1900, Hilbert was invited to attend the International Congress of Mathematicians in Paris and gave an important speech entitled "Mathematical Problems". In this historic speech, he put forward many important ideas:
Because everyone pursues the goal, because the same mathematical research also needs their own problems. It is by solving these problems that researchers will find new ideas and reach a broader realm of freedom by exercising their iron.
Hilbert especially emphasized the role in the main problems of mathematical development. He said: "If we want the closest future development of mathematical knowledge to be a concept, it must review the problems that current science hopes to solve in the future", while another said: "Undeniably, a branch of science can raise a large number of problems that have a profound impact on the general mathematical process and the work of individual researchers. The problem is that it is full of vitality and lacks signs of decline or suspension of independent development. "
He expounded the main problems related to characteristics. A good question should have the following three characteristics:
Clear and easy to understand;
Although difficult, there is hope;
Meaningful.
He analyzed the difficulties often encountered in learning mathematical problems and some methods to overcome them. At that time, his mathematician in the new century proposed that the congress should try to solve 23 problems, namely the famous "Hilbert 23 problem".
Instead of solving the problem, it promoted the situation on the field.
1 continuum assumes the development of axiomatic set theory in 1963, and Paul J.Cohen proves that the first problem is insoluble in this sense. This continuum hypothesis cannot be proved in the zermelo-Frankel axiomatic system.
The mathematical basis of the compatibility of Hilbert's two arithmetic axioms proves the compatibility axiom of arithmetic? However, in 193 1, Godel's incompleteness theorem holds that it is impossible to prove the compatibility of the arithmetic axioms of meta-mathematics. The problem of compatibility is still unsolved in mathematics.
Three volumes and two volumes are equal to tetrahedral configuration, and other high-end Hilbert students M. Dehn( 1900) gave a positive answer.
The basis of the geometric problem of the shortest distance between two points on a straight line, asking this question is too general. After Hilbert, many mathematicians devoted themselves to exploring various special structures and geometric metrics, and made great progress in the fourth study, but the problems were not completely solved.
5. Don't make long-term efforts to define the differentiable function group of the hypothetical topological Lie group theory. This problem 1952 was finally solved by Gleason, Montek Merry, Compression and others, and the answer was yes.
In the field of quantum mechanics and thermodynamic physics, the mathematical treatment and axiomatic method of axiom 6 of mathematical physics have been very successful, but in general, what this means is self-evident physics is still a problem to be discussed. AHKonmoropob and others established axiomatic probability theory.
7 Some irrational numbers and transcendental number theory 1934 transcend AOtemohm and Schneieder to solve the second half of this problem independently. The conjecture of eight prime numbers is still a conjecture on the whole. Goldbach problem, including the eighth problem that has not been solved so far. Mathematicians in China have produced a series of excellent works. Any number of domains in the most general domain-like theory, which mutually prove the resistance to the law.
True wisdom was solved by Takagi (192 1) and E.Artin( 1927).
10 Diophantine equation has a solution. The Soviet Union, the United States and mathematicians analyzed the discriminant variables. In 1970, it is proved that the general algorithm expected by Hilbert does not exist.
Quadratic quadratic H. Hasse (1929) and Cl Siegel (1936, 195 1) have made remarkable achievements in any algebraic number theory of this problem. Last/better/previous/last name
12 abelian domain kroneker theorem arbitrary algebraic rational number domain. The theory of complex multiplication has not been solved so far.
13 can't just use the seven-way solution function of binary linear equation. Equation theory and negative solution were put forward by Soviet mathematicians. This requirement is that the analytic function is a continuous function of real function in 1957, so this problem has not been solved.
14 proves that the complete theory of algebraic invariants in finite classrooms is based on the function of John Tian Yayi in 1958, and gives a negative solution.
Based on the strict symbols of Schubert calculus in algebraic geometry, due to the efforts of many mathematicians, Schubert calculus can always be treated on the basis of pure algebra, but the reasonable Schubert calculus has been solved. With algebraic geometry, the foundation established by BLVander Waerden( 1938-40) and A.Weil( 1950). analysis situs
16 topological algebraic curves and curves and surfaces, the former problem, half of the qualitative theory of ordinary differential equations, have also achieved remarkable results in recent years.
The explicit form of expression domain (real number domain) in Anting Square of 17 is solved in 1926. Partial solution of crystal structure through space group theory
18 congruent polyhedron. solution
19 periodic variational problem has a certain feeling. Elliptic partial differential equation theory solves this problem.
The research of partial differential equations is developing vigorously. The boundary value problem of elliptic partial differential equations is 20 general theories.
2 1 The theory of linear sequence-valued partial differential equations with linear existence can solve various problems in Hilbert I (1905) and H. Rolle (Germany, 1957).
In this paper, we solve the Riemann surface of the variable case of P.Koebe (Germany, 1907) 22.
Variational method Hilbert himself and many mathematicians have made important contributions to the further development of the 23 variational method.
A hundred years ago, the problem between Congress and Hilbert was crucial.
2 1 century, 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 be the direction of the first international mathematics conference in the 20th century? The reason why congressional mathematicians are always a century ago is only because of one person, because of the history of his report-"Mathematical Problems" david hilbert and his
1900, Hilbert put forward his famous mathematical problem 23 at the second international congress of mathematicians held in Paris. In the following half century, many world-class mathematical ideas turned them around. Only in another case, the famous mathematician H. Weil said, "Hilbert blew his magic flute and all the mice jumped into the river after him." No wonder his question is so clear and easy to understand. Some of them are interesting enough to make many laymen eager to try and solve any one or any major breakthrough in a problem, and we can immediately name the names of all over the world-our Chen, because in the first eight years, the Hilbert problem (that is, the prime number problem) has been solved, including Riemann conjecture and Goldbach conjecture. ), we must make the world of eyebrows a remarkable world. It summarizes the development of mathematics in the twentieth century, which is commonly called Hilbert bonfire problem, especially development.
In fact, most of these problems have already existed, and Hilbert did not raise them first. But his position, by going up one flight of stairs, is that there are clearer and simpler ways to raise these issues again and point out the direction in solving many problems.
Mathematics is a lot of problems. What is more important and basic? Making such a choice requires keen insight. Why can Hilbert be so angry? Mathematic historian, researcher of School of Mathematics and System Science of China Academy of Sciences, China, translator of the book "Hilbert Mathematical Kingdom of Alexandria" by Mr. Bird Xiangdong (translated by Mr. Li Wenlin), this is because of the Hilbert Mathematical Kingdom of Alexandria! Mathematicians can be divided into two categories. They are good at solving mathematical problems and thus make a good theoretical summary of the present situation. In addition, they can be divided into primary, secondary and tertiary categories. Hilbert's two kinds, the long journey and almost all the frontiers of modern mathematics, some left his prominent name under the background of the development of many problems mentioned in the divergence of major branches of mathematics, and he has in-depth research in the field of mathematics, "land king".
Why did Hilbert sum up the basic problems of mathematics and preach its concrete results at the meeting, instead of ordinary people? The expression of the image tells the reporter that this is related to Poincare's report on Poincare's mathematical application at the first international congress of mathematicians held in 1897. They are two international mathematicians in Gemini. Of course, the two leading figures also have some competitive psychology-his general view of physics. Poincare told Relationship that some people in Hilbert have defended pure mathematics since then.
Poincare and Hilbert in France are feuds between Germany, France and Germany, so the competition between them also smacks of competing countries. Although everyone respects each other, although it is not so obvious, as students and teachers, they often think so.
In a sense, Hilbert's teacher Felix Klein is a very powerful country. He attaches great importance to the development of German mathematics and hopes to become the oval front circle of international mathematics and the center of Paris. Now, he wants to imitate Gottingen's mathematics in their city and divide the mathematical circle into two, so that he can make the center of the ellipse?
With the help of Hilbert and his close friends hermann minkowski and Klein, he achieved his goal-1900. Hilbert and French mathematician Poincare have always been equal, and Klein himself soon came to G? Gottingen Minkowski is also a very influential mathematician. In fact, they are called "invincible professors" in Germany.
For example, you can imagine their charm.
One day, while talking about the famous theorem of topology-the four-color theorem, Minkowski suddenly had an idea, so he said to the students in the room: "This theorem has not been proved, because so far, only some third-rate mathematicians have studied it. Now I will prove it. " After that, he picked up chalk at the scene to prove this theorem. After this class, he hasn't finished his card yet. It took him several weeks to receive this certificate. Finally, on a rainy morning, he stepped onto the platform and there was a bolt from the blue in the sky. "God also angered my arrogance," he said. "I proved that it was incomplete." (This theorem was not proved by computer until 1994. )
19 12, poincare is dead. ? After the transfer of Gottingen, the center of the mathematical world in G, mathematics seems to be a circle-but the center replaces the imitation of Gottingen? At this time, the popular slogan of young mathematics, the reputation of Gottingen School, is "Take up your blanket and come to Gottingen!"
A century later, about half of the 23 problems listed by Hilbert have been solved, and most of the remaining half have made remarkable progress. But Hilbert himself didn't solve any problems. Someone asked him why he couldn't solve the problems he mentioned, such as Fermat's last theorem.
Fermat's last theorem is written in a blank book. He also claimed that he had come up with a wonderful card method, but unfortunately there was not enough space to write it down. Hilbert's answer humorously expressed the same meaning: "I don't want to kill this golden hen"-a German entrepreneur set up a foundation award for the first person to solve Fermat's Law, Hilbert served as his foundation, and on the annual interest of the chairman's fund, please make full use of outstanding scholars to give lectures in G? ttingen, so for him, Fermat's Law is just a golden hen. Fermat's law was not solved until 1997. )
Before listing 23 questions, Hilbert had recognized the international leader in the field of mathematics and made some important achievements in many fields of mathematics. His other contributions, such as his self-evident propositional formalism and the book Fundamentals of Geometry, had a far-reaching impact on the development of mathematics in the 20th century.
1265438+mathematical problems in the 7th century
Mathematical problems in 265438+7th century
Recently, on May 24th, 2000, the Clay Institute of Mathematics in Clay, Massachusetts announced a media event in Paris: seven "Millennium Mathematics Problems", each with a million dollars. Here is a brief introduction to the seven challenges. Among them, the "Millennium mystery"
: NP (non-polynomial algorithm) of P (polynomial algorithm) problem
Question, you are attending a grand party. Because they are embarrassed, you want to know if anyone else in the hall already knows. Your host hints to you that you must know who Mrs. dessert plate is, and Ross is just around the corner. It doesn't take a second, you can see there at a glance and find that your master is right. However, if there is no such hint, you should look around the room and check everyone one by one to see if there is anyone you know. Generating a solution to this 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 3607 can be decomposed into 3803, then you can easily verify this with a pocket calculator. Whether we are clever in programming can quickly determine that the answer is to verify the application of internal knowledge. There is no such hint, or it takes a lot of time to solve it, which is considered as one of the most prominent problems in logic and computer science. This was declared by StephenCook at 197 1
The second Millennium puzzle: Hodge conjecture
In the twentieth century, mathematicians found an effective method to study the complex shapes of objects. The basic idea is to what extent we can create simple geometric keys by adding dimensions to form objects with a given block shape. This technology has become so useful that it can be popularized in many different ways; Finally, it led to some powerful tools, so that mathematicians made great progress in learning the classification of various objects. Unfortunately, under this impetus, the geometric points of the departure plan become blurred. In a sense, it is not necessary to add some geometric explanations of parts. Hodge conjecture asserts that the so-called projective algebraic cluster, a particularly perfect space type, is called Hodge closed chain, which is actually called algebraic geometry closed chain component (rational linear) combination.
The Third Millennium Problem: Poincare
Think about it, if our flexible rubber band surrounds the surface of an apple, then we can tear it off, don't leave it on the surface, let it move slowly and shrink into a point. On the other hand, if we imagine the same rubber band stretching in the right direction on the tire surface, then don't tear or trample the rubber band. Is there any way to make it shrink a little? We say that the surface of apple is "simply connected", not the tread. About a hundred years ago, Poincare knew that a three-dimensional sphere is essentially a two-dimensional sphere (there is a problem that all units in a four-dimensional space correspond to each other from the origin), which is described by a single connection. This problem became very difficult at once, and mathematicians have been working hard ever since.
The Fourth Millennium Problem: Riemann Hypothesis
Some numbers are not expressed by the product of two smaller numbers with special properties, such as 2, 3, 5, 7 and so on. Such numbers are called prime numbers; They play an important role in pure mathematics and its application. In all natural numbers, this distribution of prime numbers does not follow any laws. However, the German mathematician Riemann (1826 to 1866) pointed out that the function with tight prime number frequency calls the tower correlation carefully constructed by Riemann Caiyi (the behavior of Singapore dollar). The famous Riemann hypothesis asserts that the equation Z(S)= 0 is on a straight line for all meaningful solutions, and it is always the solution of 1 500,000,000. It is proved that it is a meaningful solution for every given solution, which will bring many mysteries around the prime number of light distribution.
The fifth Millennium problem: the existence and quality gap of Yang Mill.
> The law of quantum physics is a way to establish a basic particle world based on Newton's law from classical mechanics to macro world. About half a century ago, Yang Zhenning and Mills discovered that quantum physics revealed basic particle physics. Impressive mathematics and high-energy experiments based on the geometric relationship between young objects-Mills equation in laboratories all over the world has been predicted as the fulfillment of those diagnosed: broca Wen, Stanford, CERN and Tsukuba. But they all describe heavy particles, and there is no known solution in the strictness of mathematical equations. In particular, people have realized most physicists and their respect. The explanation of "quark" invisibility is applicable to the hypothesis of "mass gap" and has never been proved to be mathematically satisfactory. The progress of this problem needs to introduce two new foundations: physics and mathematics. idea
The Sixth Millennium Puzzle: Naville-Existence and Smoothness
Mathematicians and physicists who follow us with the ups and downs of lake winds are shuttle boats. Those who follow our modern planes with the rapid current are convinced that their explanations and predictions can be solved by Naville's understanding-Stoalex equation, whether it is breeze or turbulence. Although these formulas were written in19th century, we still know little about them. The challenge is to make progress in mathematical theory so that we can solve the mystery hidden in Naville-Stokes equation.
The Seventh "Millennium Problem": The Conjecture of Berch and Svenner's Pass-Dale (Swinerton-Dale)
Because the integer solution of algebraic equation is fascinating, mathematicians have always described the problem as X 2+Y 2 = Z 2. Euclid must give a complete answer to this equation, but it becomes very difficult for more complicated equations. 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 this method has an integer solution. When the solution is a cluster of Abelian points, Berg and Svineton-Dale think that the suspect, a group of rational points, has a series of Cai functions Z with the size of S = 1 (S) near the state point. Especially this interesting conjecture, when z( 1) is equal to 0, there are infinite rational points (solutions). Conversely, when z( 1) is not equal to 0, there are only a limited number of such points.