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 Steven Kirk 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.