The full name of the famous NP problem in mathematics is NP complete problem, that is, "NP complete" problem, which is simply written as NP=P? problem The question is whether NP is equal to p or NP is not equal to p.

Whether NP problem is polynomial or non-polynomial is still inconclusive.

N in NP is not a nondeterministic N, but a nondeterministic N, and it is correct that P stands for non-polynomial. NP is a problem of nondeterministic polynomials, that is, the uncertainty of polynomial complexity.

What is the problem of uncertainty? Some calculation problems are deterministic, such as addition, subtraction, multiplication and division. As long as you follow the formula step by step, you can get the result. But some problems can't be worked out directly step by step. For example, the problem of finding large prime numbers. Is there a formula that can be worked out step by step with a set of formulas? What should be the next prime number? There is no such formula. For another example, is there a formula for decomposing prime factors with large complex numbers? If you multiply the composite numbers, you can directly calculate their respective factors. There is no such formula.

The answer to this question cannot be directly calculated, and the result can only be obtained through indirect "guessing". This is also a question of uncertainty. This kind of question usually has an algorithm, which can't directly tell you what the answer is, but can tell you whether a possible result is a correct answer or a wrong answer. This algorithm, which can tell you whether the answer to "guess" is correct or not, is called polynomial uncertainty problem if it can be solved in polynomial time. If all possible answers to this question can be found in polynomial time, it is called a complete polynomial uncertainty problem.

The problem of complete polynomial uncertainty can be answered by exhaustive method, and the results can be obtained by testing one by one. However, the complexity of this algorithm is exponential, so the calculation time increases exponentially with the complexity of the problem and soon becomes uncountable.

It is found that all complete polynomial uncertainty problems can be transformed into a kind of logical operation problems called satisfaction problems. Because all possible answers to such questions can be calculated in polynomial time, people want to know whether there is a deterministic algorithm for such questions, which can be calculated by exponent.

Time, direct calculation or search for the correct answer? This is the famous NP=P? Guess.

There are only two possibilities to solve this conjecture. One is to find such an algorithm. As long as an algorithm is found for a specific NP-complete problem, all these problems can be solved because they can be transformed into the same problem. Another possibility is that such an algorithm does not exist. Then it is necessary to prove why it does not exist from the mathematical theory.

A mathematical achievement that caused a sensation in the world some time ago is that several Indians put forward a new algorithm, which can prove whether a number is a prime number in polynomial time. Before that, people thought that the proof of prime numbers was a non-polynomial problem. It can be seen that some seemingly non-polynomial problems are actually polynomial problems, but people don't know its polynomial solution at the moment.

If the decision problem π∈NP and all other decision problems π∈NP have π' polynomials transformed into π (denoted as π '∞π), the decision problem π is said to be NP-complete.

The study of P- class, NP- class and NP- complete problems has promoted the development of computational complexity theory, produced many new concepts and proposed many new methods. But there are still many problems unsolved, P=? NP is one of them. Many scholars have speculated that P≠NP, but they can't prove it.

2. Hodge conjecture

Also called Hodge conjecture: on nonsingular complex projective algebras, any Hodge class is a rational linear combination of algebraic closed-chain classes.

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.

3. Poincare conjecture

In a three-dimensional space, if every closed curve can shrink to a point, then this space must be a three-dimensional sphere. However, in 1905, it is found that the formulation is wrong, and it is revised, which is summarized as: "Any closed manifold and N-dimensional homotopy must be homeomorphic on an N-dimensional sphere."

If you think this statement is too abstract, we might as well make an imagination like this:

We imagine such a house, and this space is a ball. Or, imagine a huge football filled with air. We got in and saw that this was a spherical house.

4. 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.

In the process of proving the prime number theorem, Riemann put forward the conclusion that all zeros of Zeta function are on the straight line Res(s) = 1/2. He gave up after his proof failed, because it had little effect on his proof of the prime number theorem. But this problem has not been solved so far, and even a simpler guess than this assumption has not been proved. Many problems in function theory and analytic number theory depend on Riemann hypothesis. The generalized Riemann hypothesis in algebraic number theory has far-reaching influence. If we can prove the Riemann hypothesis, we can solve many problems.

5. Young-Mills theory.

Gauge field theory, also known as gauge field theory, is a basic theory to study four kinds of interactions in nature (electromagnetism, weakness, strength and gravity), which was first put forward by physicists Yang Zhenning and R L Mills in 1954. It originated from the analysis of electromagnetic interaction, and the unified theory of weak interaction and electromagnetic interaction established by it was confirmed by experiments, especially the intermediate boson predicted by this theory was found in the experiments. Young-Mills theory provides a powerful tool for studying the structure of hadrons (elementary particles involving strong interaction). In a sense, the gravitational field is also a gauge field. So the role of this theory in physics is very important. Mathematicians have noticed that the gauge potential in Young-Mills field is exactly the connection on fiber bundle that mathematicians have been studying deeply since 1930s-1940s. Not only that, they also found that the Young-Mills equation in this theory is a set of nonlinear partial differential equations that have never been considered in mathematics. Since 1975, mathematicians have done a lot of in-depth research on Young-Mills equation, which has also promoted the development of pure mathematics.

6. Naville-Stokes equation

Naville-Stokes equation, also known as Naville-Stokes equation, proves or denies the existence and smoothness of solutions of three-dimensional Naville-Stokes equation (under reasonable boundary and initial conditions).

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.

7.BSD (Birch and Swinerton Dale) conjecture.

BSD conjecture belongs to the content of number theory, which is about integer and rational number solutions of equations.

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.