Vol 2 Chapter 1381: Number Theory and Geometry

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In 1940, André Weil, a French mathematician and one of the first-generation scholars of the later Bulbaki school, wrote a letter to his sister, the famous philosopher Simone Weil, in prison. In this letter, he explained his understanding of the "megatrends" of mathematics in very simple language that even philosophers can understand. In the letter, Wey talked about the role of analogy in mathematics, and clarified this problem with the analogy that he is most interested in-the analogy of number theory and geometry.
Facts have proved that the analogy between number theory and geometry has played a very important role in the development of the Langlands program.
The key point of the Langlands program is the concept of symmetry familiar to mathematicians-that is, a concept that can be handled by "group theory". The focus of the Langlands program is also the expression of the group. Related studies have found that these Galois group representations can form the "source code" of the number domain, carrying important information about the number.
Langlands himself likens this process. Symphony is formed by overlapping harmonics corresponding to sounds played by various musical instruments. Common sounds are similar, and are also formed by overlapping harmonics. Mathematically, known functions can be expressed as functions describing harmonics-such as trigonometric functions such as sine and cosine. The self-defense function can be regarded as an advanced version of these harmonics that we are more familiar with, and various analysis methods can be used to complete the calculation using the self-defense function. Langlands puts forward a jaw-dropping point: we can use self-defense functions to study much more difficult number theory problems.
In this way, he discovered that the digital composition produced an unknown "harmony".
One of the main functions of mathematics is to sort and classify information, in the words of Langlands, that is, "to get clues from clues that seem to be chaotic." The idea of ​​Langlands has extraordinary significance precisely because it can organize the seemingly messy data in number theory to form a certain law, showing symmetry and unity.
Break the gap between "number theory" and "group theory", and include this "last piece" in the layout originally planned by the Burbuck School.
The fact that these highly abstract concepts are so harmonious and harmonious, they are indeed breathtaking and unbelievable. This harmonious unification reveals the rich connotation and mysterious content behind the abstract concept, as if lifting a curtain in front of mankind, the mysterious existence that has never been known reveals its true face.
Since then, all known mathematics can be grouped into a large system.
In that famous letter, Andre Weil, one of the founders of the Bulbaki School, described this thinking in this way.
"... My research purpose is to decipher texts written in three languages. In these three fields, I only have some fragmented knowledge. I have some understanding of these three languages, but I also understand these three There are huge differences in the connotation between the tracks, and I have not fully grasped these differences so far. After several years of research, I have only accumulated some fragments of knowledge, which is not enough to compile a complete translation dictionary . "
Because of this, modern mathematicians have always compared the Langlands program to the Rosetta Stone.
A stone tablet with the same text in different languages.
"The Rosetta Stone" is an important symbol in linguistics. Its appearance makes it possible to decipher several ancient characters. It is also given the meaning of "converting several systems with different meanings to each other".
Wang Qi initially carved the stone monument, purely to create a Rosetta stone monument, and to pretend to be-the existence of the monk Gain, the exchanges between the various regions of China are very frequent, there are not many "occlusion" areas There are no conditions for forming different languages, and the "Book Tongwen" was completed very early. The same is true for the demon and dragon races. If it is not human race or mortal, the "dialect" thing is very difficult to appear.
In other words, something similar to the "Roseta Stone" would never appear in this place. In the end, Wang Qi could only engrave himself.
However, during the burning process, the content on the inscription gradually changed from a joke-like epitaph to a game of thinking.
If you can express some mathematical concepts in different ways that you have created yourself, to what extent can you achieve it?
This is the idea of ​​"symmetry".
Among the ancient word games, there is a poem called "Xuan Ji Tu", which is widely praised. The prestigious Xuanji map in China has a total of 841 characters, with 29 characters in each aspect. Vertical, horizontal, oblique, interactive, positive, reverse reading, or reversing the word or repeating the word can be written as a poem. There are three, four, five, six and seven words in the poem. From a geometrical point of view, Wang Qi ’s "play" complexity is even better than one-his stone tablet not only has symmetry and transformation between "concept" and "concept", but also a single symbol Between them, the theme of "symmetry" is also expressed with some peculiar rules.
"Concept" comes from number theory, but "symbol" is based on geometric design.
There was a period of time-around the time the Nanming Nuclear Research Center was just established, Wang Qi was particularly addicted to this game based on the Langlands program-mathematicians on the earth have little chance to splurge like he is today The computing power of industrial piles, and their limited life does not allow them to spend a lot of time and energy on this meaningless game.
Wang Qi himself did not know how this seemingly boring and wasteful mental game had any effect on his thinking. He once felt that he was only using a combination of "games" and "art" to reproduce some known content on earth.
But come back ...
"This is somewhat similar to Lianzong's approach? Use concrete images instead of concepts?" Wang Qi guessed.
Today, the process of metatheism reveals his progress vaguely.
He himself did not anticipate his current level.
"Um ... maybe I will be able to get Gushan-Shimura-Wei in a few years? Then try to impact the Fermat's theorem." Wang Qi said to himself, then smiled bitterly.
not that simple.
Langlands himself spent thirty years on the joint between number theory and group theory, which is still under the condition that a whole generation of mathematicians can serve as his help-the rise of Robert Langlands , The retreat with Alexander Grothendieck occurred in the same era, only a few years between the two events. The two generations of geniuses have completed some kind of "hand-off" like tacit understanding. The Pope of Mathematics gave the mathematics community he had transformed to the next person who was determined to unify academics. Langlands held the position of "Martial Leader" for a long time, even if the Soviet school was discussing its work.
But that's it. It still took him thirty years to complete his work.
Even if Wang Qi knows most of the details of the other party's work, it is impossible to repeat the engraving in a short time, and it is impossible to complete the proof of Fermat's theorem from scratch in just a few years.
What's more, his soul is imminent, and "in a few years' time" is no longer meaningful.
on the other hand……
Wang Qi ’s previous life was a physicist after all. If according to a certain argument, he can also be regarded as a "Bacon believer"-a person who firmly believes that "everything is based on the hard facts of nature seen by the eyes."
According to Bacon's point of view, scientists need to travel around the world to collect facts, or repeatedly test and reproduce phenomena until the accumulated facts can reveal the natural way of movement. Scientists deduced from these facts the laws followed by natural operations.
According to the concept of a physicist ... "One of the major deficiencies of the Bulbaki program is the missed element of surprise."
The development of the history of mathematics is full of non-logical jumps. The order of the invention of the index and the logarithm is a typical example-the logical order is completely different from the actual order. The Bulbaki school tries to put everything into a logical process.
This is not to say that the Bulbaki school is not good. As the soul of a mathematician, Wang Qi has always been intoxicated in his research. But his inner tendency made him think-maybe I can try to use other roads to break through the heavens and become the soul?
What else? Based on the "sixteenth state" of this world?
The so-called "octet" is a theory when humans first discovered only three kinds of quarks. In this model, the concept of "generation" has not yet appeared. The upper and lower quarks are regarded as two completely different quarks-in fact, they should be regarded as the same generation of quarks. In this symmetric model, there are eight types of quark, lepton, and neutrino. They are included in a model called "octet"
And a year later, someone made a new extension on this model and proposed the possibility of the fourth kind of quark. Then, more quarks outside this model were found.
In the era of Wang Qi, the number of quarks, leptons and neutrinos should be like this: three generations of neutrinos, three generations of leptons, three generations of quarks, and each generation of quarks has both positive and negative. Six kinds of quarks.
Without calculating the "flavor" of the quark, a model of the twelve states is established.
In this universe, it should be the sixteenth state.
Neutrino, four generations, four.
Lepton, four generations, four.
Quark, four generations, eight species.
A total of sixteen species.
"If you don't use your brains, you can make a whole thing."
"In this case, it won't even touch the process of metatheism ... It is also possible to publish a paper and let others help me complete the follow-up work."
"However, it cannot be ruled out that there is a fifth-generation quark in this world. That is to say ... there is still some danger in using this theory as the foundation of the primal magic."
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