1900, Hilbert was invited to attend the International Congress of Mathematicians held in Paris and delivered an important speech entitled "Mathematical Problems". In this historic speech, first of all, he put forward many important points:

Just like every career of human beings pursues a certain goal, mathematical research also needs its own problems. It is through the solution of these problems that researchers have exercised their iron will, discovered new ideas and reached a broader realm of freedom.

Hilbert particularly emphasized the role of major problems in the development of mathematics. He pointed out: "If we want to have an idea of the possible development of mathematical knowledge in the near future, we must review the problems raised by science today and hope to solve them in the future." At the same time, it is pointed out: "The profound significance of some problems to the general mathematical process and their important role in the personal work of researchers are undeniable. As long as a branch of science can ask a lot of questions, it is full of vitality, and no problem indicates the decline or suspension of independent development. "

He expounded the characteristics of major issues. A good issue should have the following three characteristics:

Clear and easy to understand;

Although difficult, it gives people hope;

Very profound.

At the same time, he analyzed the difficulties often encountered in learning mathematics problems and some methods to overcome them. It was at this meeting that he put forward 23 problems that mathematicians should try to solve in the new century, namely the famous "Hilbert 23 problem".

The situation of solving the numbering problem in the field of promoting development

1 continuum hypothesis axiom set theory 1963 and Paul J.Cohen proved that the first problem is insoluble in the following sense. That is to say, in the zermelo-Frankel axiom system, the truth of the continuum hypothesis cannot be judged.

2 Compatibility of Arithmetic Axioms Mathematical Basis Hilbert's idea of proving the compatibility of arithmetic axioms later developed into a systematic Hilbert plan ("meta-mathematics" or "proof theory"), but Godel pointed out in the "incomplete theorem" of 193 1 year that it is impossible to prove the compatibility of arithmetic axioms with "meta-mathematics". The compatibility problem of mathematics has not been solved so far.

The geometric basis of equal volume of two kinds of high-base tetrahedrons is very fast (1900), and M.Dehn, a student of Hilbert, definitely answered it.

The problem that a straight line is the geometric basis of the shortest distance between two points is too general. After Hilbert, many mathematicians devoted themselves to constructing and exploring all kinds of special metric geometry, and made great progress in studying the fourth problem, but the problem was not completely solved.

5 The concept of Lie group does not define the topological group theory of the assumption of differentiability of group functions. After a long period of efforts, this problem was finally solved by Gleason, Montqomery, Zipping and others in 1952, and the answer was yes.

6 Mathematic treatment of physical axioms Mathematical physics has achieved great success in quantum mechanics, thermodynamics and other fields, but in general, what axiomatic physics means is still a problem that needs to be discussed. Axiomatization of probability theory was established by A.H.Konmoropob and others.

7 Irrational Numbers of Some Numbers and Transcendence Transcendental Number Theory 1934 A.O.temohm and Schneieder solved the second half of this problem independently.

Riemann conjecture is still a conjecture in the general case of number theory of 8 prime numbers. The Goldbach problem contained in the eighth question has not been solved so far. Mathematicians in China have done a series of excellent work in this field.

Proof of the most general reciprocal law in arbitrary number field. The field-like theory has been solved by Takagi Sadako (192 1) and E.Artin( 1927).

10 discriminant uncertainty analysis of the solvability of Diophantine equation 1970 mathematicians in the Soviet Union and the United States proved that the general algorithm expected by Hilbert did not exist.

The quadratic quadratic theory H. Hasse (1929) and C. L. Siegel (1936, 195 1) with arbitrary algebraic coefficients have obtained important results on this issue.

The Kroneker theorem on 12 Abel field is extended to any algebraic rational number field. The theory of complex multiplication has not been solved.

13 It is impossible to solve the ordinary seventh-order equation with a function with only two variables. The continuous function of equation theory and real function theory was denied by Soviet mathematicians in 1957. If you need to parse the function, the problem remains unsolved.

14 proves that the finite algebraic invariant theory of a complete function system gives a negative solution.

15 strict basic algebraic geometry of Schubert counting calculus Due to the efforts of many mathematicians, it is possible to treat the foundation of Schubert calculus purely by algebraic method, but the rationality of Schubert calculus remains to be solved. As for the foundation of algebraic geometry, it has been established by B.L. Vander Waals Deng (1938-40) and A.Weil (1950).

Topological curves and surfaces of 16 algebra, surface topology and the first half of qualitative theoretical problems of ordinary differential equations have obtained important results in recent years.

The theory of square expression domain (real domain) in positive definite form was solved by Artin in 1926.

18 is partially solved by the theory of space crystal group of congruent polyhedron.

Whether the solution of 19 regular variational problem must analyze the theory of elliptic partial differential equations has been solved in a sense.

20 general boundary value problems elliptic partial differential equation theory The research on boundary value problems of partial differential equations is developing vigorously.

2 1 Existence of linear partial differential equations with given value groups The large-scale theory of linear ordinary differential equations has been solved by Hilbert himself (1905) and H.Rohrl (Germany, 1957).

P.Koebe (Germany, 1907) has solved the case of univalent Riemannian surfaces with one variable of analytic relation.

23 Further Development of Variational Method Hilbert himself and many mathematicians have made important contributions to the development of Variational Method.

Congress of Mathematicians a Hundred Years ago and Hilbert Problem

Xiong Weimin

2/kloc-0 The first international congress of mathematicians will be held in Beijing soon. What will it bring to the development of mathematics in this century? Can it influence the direction of mathematics development like the first international congress of mathematicians in the 20th century? A century ago, the Congress of Mathematicians went down in history forever because of a man, because of his report-David Hilbert and his mathematical problems.

1900, Hilbert put forward his famous 23 mathematical problems at the second international congress of mathematicians held in Paris. In the following half century, many world-class mathematical minds surrounded them. As another very famous mathematician H. Weyl said, "Hilbert blew his magic flute, and groups of mice jumped into the river after him." No wonder his questions are so clear and easy to understand, and some of them are so interesting that many laymen are eager to try, and solving any one of them, or making a major breakthrough on any one of them, will immediately become famous all over the world-Chen Jingrun of China has made great contributions to solving the eighth problem of Hilbert (that is, the prime number problem, including Riemann conjecture and Goldbach conjecture, etc.). ) and attracted worldwide attention. When people summarize the development of mathematics in the twentieth century, especially in the first half of the twentieth century, Hilbert's problem is usually used as a navigation mark.

In fact, most of these problems have already existed, and Hilbert did not raise them first. But he stood at a higher level, raised these problems again in a sharper and simpler way, and pointed out the direction to solve many of them.

There are many problems in the field of mathematics. Which are more important and basic? Making such a choice requires keen insight. Why can Hilbert be so angry? Mr. Yuan Xiangdong (co-translated with Mr. Li Wenlin), a historian of mathematics, a researcher at the Institute of Mathematics and Systems Science of China Academy of Sciences, and a translator of the book Hilbert-Alexander in the Kingdom of Mathematics, believes that this is because Hilbert is the Alexander in the Kingdom of Mathematics! Mathematicians can be divided into two categories, one is good at solving mathematical problems, the other is good at summarizing the existing situation theoretically, and both categories can be subdivided into first-class, second-class and third-class. Hilbert is good at both. He traveled almost all the front positions of modern mathematics, left his distinguished name in many different branches of mathematics, knew the background of mathematics development like the back of his hand, and made in-depth research on many problems mentioned. He is the king in the field of mathematics.

Why did Hilbert sum up the basic problems of mathematics at the conference instead of preaching one of his achievements like ordinary people? Yuan Xiangdong told reporters that this is related to another great mathematician, Henri Poincaré, who gave a report on applied mathematics at the first international congress of mathematicians held in 1897. Both of them were Gemini in the international mathematics field at that time, and they were both leading figures. Of course, there is also a certain competitive psychology. Because Poincare talked about his general view on the relationship between physics and mathematics, Hilbert made some defense for pure mathematics.

Poincare is French, Hilbert is German, and France and Germany have feuds, so the competition between them has the flavor of national competition. Although they respect each other very much, which is not obvious to them, their students and teachers often see it this way.

Hilbert's teacher, Felix Klein, is a man with strong national consciousness. He attached great importance to the development of German mathematics and wanted to turn the international mathematics circle into an ellipse-a circle with Paris as the center; Now he wants to make his city of G? ttingen the center of world mathematics and the world of mathematics an ellipse with two centers.

With the help of Hilbert and his close friend hermann minkowski, Klein achieved his goal-by 1900, Hilbert was as famous as Poincare, the greatest French mathematician, and Klein himself and Minkowski, who was coming to G? ttingen, were also very influential mathematicians. In fact, they are called "invincible three professors" in Germany.

You can imagine their charm from an example.

One day, when talking about the four-color theorem, a famous theorem of topology, Minkowski suddenly had a brainwave, so he said to the students in the room, "This theorem has not been proved, because only some third-rate mathematicians have studied it so far. Now it's up to me to prove it. " Then he picked up the chalk and proved this theorem on the spot. After this course, he hasn't finished his certificate. He continued to testify in the next class for several weeks. Finally, on a rainy morning, as soon as he stepped onto the platform, there was a thunderbolt in the sky. "God was angered by my arrogance," he said. "My proof is also incomplete." This theorem was not proved by computer until 1994. )

19 12, poincare is dead. The center of world mathematics has further shifted to Gottingen, and the mathematical world seems to have become a circle again-only the center of the circle has been changed to Gottingen. At this time, the reputation of Gottingen School is in full swing, and the popular slogan among young mathematicians is "Pack your bags and go to Gottingen!"

A century later, about half of the 23 problems listed by Hilbert have been solved, and most of the remaining half have also made significant progress. But Hilbert himself didn't solve any problems. Someone asked him why he didn't solve his own problems, such as Fermat's last theorem.

Fermat wrote the theorem in the margin of a page, and claimed that he had come up with a wonderful proof, but the margin was not big enough to write it down. Hilbert's answer is also humorous: "I don't want to kill this hen that can only lay golden eggs"-a German entrepreneur set up a foundation to reward the first person who solved Fermat's Law. Hilbert was the chairman of the foundation at that time, and used the interest of the fund to invite outstanding scholars to give lectures in G? ttingen every year, so for him, Fermat's Law is a hen that can only lay golden eggs. Fermat's law was not solved until 1997. )

Before listing 23 questions, Hilbert has been recognized as a leading figure in the international mathematics field and has made many important achievements in many fields of mathematics. His other contributions, such as his axiomatic thought, formalism thought, and the book Geometry Basis, had a far-reaching impact on the development of mathematics in the 20th century.

1265438+Seven Mathematical Problems in the 20th Century

Seven mathematical problems in 2 1 century

Recently, on May 24th, 2000, the Clay Institute of Mathematics in Massachusetts, USA, announced an event that was heated up by the media at the Institut de France in Paris: offering a reward of $/kloc-0,000,000 to collect seven "Millennium Mathematical Puzzles". Here is a brief introduction to these seven difficult problems.

One of the Millennium Problems: P (polynomial algorithm) versus NP (non-polynomial algorithm)

On a Saturday night, you attended a grand party. It's embarrassing. You want to know if there is anyone you already know in this hall. Your host suggests that you must know Ms. Ross sitting in the corner near the dessert plate. You don't need a second to glance there and find that your master is right. However, if there is no such hint, you must look around the whole hall and look at everyone one by one to see if there is anyone you know. Generating a solution to a problem usually takes more time than verifying a given solution. This is an example of this common phenomenon. Similarly, if someone tells you that the numbers 13, 7 17, 42 1 can be written as the product of two smaller numbers, you may not know whether to believe him or not, but if he tells you that you can factorize it into 3607 times 3803, then you can easily verify this with a pocket calculator. Whether we write a program skillfully or not, it is regarded as one of the most prominent problems in logic and computer science to determine whether an answer can be quickly verified with internal knowledge, or it takes a lot of time to solve it without such hints. It was stated by StephenCook in 197 1.

The second Millennium puzzle: Hodge conjecture

Mathematicians in the twentieth century found an effective method to study the shapes of complex objects. The basic idea is to ask to what extent we can shape a given object by bonding simple geometric building blocks with added dimensions. This technology has become so useful that it can be popularized in many different ways; Finally, it leads to some powerful tools, which make mathematicians make great progress in classifying various objects they encounter in their research. Unfortunately, in this generalization, the geometric starting point of the program becomes blurred. In a sense, some parts without any geometric explanation must be added. Hodge conjecture asserts that for the so-called projective algebraic family, a component called Hodge closed chain is actually a (rational linear) combination of geometric components called algebraic closed chain.

The third "Millennium mystery": Poincare conjecture

If we stretch the rubber band around the surface of the apple, then we can move it slowly and shrink it into a point without breaking it or letting it leave the surface. On the other hand, if we imagine that the same rubber belt is stretched in a proper direction on the tire tread, there is no way to shrink it to a point without destroying the rubber belt or tire tread. We say that the apple surface is "single connected", but the tire tread is not. About a hundred years ago, Poincare knew that the two-dimensional sphere could be characterized by simple connectivity in essence, and he put forward the corresponding problem of the three-dimensional sphere (all points in the four-dimensional space at a unit distance from the origin). This problem became extremely difficult at once, and mathematicians have been fighting for it ever since.

The fourth "billion billion puzzles": Riemann hypothesis

Some numbers have special properties and cannot be expressed by the product of two smaller numbers, such as 2, 3, 5, 7, etc. Such numbers are called prime numbers; They play an important role in pure mathematics and its application. In all natural numbers, the distribution of such prime numbers does not follow any laws; However, German mathematician Riemann (1826~ 1866) observed that the frequency of prime numbers is closely related to the behavior of a well-constructed so-called Riemann zeta function z(s$). The famous Riemann hypothesis asserts that all meaningful solutions of the equation z(s)=0 are on a straight line. This has been verified in the original 1, 500,000,000 solutions. Proving that it applies to every meaningful solution will uncover many mysteries surrounding the distribution of prime numbers.

The fifth of "hundreds of puzzles": the existence and quality gap of Yang Mill.

The laws of quantum physics are established for the elementary particle world, just as Newton's classical laws of mechanics are established for the macroscopic world. About half a century ago, Yang Zhenning and Mills discovered that quantum physics revealed the amazing relationship between elementary particle physics and geometric object mathematics. The prediction based on Young-Mills equation has been confirmed in the following high-energy experiments in laboratories all over the world: Brockhaven, Stanford, CERN and Tsukuba. However, they describe heavy particles and mathematically strict equations have no known solutions. Especially the "mass gap" hypothesis, which has been confirmed by most physicists and applied to explain the invisibility of quarks, has never been satisfactorily proved mathematically. The progress on this issue needs to introduce basic new concepts into physics and mathematics.

The Sixth Millennium Problem: Existence and Smoothness of Navier-Stokes Equation

The undulating waves follow our ship across the lake, and the turbulent airflow follows the flight of our modern jet plane. Mathematicians and physicists are convinced that both breeze and turbulence can be explained and predicted by understanding the solution of Naville-Stokes equation. Although these equations were written in19th century, we still know little about them. The challenge is to make substantial progress in mathematical theory, so that we can solve the mystery hidden in Naville-Stokes equation.

The seventh "Millennium Mystery": Burch and Swinerton Dale's conjecture.

Mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as x 2+y 2 = z 2. Euclid once gave a complete solution to this equation, but for more complex equations, it became extremely difficult. In fact, as a surplus. V.Matiyasevich pointed out that Hilbert's tenth problem is unsolvable, that is, there is no universal method to determine whether such a method has an integer solution. When the solution is a point of the Abelian cluster, Behe and Swenorton-Dale suspect that the size of the rational point group is related to the behavior of the related Zeta function z(s) near the point s= 1. In particular, this interesting conjecture holds that if z( 1) is equal to 0, there are infinite rational points (solutions); On the other hand, if z( 1) is not equal to 0, there are only a limited number of such points.