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This article comes from the official account of Wechat: ID:fanpu2019, author: Ian Stewart

Because of his poor eyesight, the Norwegian mathematician Sofius Lee "abandoned medicine" and embarked on the academic road. Along the way, he met great mathematicians such as Klein, Kumer, Weir Strauss and so on. He also created his own glory, and even its significance went beyond his original imagination: in order to find a differential equation theory analogous to Galois theory and determine whether the differential equation can be solved, Sofius Lee discovered the continuous group-lie group. this epoch-making tool has completely changed the development of mathematics and physics.

Write | Ian Stewart (Ian Stewart)

Translators: Li Sizheng, Zhang Bingyu

A new way of looking at geometry Marius Sophie Lee (Marius Sophus Lie) embarked on the road to science simply because his eyesight was too poor to hold any military post. When Sofius, as he was later called, graduated from the University of Cristiania in 1865, he took only a few math courses, including Galois theory taught by Norwegian Ludwig Sylow, but he did not show any extraordinary talent in this field. He hesitated for a time-he knew he wanted to follow the path of academic research, but he was not sure which field to study, botany, zoology or astronomy.

Photo 1 Norwegian mathematician Sofius Lee (Sophus Lie,1842-1899). Photo Source: wiki, but the borrowing records of the university library show that he has borrowed more and more math books. In the middle of the night in 1867, he opened up and saw the cause he would work for all his life. His friend Ernst Motzfeldt (Ernst Motzfeldt) was awakened by the excited Lee in his sleep and saw him shouting: "I found it, it's so simple!"

What he found was a new way of looking at geometry.

Li began to study the work of great geometries, such as the German mathematician Julius Plcker and the French mathematician Jean-Victor Jean-VictorPoncelet. From Plucker, he learned that the basic elements of geometry are not the familiar points proposed by Euclid, but other objects such as lines, planes and circles.

In 1869, he published a paper at his own expense, summarizing his main ideas. He realized that his ideas, like Galois and Abel before him, were so radical for conservatives that ordinary journals were reluctant to publish his research. But Ernst encouraged him to stick to his own geometric research, and with his support, Lee was not discouraged. In the end, one of Li's papers was published in a well-known journal and received an enthusiastic response. This gave Li a grant, and he finally had the money to visit top mathematicians everywhere and discuss his ideas with them. He went to Gottingen and Berlin, the cradle of countless Prussian and German mathematicians, to discuss with the algebraists Leopold Kronecker and Ernst Kummer, and the analyst Karl Weierstrass. He was impressed by Cuomer's mathematical research method, while Weir Strauss's method did not attract much attention.

The most important meeting was to visit Felix Klein (Felix Klein) in Berlin, who happened to be a student of Plucker whom Lee admired and hoped to emulate. Lee and Klein have very similar mathematical backgrounds, but their styles are very different. Klein is basically an algebraist who tends to geometry and likes to delve into specific problems full of inner beauty, while Li is an analyst who likes the comprehensiveness and vastness of general theories. Ironically, it is Li's general theory that provides some of the most important special structures for mathematics, which are still particularly beautiful and profound today, most of which are algebraic structures. If Li hadn't made the theory universal, these structures might not have been discovered at all. If you try to understand all the mathematical objects in a category and succeed, you will inevitably find that many of them have special properties.

Lee and Klein met again in Paris in 1870. There, under the influence of Camille Jordan, Li turned his research goal to group theory. More and more mathematicians realize that geometry and group theory are two sides of the same coin, but this idea took a long time to fully take shape. Lee and Klein have done some research together in an attempt to further clarify the relationship between groups and geometry. Finally, Klein clearly put forward this idea in his "Erlangen Program" in 1872, which showed that geometry and group are the same in nature.

In today's language, this idea sounds too simple and should have been clear at a glance. The group corresponding to any given geometry is the symmetry group of that geometry. On the contrary, the geometry corresponding to a group is any geometry in which the group is a symmetric group. In other words, geometry is defined by something that remains unchanged under the transformation of a group.

For example, symmetry in Euclidean geometry is a transformation in a plane that keeps the length, angle, line, and circle constant. What they make up is the group of all the rigid bodies in the plane. On the contrary, all objects that remain unchanged in rigid body motion naturally fall within the scope of Euclidean geometry. Non-Euclidean geometry only uses different transformation groups.

So why bother to convert geometry into group theory? Because in this way you can look at geometry in two different ways, and there are also two different ways to look at groups. Sometimes it's easier to understand in one way, sometimes it's another. It's better to have two perspectives than only one.

Exploring differential equations the relationship between France and Prussia deteriorated sharply. Napoleon III thought he could support his declining popularity by waging war on Prussia. Prussian Prime Minister Bismarck sent a stern and sharp telegram to France, and the Franco-Prussian war officially broke out on July 19, 1870. Klein, a Prussian living in Paris, wisely chose to return to Berlin.

However, Li was Norwegian and enjoyed his visit to Paris so much that he decided to stay. But when he realized that France was about to lose and the Germans were advancing on Mace, he changed his mind. Although he is a national of a neutral country, it is not safe to stay in a potential war zone.

Li decided to start quickly and set out for Italy. But he didn't get far: the French authorities caught him in Fontainebleau, about 25 miles southeast of Paris, and he carried many documents full of incomprehensible symbols. Because the symbols appeared to be ciphertext, he was arrested and jailed as a German spy. It was only after Gaston Gaston Darboux, France's top mathematician, stepped in that he convinced the authorities that the symbols were mathematical deductions. Li was released, then the French surrendered, the Germans sealed off Paris, and Li went to Italy again-this time he succeeded. He returned to Norway from Italy. On the way, he also dropped in on Klein, who had safely stayed in Berlin.

In 1872, Li received his doctorate. Li's research so shocked Norwegian academia that the University of Kristiania created a position for him in the same year. Together with his former teacher, Ludwig Silo, he began to edit a collection of Abel's research. He married Anna Bilk in 1874 and they had three children.

At this point, Li has focused his research on a specific topic that he thinks is ripe enough for development. There are many different kinds of equations in mathematics, but two of them are particularly important. The first is algebraic equations, which have been fully studied by Abel and Galois. The second category is the differential equation, which was introduced by Newton in his study of the laws of nature. This kind of equation involves the concept of calculus. Unlike describing physical quantities directly, they describe how physical quantities change with time. More precisely, they give the rate of change of this quantity. For example, Newton's most important law of motion states that the acceleration of an object is proportional to the resultant force acting on it. Acceleration is the rate of change of velocity. The law does not directly tell us the speed of the object, but the rate of change of the velocity. Similarly, another equation proposed by Newton to explain how the temperature of an object changes as it cools is that the rate of change of temperature is proportional to the difference between the temperature of the object and the temperature of its surroundings.

Many important equations in physics-about fluid flow, gravity, planetary motion, the transfer of heat, the motion of waves, magnetism, and the propagation of light and sound-are differential equations. Newton was the first to realize that if we pay attention to the rate of change of the quantities we want to observe, rather than just looking at the quantities themselves, the laws of nature tend to become simpler and easier to detect.

Li asked himself a big question: is there a theory of differential equations analogous to Galois's theory of algebraic equations? Is there a way to determine when a differential equation can be solved in a specific way?

Once again, the crux of the problem comes back to symmetry. Li now realizes that some of his geometric results can be reinterpreted in the language of differential equations. Once there is a solution to a particular differential equation, Li can apply some transformation to it and prove that the result is also a solution of the equation. Many solutions can be obtained from one solution, all of which are related by this group. In other words, this group is made up of the symmetry of differential equations.

This is an obvious hint that there is some kind of beautiful theory to be discovered. Think back to what Galois's application of symmetry brought to algebraic equations-now imagine what if the same thing happened to a much more important differential equation.

Lie groups and lie Algebras: the groups studied by Galois are all finite groups. In other words, the number of transformations contained in the group is a finite integer. For example, a group consisting of all permutations of the five roots of the quintic equation has 120 elements. However, there are many meaningful groups that are infinite groups, including symmetric groups of differential equations.

A common infinite group is the symmetry group of a circle, which contains all the transformations that rotate the circle at any angle. Because there are infinitely many possible rotation angles, the rotation group of a circle is an infinite group. The symbol for this group is SO (2). Here O means "orthogonal", meaning that these transformations are rigid body motions in the plane, while S means "special"-rotation does not flip the plane.

The circle also has an infinite number of reflection axes of symmetry. If you reflect the circle along any diameter, you will get the same circle. By adding reflection transformation to the rotation group, we get a larger group, O (2).

SO (2) and O (2) are infinite groups, but they belong to the category that is easy to manipulate. As long as a specific number, the rotation angle, is given, all kinds of rotations can be determined. When the two rotations are combined, you only need to add their corresponding angles. Li calls this situation "continuous", and in his terms, the SO (2) group is a continuous group. Because only one number is needed to determine the angle, the SO (2) group is one-dimensional.

O (2) is also one-dimensional, because all we need is a way to distinguish between reflection and rotation, which is a problem of positive and negative signs in algebra.

SO (2) group is the simplest lie group. Lie group has two kinds of structure at the same time: it is not only a group, but also a manifold-a multi-dimensional space. For SO (2), a manifold is a circle, and the group operation that connects two points on a circle is to add two corresponding rotation angles.

Fig. 2 there are infinitely many rotational symmetries (left) and infinite reflection symmetries (right). Li discovered a beautiful feature of lie groups: the group structure of lie groups can be "linearized". That is to say, the curved manifold in which the lie group is located can be replaced by a flat Euclidean space. This space is the tangent space of the manifold. For SO (2), as shown in figure 3.

Fig. 3 from lie group to lie algebra: the group structure of the linearized tangent space of a circle gives the tangent space an algebraic structure of its own, which is an "infinitesimal" version of the group structure. it describes how a transformation very close to an identity transformation behaves. This structure is called the lie algebra of the group. It has the same dimension as the group, but its geometry is much simpler and flat.

Of course, there is a price to pay for such simplification: lie algebras can capture the most important properties of the corresponding group, but they will lose some small details, and these captured properties will change slightly. In spite of this, you can still learn many properties of a lie group by switching to lie algebra, and most of the problems are easier to solve with lie algebra.

It can be proved-- this is one of Li's great insights-- that the natural algebraic operation on lie algebras is not the product AB, but the difference of AB-BA, which is called commutator (called commutator in physics). For groups such as SO (2) which satisfy AB = BA, the commutator is equal to 0. But for groups such as rotation group SO (3) on three-dimensional linear space, AB-BA will not be zero unless the rotation axes of An and B coincide or are perpendicular to each other. Therefore, the geometric characteristics of the group are reflected in the performance of the commutator.

At the beginning of the 20th century, with the birth of "differential domain" theory, Li's dream of establishing differential equation version of "Galois theory" finally came true. However, it has been proved that lie group theory is far more important than Li expected, and its application is more extensive. The theory of lie group and lie algebra is no longer just a tool to judge whether a differential equation can be solved by a specific method, but has spread to almost all branches of mathematics. "lie theory" has surpassed its creator and has become greater than he could have imagined.

In hindsight, the reason is symmetry. Symmetry has gone deep into every field of mathematics, and it is also the foundation of most basic ideas of mathematical physics. Symmetry expresses the laws contained in the world, and it is these laws that push physics forward. Continuous symmetries such as rotation are closely related to the properties of space, time and matter; they imply the existence of various conservation laws, such as the law of conservation of energy, which says that closed systems can neither gain nor lose energy. This connection between symmetry and conservation laws was discovered by Hilbert's student Emmy Noether.

The next step, of course, is to understand these possible lie groups, just as Galois and his successors sorted out a variety of properties from finite groups.

At this point, another mathematician joined the quest.

A brief introduction to the author

Ian Stewart (Ian Stewart) is an emeritus professor of mathematics at the University of Warwick and a member of the Royal Society. He has won the Faraday Medal of the Royal Society, the Public understanding of Science and Technology Award of the American Association for the Advancement of Science and the Zeeman Medal awarded by the Mathematical Society of London and the British Institute of Mathematics and Applied Research. Stewart has written a lot, especially in the field of popular science, "incredible numbers", "who is rolling the dice: uncertain mathematics", Fearful Symmetry: Is God a Geometer? And so on.

This article is an authorized excerpt from Chapter 10, "nearsightedness and frail nerds determined to join the army" (Why Beauty Is Truth: The History of Symmetry) (Nautilus, Citic Publishing House, September 2022). The title is added by the editor.

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