The Science Fiction World of Xueba

(The five-thousand-word chapter, there is a lot of science history content in the second half of the history, which was originally intended to be skipped directly. After all, the content of nearly two thousand words may affect everyone's reading experience, but think about it, 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 understands why Liu Tingbo is so excited. In the domestic mathematics world, Peking University is No. 1 and Fudan Tsinghua University is slightly inferior. The remaining Mathematics Departments of HKUST, Jiaotong University and NTU are almost above Jiangda University.

In recent years, the Department of Mathematics of Jiangda University has been developing tepidly in the country, and few mathematicians have been influential. Pang Xuelin is already one of the best among them.

If Pang Xuelin proves the BSD conjecture this time, his academic achievements will be enough to stand alone in the domestic mathematical community.

By that time, as long as Pang Xuelin is still at Jiangda University, it is a towering tree that can compete with Beijing University of Mathematics. After seeing Qiu Chengtong coming to Mizuki, in recent years, there have been many outstanding young scholars at Mizuki's Qiu Chengtong Mathematical Center. .

Although there is still a gap between Mizuki's Department of Mathematics and Peking University, it is no longer under Fudan, an old math strong school in China.

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

And this BSD conjecture report will be the first shot of Jiangda Mathematics Institute in the international mathematics community!

In the next few days, a dark tide surged in mathematics. Pang Xuelin's thesis on arXiv entered the field of view of most mathematicians who are interested in this field.

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

"Have you heard? BSD is guessed by a professor from Jiang University called Pang Xuelin!"

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

"Don't you watch the news? A newly emerged genius mathematician in China in recent years has just graduated with a doctoral degree at the age of 22 and published a top-level dissertation, studying under Professor Tao Zhexuan. This year, he returned directly to Jiangda University as a professor and Ph.D.

"It's pretty cool to look at your resume, but BSD guesses that it is one of the seven major problems in the millennium. How can it feel unreliable!"

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

"Who are you teaching?"

"Aksay Venkatesh! The 2018 Fields Prize winner, his main research areas are analytical number theory, topology, and representation theory."

"Well, it turned out to be Stanford's elder brother. The elder brother is on the top. Please worship me!"

Uh ...

Such discussions are endless, and even on the campus of Jiang University, there are similar rumors.

上午 In the morning of September 11, 2019, today is the first class that Pang Xuelin has taken since joining Jiangda University.

At seven o'clock in the morning, he came to the ladder classroom early.

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

About twenty minutes later, Pang Xuelin suddenly felt a scent of fragrance coming from her nose, raised her head, and saw a tall girl with a tall nose, a deep nose, a delicate face, and a bit of a mixed-race girl sitting in the first row of the classroom. under.

的 The classroom they were in was a large stepped classroom. 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 eyes stared at each other. The girl smiled slightly at Pang Xuelin and said earlier.

Xun Pang Xuelin nodded toward her, without speaking.

After a while, the girl saw Pang Xuelin crackling on the keyboard, and she got up and walked to the side of Pang Xuelin, curiously: "Classmate, what are you doing?"

"Be a PPT!"

"Are you doing this report? By the way, why haven't I seen you before and you are here for the lesson? Today is the first class of Pangshen. Many students from other majors come to worship God. "

Pang Xuelin took a moment's notice and raised her head: "What about you?"

The girl said: "I'm from the Department of Mathematics." Abstract Algebra "is a basic course. Of course, there is no need to teach it."

Pang Xuelin couldn't help it. The ratio of male and female students in the department of mathematics has always been very different. Such a beautiful girl actually chose to study mathematics, and the boys in their class are blessed.

At this time, the girl said again, "But I can see the idol today, and it's exciting to think about it. Hee hee, have you heard of a news yet? Now all the forums are saying that Pang Shen proved the BSD conjecture, that But one of the seven major problems of the millennium, the Clay Institute is said to have given out a million dollar bonus. "

"I don't think Professor Pang should do research for that money!"

The girl gave him a blank look and said: "Of course it is not, but don't you think Pang Shen is very powerful? People are only 23 years old. They are the same age as us. They are already full professors in our school. Compared with others, I feel like I have been alive for the past 20 years. "

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

The girl raised her small hand and said, "Yes, you must work harder. My goal is to be a graduate student of Pangshen two years later, hee hee!"

Xi Pang Xuelin glanced at the girl silently. The girl was pretty eye-catching, but why didn't she look smart?

In the future, if she really applied for her postgraduate examination, she would have to consider it carefully. Such a nonsense apprentice would probably be annoyed by her.

But I was okay. The girl saw Pang Xuelin turn her attention back to the computer, and she was embarrassed to continue to disturb Pang Xuelin, sitting beside her and reading the book intently.

After a while, someone came in one after another. The girl seemed to be quite well-known. Maybe it had something to do with her carefree personality. Many people came to say hello to her.

After the arrival of her girlfriend, especially the girl, the girl resumed the nature of her words, and she kept talking.

Maybe it was really to come here to worship God. Today, the ladder classroom of more than 300 people is crowded, and even the aisle is full of people.

The time elapsed minute by minute, and it was eight o'clock unknowingly.

咦 "Oh, time is up, why isn't Pang Shen yet?"

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

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

Uh ...

The students talked a lot, Pang Xuelin didn't change his face, closed his notebook, went to the podium, put on a microphone for sound reinforcement, and said lightly, "Hello everyone, I am the Pang Xuelin in your mouth. I am in charge of this semester." The teaching work of "Abstract Algebra" course. The first class started today, and there are many people. In order to ensure the quality of teaching, please also try to keep quiet as much as possible. I will reserve twenty minutes for questions before class. What is there? You can ask questions at that time! "

Noisily, the classroom was suddenly noisy. No one expected that Pang Xuelin was sitting in the first row of the classroom.

The students who were sitting next to him were even stunned. They didn't even think that this guy who looked a little handsome and handsome, was the legendary Professor Pang?

In particular, the pretty girl who had had a chat with Pang Xuelin was dull, and then cried to her girlfriend next to her: "It's over, Xiaoyue, I just talked to Professor Pang just now, and I said I would apply for his graduate student. You said, would he leave a bad impression on me? "

Xiao Xiaoyue laughed: "Ai Ai, you are so beautiful, how could Professor Pang refuse you to be his student?"

"Okay, dare you say I have a face value but no IQ!"

"Hush, what about class ... Professor Pang is on the podium, shouldn't you really want him to make a bad impression on you? Professor Pang is really handsome, especially on the podium, especially charming!"

"You ... hmm!"

Ai Ai glared at her girlfriend, and saw Pang Xuelin's eyes glanced over, quickly shrinking her head, her mouth slightly raised: "Pang Shen is really handsome!"

Pang Xuelin didn't pay attention to the noise of the audience and said, "Okay, let's get to the topic. The course of abstract algebra that everyone wants to learn this semester, also known as modern algebra, was born in the 19th century. It contains group theory. , Ring theory, Galois theory, lattice theory, linear algebra and many other branches, and combined with other branches of mathematics have produced algebraic geometry, algebraic number theory, algebraic topology, topological groups and other new mathematical disciplines, abstract algebra is also modern computer theory One of the foundations. "

"In a way, abstract algebra has become the universal language of most contemporary mathematics. Therefore, although abstract algebra is a basic course, it has a very important role for everyone in the future, whether in the academic field or the application field."

"But the concept of abstract algebra is complicated and difficult to understand. If you start from the concept in the textbook, I guess everyone will feel confused. But since abstract algebra has the word algebra, it must be related to solving the equation. Today this In class, I start with the history of solving equations. "

"Equations have a very high status in the history of mathematics. As early as more than 2,000 BC, people in the Babylonian period had listed equations once, 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 Babylonian era, there were only integers at that time, how did the ancients understand b / a? So the concept of fractions was introduced. Fractions and integers are added together, collectively called rational numbers. "

"Not only that, but the Babylonians also listed univariate quadratic equations, namely ax ^ 2 + bx + c = 0, but this type of equations were not thoroughly studied by the Babylonians, and they could only give some specific integer and fractional solutions. By the time of the ancient Greek Pythagoras school, although they did not find the root solution of the one-variable quadratic equation, they found some specific equations. For example, when studying the Pythagorean theorem, they found that the side length is A square of 1 whose diagonal length can be solved by a column equation. "

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

"So irrational numbers were introduced in order to solve the one-quadratic equation. But have you found that there is no negative root here? The reason is simple. Europeans thought that negative numbers were meaningless until the Newton and Leibniz period of the seventeenth century. They just accepted the concept of negative numbers. In China, as early as the pre-Qin BC, there was the concept of negative numbers, which is related to cultural traditions ... "

"The root solution of the quadratic equation was first given by the Persian mathematician Hua Lazimi in the eighth century AD, but he only gave the positive root. Later, his solution was transmitted to Europe. In the concept of negative numbers, A little improvement after the introduction is the root solution of the unary quadratic equation we now know. "

Pang Xuelinton paused, took a bottle of mineral water on the podium and moistened his throat.

There was a buzzing discussion in the quiet classroom. Many students today are not even mathematics students. They come to see the fun, worship the gods, but did not expect that Pang Xuelin's lecture did not let everyone I feel jerky, but I have a feeling of not being ill, not being able to catch on.

This surprised everyone.

After all, many Daniels are academically strong, but they don't perform well when they teach, they either follow the original instructions or they are obscure.

On the contrary, Pang Xuelin opened up the history of mathematics and gave everyone a refreshing feeling.

Xi Pang Xuelin ignored the noise below and continued:

"With the root solution, as long as you put in the coefficients, you can easily solve it, so mathematicians began to find the root solution of the cubic equation and quartic equation one after another."

"The root solution of the cubic equation was given by 16th-century Italian mathematicians Ferro and Tartaglia. Ferro gave the root solution of x ^ 3 + px = q. Here you may think This cubic equation has no general meaning, but if p and q are represented by complex numbers, all cubic equations can be represented in this form. But at that time there was no concept of complex numbers, so another Italian digitist, Tartali Ya gave the root solution of the general one-ary cubic equation ax ^ 3 + bx ^ 2 + cx + d = 0, which is the so-called Cardano formula ... "

Xi Pang Xuelin got up and wrote with a chalk brush on the blackboard. After spending more than half of the blackboard, he expressed the Cardano formula.

"Did you find out that in the Cardano formula, the problem of -3 needs to be slashed, but there was no plural at that time, so people began to be interested in the problem of scoring negative values. With the later complex field. To some extent, in order to solve the one-ary cubic equation, people introduced the concept of complex numbers. Within a few years after the Cardano formula came out, a student of Cardano Ferrari A solution formula for the quartic equation of one element is also given. So far, the root solutions of the quartic equation of one, two, three, and four have all appeared. "

"So people think that the formula for finding the root of the fifth-degree equation is not far away, but did not expect that in the next hundred years, people have never found the answer. So everyone began to think of ways to simplify this problem. There is no root for the secondary equation. This is what the famous little prince of mathematics Gauss did. Gauss proved that for any non-zero unary n-th complex coefficient equation, there are exactly n complex roots. This is the basic theorem of algebra Even if the discriminant of a quadratic equation is less than zero, it has two complex roots. Then a fifth-degree equation should have five roots. "

"Since there is a root, there should be a root solution, so people continue to look for this problem, which is solved by Norwegian talented mathematician Niels Abel. If you do n’t know who Abel is, you should also I've 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 fifth-order equation. Instead, he proved that there is no root solution for the fifth-order equation. This is very powerful. In the mathematical world, it is often better to prove that something does not exist than to prove it. It's much harder to exist. "

"This Abel is a character we want to focus on today. Abel was born in Norway in 1802. When he was 17 years old, he wrote a dissertation that he discovered the root solution of the fifth degree equation. Later He found several errors in this paper, so he studied it and continued to revise it. Four years later, he came to a new conclusion. There was no root solution for the quintic equation. He also introduced a theorem called Abel Lu Fini's theorem, but because this thesis was too advanced and was not understood by professional mathematicians at the time, it did not attract people's attention at the beginning. "

"Abel's paper has also been shown to Gauss. Gauss believes that this is just an unreasonable trouble of a 21-year-old child, which means that it is difficult for Abel to publish his paper. Fortunately, Abel also has A friend called Cralle, he founded a mathematical journal, so Abel published this paper on it. In the following years, Abel has made contributions in many fields, but mainstream mathematicians None of them accepted it. He even sent his dissertation to the famous mathematician Cauchy. As a result, Cauchy was even more cold and never looked at it. Abel died at the age of 27 at the age of 27, until his death. It was only discovered that his dissertation was classic. "

"Abel proved that the fifth-order equation has no root solution. In fact, it is the group theory that we will learn in abstract algebra, but he did not propose it systematically, but just used the idea of ​​group theory to solve this problem. In a few years, another genius appeared, that is, Galois. Galois used the same group theory to find that there is no root solution to the fifth degree equation ~ www.NovelMTL.com ~ He also gave Arbitrary higher-order equations have a root solution under what circumstances and no root solution. In addition, Galois first proposed the concept of a group, setting 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. He also thought he found a dollar five at the age of 16. The root solution of the secondary equation, but later proved that there is no root solution of the fifth equation. At the age of nineteen, Galois joined the French Revolution. He was arrested and jailed at the age of 20, and dueled at the age of 21. He died. Three days before the duel, Galois seemed to realize that he could not survive the duel, so he wrote the next thesis, which was the beginning of group theory and was again Called Galois theory, but this paper was never accepted by the mathematics community. Until 1843, eleven years after Galois' death, mathematician Liu Weier found this paper and published it, causing a great sensation. The Galois theory ended the history of equation solving in the mathematical world for more than 4,000 years, and also started the beginning of group theory. "

"The Galois Group Theory also gives a general method of judging whether a geometric figure can be drawn with a ruler and a compass. It satisfactorily solves the problem of three-division arbitrary angles or double cubes. The most important thing is, Group theory has opened up a whole new field of study, replacing calculations with structural studies, transforming the thinking mode that emphasizes computational research into a thinking mode that uses structural concepts, and categorizing mathematical operations, so that group theory has rapidly developed into a brand new The branch of mathematics had a huge impact on the formation and development of modern algebra. At the same time, this theory had a huge impact on the development of physics, chemistry, and even the production and development of structuralist philosophy in the twentieth century. "