"Strange" Part II: Hodge suspects that mathematicians in the 20th century have found a powerful 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 of "hundreds of puzzles": Poincare guessed that if we stretch the rubber band around the surface of the apple, then we can make it move 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 mystery of the Millennium: Riemann assumes that some numbers have special properties and cannot be expressed as the product of two smaller numbers, 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, 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 laws of Yang-Mills existence and mass gap are established for the elementary particle world in the way of Newton's law of classical mechanics for the macro 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 6th Millennium Problem: Existence and Smoothness of Naville-Stokes Equation. The ups and downs of the 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 Problem: Birch and Swinerton-Dale conjecture Mathematicians are always fascinated by the problem of describing 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.
Eight: the problem of drawing geometric ruler "The problem of drawing geometric ruler" here refers to the restriction that only rulers and compasses can be used in drawing. The ruler here refers to the ruler that can only draw straight lines without scale. The problem of geometric ruler drawing includes the following four problems: 1. Turn a circle into a square-find a square so that its area is equal to the known circle; 2. Divide any corner into three equal parts; 3. Double Cube-Find a cube, and make its volume twice that of the known cube. 4. Make a regular heptagon. The above four problems have puzzled mathematicians for more than two thousand years, but in fact, the first three problems have been proved impossible to be solved with a ruler and compass in a limited number of steps. The fourth problem was solved by Gauss in algebra. He also regarded it as a masterpiece of his life and told him to carve the regular heptagon on his tombstone. But later, his tombstone was not engraved with a heptagon, but with a 17 star, because the sculptor in charge of carving the monument thought that the heptagon was too similar to the circle, so everyone would be confused.
Nine: Goldbach Conjecture 1742 On June 7, Goldbach wrote to the great mathematician Euler at that time and put forward the following conjectures: (a) Any one >; Even number =6 can be expressed as the sum of two odd prime numbers. (b) Any odd number > 9 can be expressed as the sum of three odd prime numbers. Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 200 years have passed and no one has proved it. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics.
Ten: Four-color conjecture 1852. When Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, which makes countries with the same border painted with different colors." 1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. 1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. The computer proof of the four-color conjecture has caused a sensation in the world.