"

Pang Xuelin looked at the girl speechlessly. This girl is quite attractive, but why doesn't she look very smart?

In the future, if she really applies for her own graduate school, I'm afraid I'll have to think about it carefully. I guess I'll be annoyed to death by such a chattering apprentice.

But fortunately, the girl saw that Pang Xuelin turned his attention back to the computer, and she was too embarrassed to continue to disturb Pang Xuelin, so she sat aside and concentrated on reading the book.

After a while, people began to come in one after another. The girl seemed to be quite well-known, maybe it had something to do with her carefree personality, and many people came to say hello to her.

Especially after her best friend arrived, the girl resumed her chatty nature, chatting non-stop.

Perhaps it was really to come to worship the God of Learning. Today, the lecture theater with more than 300 people was packed to the brim, and even the aisles were full of people.

The time passed by one minute and one second, and it was eight o'clock before I knew it.

"Hey, it's time, why hasn't Pang Shen come yet?"

"No way, Pang Shen was late for class on the first day?"

"It is said that abstract algebra is very difficult and there are so many concepts. I hope that Pang Shen will not follow the script in class, otherwise the final exam will be troublesome."

...

The students talked a lot, Pang Xuelin didn't change his face, closed his notebook, walked up to the podium, put on the microphone for sound reinforcement, and said indifferently: "Hello everyone, I am Pang Shen Pang Xuelin in everyone's mouth. I will be in charge of " The teaching work of the course "Abstract Algebra". Today is the first time to start the class, and there are a lot of people. In order to ensure the quality of teaching, please keep quiet as much as possible. I will reserve 20 minutes for questions before class ends. You can ask questions later!"

With a bang, the classroom suddenly became noisy. No one expected that Pang Xuelin was sitting in the first row of the classroom.

The students sitting next to him were all stunned. They never thought that this gentle and handsome boy was the legendary Professor Pang?

Especially the pretty girl who had struck up a conversation with Pang Xuelin, with a dull expression on her face, and then cried to her best friend beside her: "It's over, Xiaoyue, I was talking to Professor Pang just now, and I also said that I want to apply for his graduate school. Do you think he will leave a bad impression on me?"

Xiaoyue smiled and said: "Ai Ai, you are so beautiful, how could Professor Pang refuse you to be his student?"

"Okay, dare you say that I have good looks but no IQ!"

"Hush, it's class...Professor Pang is on the podium, don't you really want him to leave a bad impression on you, right? By the way, Professor Pang is really handsome, especially the way he looks on the podium, especially charming!"

"You... hum!"

Ai Ai glared at her best friend, saw Pang Xuelin's gaze swept over, quickly shrank her head, and slightly raised the corners of her mouth: "Pang Shen is really handsome!"

Pang Xuelin didn't care about the noise in the audience, and said: "Okay, now let's get to the point. The course of abstract algebra that everyone will learn this semester, also known as modern algebra, was born in the nineteenth century. It includes group theory. , ring theory, Galois theory, lattice theory, linear algebra, and many other branches, and combined with other branches of mathematics to produce new mathematical disciplines such as algebraic geometry, algebraic number theory, algebraic topology, and topological groups. Abstract algebra is also a modern computer theory. One of the foundations."

"To some extent, abstract algebra has become the common language of most contemporary mathematics. Therefore, although abstract algebra is a basic course, it plays a very important role in the future, whether in academic or applied fields."

"However, the concept of abstract algebra is complicated and difficult to understand. If you start from the concept directly in the textbook, I guess everyone will feel at a loss. But since abstract algebra has the word algebra, it must be related to solving equations. Today's In the first class, I will start with the history of solving equations."

"Equations have a very high status in the history of mathematics. As early as two thousand years BC, people in the Babylonian period could already formulate linear equations in one variable, that is, ax+b=0. Needless to say, everyone should know that it The formula for finding the root is -b/a. But in the ancient Babylonian period, there were only integers at that time, so how did the ancients understand b/a? So the concept of fractions was introduced, and fractions and integers are added together, collectively called rational numbers.”

"Not only that, but the ancient Babylonians also formulated quadratic equations in one variable, that is, ax^2+bx+c=0, but the ancient Babylonians did not study this type of equation thoroughly, and could only give some specific integer solutions and fractional solutions. When the Pythagorean school in ancient Greece, although they did not find the root solution of the quadratic equation in one variable, they found the solution of some specific equations. For example, when studying the Pythagorean theorem, they found that the side length is 1, the length of its diagonal can be solved by the equation."

Pang Xuelin got up and wrote on the blackboard: 1^2+1^2=x^2, x=√2.

"So in order to solve the quadratic equation in one yuan, irrational numbers were introduced. But have you noticed that there is no negative root here? The reason is very simple. Europeans think that negative numbers are meaningless. Until the time of Newton and Leibniz in the seventeenth century, They accepted the concept of negative numbers. In China, as early as the pre-Qin period before BC, there was a concept of negative numbers, and this has something to do with cultural traditions..."

"The root solution formula of a quadratic equation in one variable was first given by the Persian mathematician Khwarizmi in the eighth century AD, but he only gave positive roots. Later, his solution was spread to Europe. In the concept of negative numbers, A little improvement after the introduction is the root solution of the quadratic equation we know now."

Pang Xuelin paused, picked up a bottle of mineral water on the podium and moistened his throat.

There was a buzz of discussion in the originally quiet classroom. Today, many students are not from the Department of Mathematics at all. It feels jerky, but it feels like it's not too fast or slow, and it feels like it's coming at your fingertips.

This surprised everyone.

After all, many big cows are very academically strong, but they don't perform very well when teaching, either following the script or obscure.

Instead, Pang Xuelin talked about the history of mathematics separately, giving everyone a refreshing feeling.

Pang Xuelin ignored the commotion below and continued:

"With the root solution formula, you can easily solve it by simply substituting the coefficients into it, so mathematicians began to look for the root solution formula of the cubic equation and the quartic equation one after another."

"The root solution of the cubic equation was given by Italian mathematicians Ferro and Tartaglia in the sixteenth century Renaissance. Ferro gave the root solution of x^3+px=q. Here you may think This cubic equation does not have a general meaning, but if p and q are expressed in complex numbers, all cubic equations can be expressed in this form. But there was no concept of complex numbers at that time, so another Italian mathematician Tartari Ya gave the root solution formula of the general one-dimensional cubic equation ax^3+bx^2+cx+d=0, which is the so-called Cardano formula..."

Pang Xuelin got up and wrote on the blackboard with a brush of chalk. It took more than half of the blackboard to express the Cardano formula.

"Have you noticed that in Cardano's formula, there was a problem of taking the square root of -3, but there were no plurals at that time, so people began to be interested in the problem of taking the square root of negative values, which is why With the later field of complex numbers. To some extent, in order to solve the one-dimensional cubic equation, people introduced the concept of complex numbers. A few years after Cardano's formula came out, one of Cardano's students Ferrari He also gave the solution formula of the quartic equation in one variable. So far, the root solution formulas of the 1, 2, 3, and 4 equations have appeared."

"So people thought that the formula for finding the root of the quintic equation is not far away, but they didn't expect that in the next few hundred years, people have not found the answer. So everyone began to think of ways to simplify this problem. Does the quadratic equation have any roots? This was done by the famous little prince of mathematics, Gauss. Gauss proved that for any non-zero one-variable n-degree complex coefficient equation, there are exactly n complex roots. This is the fundamental theorem of algebra. Even if the discriminant of a quadratic equation is less than zero, it has two complex roots. Then a quintic equation should have five roots."

"Since there is a root, there should be a root solution, so people continue to look for it. This problem was solved by the Norwegian mathematician Niels Abel. If you don't know who Abel is, you should I heard that the Abel Prize, one of the highest awards in mathematics, is named after him."

"Abel did not give the root solution of the quintic equation. Instead, he proved that the root solution of the quintic equation does not exist. This is very powerful. In the field of mathematics, it is often more difficult to prove that something does not exist than to prove It's much harder to exist."

"This Abel is the one we want to focus on today. Abel was born in Norway in 1802. When he was 17 years old, he wrote a paper about his discovery of the root solution of the quintic equation. Later He found several mistakes in this paper, so he studied with great concentration and continued to revise it. Four years later, he came to a new conclusion that the quintic equation in one variable has no radical solution. He also introduced a theorem called Abel Roux Feeney's theorem, but because this paper was too advanced, professional mathematicians at the time could not understand it, so it did not attract people's attention at first."

"Abel's paper was once shown to Gauss. Gauss thought it was just a 21-year-old child's unreasonable trouble, which meant that it was very difficult for Abel to publish his paper. Fortunately, Abel still has A friend named Kreller founded a mathematics journal, so Abel published this paper on it. In the following years, Abel made contributions in many fields, but mainstream mathematicians He even sent his paper to the famous mathematician Cauchy, but Cauchy was even more cold and cold, and he didn’t even read it. Abel died young at the age of 27, and until after his death, Only then did people discover that his papers are all classics."

"Abel's method of proving that the quintic equation has no root solution is actually the group theory we will learn in abstract algebra, but he did not put it forward systematically, but only used the idea of ​​group theory to solve this problem. The result was unsuccessful. In a few years, another genius appeared, that is, Galois, who used the same idea of ​​group theory to obtain the solution formula of the quintic equation that does not exist. He also gave for any high-degree equation, in Under what circumstances is there a root solution, and under what circumstances is there no root solution. Moreover, Galois also proposed the concept of group for the first time, creating a precedent for modern algebra."

"In my personal opinion, Galois' achievements are enough to rank among the top three in the history of mathematics. Galois was born in France in 1811, similar to Abel. At the age of 16, he also thought that he had discovered one dollar five The root solution of the quadratic equation, but later proved by himself that there is no root solution of the quintic equation. At the age of 19, Galois joined the French Revolution, was arrested and imprisoned at the age of 20, and after being released from prison at the age of 21, he fought a duel Three days before the duel, Galois seemed to realize that he would not be able to survive the duel, so he wrote the next paper, which was the beginning of group theory, and was again It is called Galois theory. But this paper has not been accepted by the mathematical community until 1843, eleven years after Galois died, the mathematician Liouville discovered this paper and published it, which caused a huge sensation The Galois theory ended the history of solving equations in mathematics for more than 4,000 years, and also opened the beginning of group theory."

"Galois group theory also gives a general discriminant method for judging whether a geometric figure can be constructed with a straightedge and a compass, and satisfactorily solves the problem of trisecting any angle or doubling a cube. Most importantly, Group theory has opened up a new field of research, replacing calculation with structure research, changing the way of thinking that focuses on calculation research into a way of thinking that uses structural concepts to study, and classifying mathematical operations, so that group theory has rapidly developed into a brand-new discipline. The branch of mathematics has had a huge impact on the formation and development of modern algebra. At the same time, this theory has had a huge impact on the development of physics and chemistry, and even on the emergence and development of structuralist philosophy in the 20th century."