The university is divided into pure mathematics college and applied mathematics college, is this lucky or unlucky?

Mathematics, is it science or art? It can be said that it is both science and art. Mathematics is a very old discipline that both drives technological progress and advances along with technological progress. Human beings need to measure land, so geometry was invented; the development of electrodynamics also benefited from the introduction of partial differential equations by mathematicians; and the development of mathematical fields such as complex variable analysis was born in plane and fluid mechanics.

But the importance of mathematics is also reflected in another level, that is, proof, especially for experimental science. The criterion for testing truth is the reproducibility of experiments, so whatever theory is, it must be independently demonstrated through mathematical experiments. Von Neumann once said that the idea had changed many times in his life, but it remained so. Mathematics has amazing stability, and the proof in Euclid's book more than 2,000 years ago is still very valid today. The purpose of proof is to understand. It's not enough for mathematicians to just know whether something is right or wrong, they want to know why it's right and what the thinking behind it is.

The beauty of mathematics is consistent with the beauty of physics and even the beauty of science

The process of proof can be beautiful or ugly. If you've heard a conversation between two mathematicians, you're likely to think that they should be artists, because they often talk about this proof of beauty and that proof of not being very good.

So, what does "beauty" mean in mathematics? "Beauty" cannot be defined. Of course, mathematicians will show this using very simple multipliers, profound truths, and complex proofs, like the great Fermat's Last Theorem. They like and expect ideas from different fields, and they also like universality, where the same idea comes up in different forms in different contexts.

In mathematics, the gold standard for measuring "beauty" is Galois's theory. Galois, a Native of France who died in a duel at the age of 20, never went to school, received no formal mathematics education, and did not even pass university entrance exams. At that time, he was just a young man who no one was interested in, and even how he looked, so far he only relied on some speculations from artists. It was not until 100 years later that people knew that he was a mathematical genius that no one could replace, and the group theory he founded at the age of 18 changed the course of mathematics in later generations.

Now everyone knows that quadratic equations, cubic equations, and even quadratic equations have solution formulas for finding roots. For years, mathematicians have been looking for a similar formula to solve an equation of five or higher orders, but it has never been successful because such a formula does not exist and never appears.

The first to try to prove this phenomenon was the Italian mathematician Ruffini, but his proof was incomplete, and he spent his life trying to perfect it. The first correct proof came from the mathematical genius Nils Henrik Abel. He proved that equations with five or more degrees have no root-finding formula. Galois explains why this result occurs and relates this phenomenon to symmetry. This proof can be traced back to Lagrange, where any object, from macroscopic to microscopic elementary particles, is symmetrical. This so-called symmetry is called group theory in the field of mathematics.

It was not until the beginning of the 20th century that physicists began to study elementary particles, and symmetry became an important basis for their research. Glashow, a Nobel laureate and father of the Standard Model of particle physics, said, "I don't know if God exists, but if he does, then he must know about group theory." ”

Speaking of beauty and mathematics, Hermann Weil said: "My work always tries to combine truth and beauty, and when I have to make a choice, I usually choose beauty." Beautiful and of the highest standard I chose. ”

After Einstein proposed the theory of relativity, Weil wrote a high-quality mathematical paper on unified field theory and sent it to a physics journal, and Planck was the editor of the journal and sent it to Einstein for review. Einstein noticed that this model is indeed beautiful, but the experimental data are different from the theoretical model. Einstein believed that the model was wrong. Planck did a remarkable job of publishing the paper and Einstein's review conclusions under the same column. Ten years later, when physicists began to study quantum mechanics and gauge invariance, Weyl's model fit perfectly. Therefore, choosing beauty is often worth it.

Mathematics, the "strange" art, has the most practical value

So, how do mathematicians know what beauty is? Just as students who study art know that this is beauty by listening to the works of Beethoven and Mozart, and cultivate their own appreciation by copying the paintings of great artists, mathematicians also learn beauty from great works.

But hundreds of millions of people can enjoy a beautiful picture and beautiful music, and fewer than 10 people can read papers on number theory. So, is mathematics a strange art? Not really. Mathematics is the art that can be best supported.

Mathematics is not only beautiful, but also practical. For example, the finite domain invented by Galois is now used in the field of satellite launches. Another example is the application of mathematics in cryptography. In the last century, if two generals wanted to communicate with MeSince, they would jointly confirm a page on a thick book, which was used for encoding. It's very difficult to crack this code, and of course it's very difficult to code. Nowadays, when you send and receive emails or withdraw money from an ATM, you exchange passwords with the Internet or bank. Because it was a common device for billions of people, passwords had to be changed frequently, and mathematicians came up with a brilliant new method based entirely on Galois's finite domain.

There is another mathematical example. In 1917, austrian scientist Johann Ladon published a paper, when he was already a member of the Austrian Academy of Sciences, and he discovered a way to derive the original function from the integral of straight lines. By 1970, Cormac and Huntsfield had invented the tomography scanner (CT) and won the Nobel Prize in Science for it. This invention principle is based entirely on Galois's group theory and Raton's function. X-rays illuminate the human brain, take the entry value and the injection value, obtain a density integral, and then calculate the linear density integral to obtain an image of the brain. Of course, when the Nobel Prize is awarded, if Raton is still alive, he can also share this Nobel Prize.

Many years ago, when I was a student, I was told that the real world is made up of real and complex numbers. The so-called finite domain, some of Galois's theories, number theory, only a very small number of professionals will be interested. But now that has completely changed. All financial transactions in the world are actually operated based on such a formula.

For example, how do you build a good network? In the Internet age, it takes very little time to transmit information from one point to another, and it takes the shortest way to go on the network. The first network based on Raton's theory was constructed by Gregory Margulis in 1980, using very profound conclusions from group theory and functional analysis, as well as number theory, to construct a network. This network is called a Ladon measurement image. The paper is only 3 pages. Since then, Margullis has also proved many other theories and received many awards, including the Fields Medal. But so far, his most cited paper is the short 3-page paper.

Mathematics, which is divided into pure mathematics and applied mathematics, is a kind of "misfortune"

By the early 20th century, mathematics had begun to be rewritten, and the development of abstract symbols or other abstract ideas made mathematics something that only a few people could understand. Today's mathematics has been divided into applied mathematics and pure mathematics, which is very "unfortunate". No one asks Euler or Gauss whether you are a theoretical mathematician or an applied mathematician, because they are both. But now, many universities have pure mathematics colleges and applied mathematics colleges.

But the question is, if you ask Galois or Radon or Margulis, or even Jim Simmons, the founder of a well-known hedge fund, are you pure mathematicians or applied mathematicians? They'll tell you that they're pure mathematicians.

Jim Simmons talked about when they hire young mathematicians, they don't ask you if you're pure mathematicians or applied mathematicians, they just need smart brains.

Mathematics is like a plant, its parts are connected to each other. If you want to cut out what you find useless and leave only the parts that are useful to humans, the plant will definitely die. The same metaphor can also be used in mathematics, where every part is indispensable.

The so-called information revolution that we are now experiencing is rooted in mathematics and derived from mathematics. For big data, it also brings huge impacts and challenges to mathematics, in fact, this problem has not been fully considered before. Whether it's artificial intelligence, machine learning, genomics, or even predictions of consumer sentiment... The same problems and challenges have arisen in these areas. Because now we have huge data matrices that are too big to even fit in memory. We know that a matrix is a way of calculating the elements within it, calculating a possibility, and we don't even need to know a precise value.

If you want to solve some common problems, such as problems with systems of equations, you might want a better approach than the old Gaussian elimination method. But in the era of big data, these old computing methods are no longer available, which is a huge challenge and problem for mathematics, and it may exist for centuries. At the moment we don't know which branch of mathematics will ultimately help us solve this problem. Perhaps, we need very deep mathematical theories to really solve this problem.

Author: Effie Jemanov (The author is a foreign academician of the Chinese Academy of Sciences, an academician of the American Academy of Sciences, a winner of the Fields Medal, and a distinguished visiting professor of the Department of Mathematics of Southern University of Science and Technology, this article is from the Institute for Advanced Academic Research of Fudan University)

Editor: Chu Shuting