Eight major problems

The first seven puzzles are recognized as the seven major problems, and the eighth puzzle is one of the three major conjectures in the world.

One:

P (polynomial algorithm) problem vs. NP (non-polynomial algorithm) problem

On a Saturday night, you attend a gala party. Feeling cramped and uneasy, you wonder if there are people in this hall that you already know. Your host proposes to you that you must know the lady Rose who is in the corner near the dessert plate. It doesn't take a second for you to glance there and see that your master is right. However, without such a hint, you have to look around the hall and look at each one one one to see if there is anyone you know. It usually takes much more time to generate a solution to a problem than it does to validate a given solution. This is an example of this general phenomenon.

Similarly, if someone tells you that the number 13,717,421 can be written as the product of two larger numbers, you may not know whether you should trust him or not, but if he tells you that it can be factorized into 3607 times 3803, then you can easily verify that this is correct with a pocket calculator. Regardless of whether we are clever in writing a program, determining whether an answer can be quickly verified with internal knowledge or takes a lot of time to solve without such a hint is considered one of the most prominent problems in logic and computer science. It was stated by Steven Cook in 1971.

Two:

Hodge conjecture

Mathematicians in the twentieth century discovered powerful ways to study the shape of complex objects. The basic idea is to ask to what extent we can take the shape of a given object by gluing together simple geometric building blocks of increasing dimensions. This technique has become so useful that it can be promoted in many different ways; Eventually, some powerful tools have enabled mathematicians to make great strides in classifying the wide variety of objects they encounter in their research. Unfortunately, in this promotion, the geometric starting point of the program becomes blurred. In a sense, certain parts must be added without any geometric explanation. The Hodge conjecture asserts that for a particularly perfect type of space, such as projective algebraic clusters, the components called Hodge closures are actually (rational linear) combinations of geometric components called algebraic closures.

Three:

Poincare's conjecture (which has been proven) is that if we retract the rubber belt around the surface of an apple, then we can neither tear it off nor let it off the surface, causing it to slowly move and shrink into a point. On the other hand, if we imagine that the same rubber belt is stretched on a tire face in the appropriate direction, then there is no way to shrink it to a point without tearing off the rubber belt or tire face. We, the apple surface is "single-connected", while the tread is not. About a hundred years ago, Poincaré already knew that a two-dimensional sphere could essentially be characterized by a single connection, and he proposed the correspondence of a three-dimensional sphere (the whole of points in four-dimensional space that are at a unit distance from the origin). The problem immediately became extremely difficult, and mathematicians have been fighting over it ever since.

Four:

Riemann hypothesis

Some numbers have special properties that cannot be expressed as the product of two more numbers, e.g., 2, 3, 5, 7, and so on. Such a number is called a prime number; They play an important role in both pure mathematics and its applications. In all natural numbers, the distribution of such prime numbers does not follow any regular pattern; However, the German mathematician Riemann (1826~1866) observed that the frequencies of prime numbers are closely related to the properties of a well-constructed so-called Riemann Zeita function z(s$. The famous Riemann hypothesis asserts that all meaningful solutions to the equation z(s)=0 are in a straight line. This has been verified for the first 1,500,000,000 solutions. Proving that it holds true for every meaningful solution will shed light on many mysteries surrounding the distribution of prime numbers.

Five:

Yang-Mills's laws of existence and mass gap quantum physics are held for the world of elementary particles in the same way that Newton's laws of classical mechanics apply to the macroscopic world. About half a century ago, Yang and Mills discovered that quantum physics revealed a remarkable relationship between elementary particle physics and the mathematics of geometric objects. The prediction based on the Young-Mills equation has been confirmed in high-energy experiments performed in laboratories around the world: Brockhaven, Stanford, the European Institute for Particle Physics, and Tsukuba. Despite this, there is no known solution to their mathematically rigorous equation that describes both heavy particles. In particular, the "mass gap" hypothesis, confirmed by most physicists and applied in their explanations of the invisibility of "quarks", has never been mathematically satisfactorily confirmed. Progress on this issue requires the introduction of fundamentally new ideas, both physical and mathematical.

Six:

The existence and smoothness of the Navier-Stokes equations, the undulating waves that follow our boat as it winds its way through the lake, and the turbulent air currents that follow our modern jet planes, the mathematicians and physicists of the flight school are convinced that both the breeze and the turbulent currents can be explained and predicted by understanding the solutions to the Navier-Stokes equations. Although these equations were written in the 19th century, we still know very little about them. The challenge is to make substantial progress in mathematical theory that will enable us to unravel the mysteries hidden in the Navier-Stokes equations.

Seven:

Birch and Sinnerton-Dyer conjecture

Mathematicians have always been fascinated by the problem of characterizing 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, this becomes extremely difficult. In fact, as Mattyasievich (. Matiyasevich points out that Hilbert's tenth problem is unsolvable, i.e., there is no general way to determine whether such a method has an integer solution. When the solution is a point of an Abelian cluster, the Behe and Swinaton-Dale conjectures that the large size of the group of rational points is related to the properties of the Zeita function z(s) near the point s=1. In particular, this interesting conjecture holds that if z(1) is equal to 0, then there are infinitely many rational points (solutions), and conversely, if z(1) is not equal to 0, then there are only a finite number of such points.

Eight:

Goldbach conjecture

In a letter to Euler dated 7 June 1742, Goldbach proposed the following conjecture: a) any even number other than 6 can be expressed as the sum of two odd prime numbers; b) Any odd number other than 9 can be expressed as the sum of three odd prime numbers. In his reply, Euler also proposed another equivalent version, that is, any even number greater than 2 can be written as the sum of two prime numbers. These two propositions are often referred to collectively as the Goldbach conjecture. The proposition "any large even number can be expressed as the sum of the number of prime factors not exceeding a and another number of prime factors not exceeding b" is recorded as "a+b", and the Goehl's conjecture is to prove that "1+1" is true. In 1966, Chen Jingrun proved the establishment of "1+2", that is, "any large even number can be expressed as the sum of a prime number and another number with no more than 2 prime factors". [7]