The 20th century is a century of great development of mathematics. Many important problems in mathematics have been satisfactorily solved, such as the proof of Fermat's last theorem and the completion of the classification of finite simple groups, thus making the basic theory of mathematics develop unprecedentedly.
The appearance of computer is a great achievement in the development of mathematics in the 20th century, which greatly promoted the deepening of mathematical theory and the direct application of mathematics in the front line of society and productivity. Looking back on the development of mathematics in the 20th century, mathematicians deeply thank david hilbert, the greatest master of mathematics in the 20th century. 1900 At the 2nd World Congress of Mathematicians held in Paris on August 8th, Hilbert put forward 23 mathematical problems in his famous speech. Hilbert problem has inspired the wisdom of mathematicians and guided the direction of mathematics in the past hundred years, and its influence and promotion on the development of mathematics is enormous and immeasurable.
Taking Hilbert as an example, many famous mathematicians in the contemporary world have sorted out and put forward new mathematical problems in the past few years, hoping to point out the direction for the development of mathematics in the new century. These mathematicians are very famous, but their actions have not attracted the attention of the world mathematics community.
At the beginning of 2000, the Scientific Advisory Board of the Clay Institute of Mathematics in the United States selected seven "Millennium Prize Questions", and the board of directors of the Clay Institute of Mathematics decided to set up a grand prize fund of 7 million dollars, and each "Millennium Prize Question" could be awarded10 million dollars. The purpose of the selection of "Millennium Prize Problem" in Clay Institute of Mathematics is not to form a new direction of mathematics development in the new century, but to focus on the center of mathematics development and the major problems that mathematicians dream and expect to solve.
On May 24th, 2000, the Millennium Mathematics Conference was held in the famous French Academy. At the meeting, Gavos, winner of the 1998 Faldts Prize, gave a speech on the topic of "the importance of mathematics". Later, Tate and Atia announced and introduced these seven "Millennium Prize Issues". Clay Institute of Mathematics also invited experts in related research fields to elaborate on each issue. Clay Institute of Mathematics has made strict regulations on the answer and award of the "Millennium Prize Question". Every "Millennium prize problem" is not solved immediately. Any solution must be published in a world-renowned mathematical magazine for two years and recognized by the mathematical community before it can be examined and decided by the scientific advisory Committee of Clay Institute of Mathematics whether it is worth winning a million dollars.
Seven mathematical problems in the world
These seven "Millennium Prize problems" are NP complete problem, Hodge conjecture, Poincare conjecture, Riemann hypothesis, Jan-Mills theory, Naville-Stoke equation and BSD conjecture.
Among them, Poincare conjecture has been cracked by Professor Zhu Xiping of Sun Yat-sen University in China and Cao Huaidong, an American mathematician and part-time professor in Tsinghua University.
Since its publication, Millennium Prize Magazine has had a strong response in the field of mathematics. These problems are all about the basic theory of mathematics, but the solution of these problems will greatly promote the development and application of mathematical theory. Understanding and studying the "Millennium Prize" has become a hot spot in mathematics. Mathematicians in many countries are organizing joint research. It can be expected that the "Millennium Prize Problem" will change the historical process of mathematics development in the new century.
One of the Millennium Problems: P (Polynomial Algorithm) Problem vs 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 number 13, 7 17, 42 1 can be written as the product of two smaller numbers, you may not know whether to believe him, but if he tells you that this number can be factorized into 3607 times 3803, then you can easily verify that it is correct 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. This was put forward by Steven Kirk in 197 1 year.
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, some powerful tools were used to make mathematicians make great progress in classifying various objects they encountered 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 puzzle": 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.
On June 3rd, Xinhua News Agency reported that Professor Zhu Xiping of Sun Yat-sen University cooperated with Cao Huaidong, an American mathematician and an adjunct professor in Tsinghua University, and solved the Poincare conjecture, a major problem that has been concerned by the international mathematics community for hundreds of years.
The fourth Millennium puzzle: 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 "Millennium Problem": 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 Puzzle: The Conjecture of Burch and Swinerton Dale.
Mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as x2+y2=z2. Euclid once gave a complete solution to this equation, but for more complex equations, it became extremely difficult. In fact, as Matthiasevich pointed out, 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.