1, NP complete problem
On a Saturday night, I attended a grand party. Embarrassed, I wonder if there is anyone you already know in this hall. The host of the party suggested that you must know Ms. Ross sitting in the corner near the dessert plate. It doesn't take you a second to find out that the host of the party is right.
If there is no such hint, you must look around the whole hall and check 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.
2. 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 the shape of a given object can be formed by bonding simple geometric building blocks with added dimensions. This technology has become very useful and 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 a component called Hodge closed chain is actually a (rational linear) combination of geometric components called algebraic closed chain for so-called projective algebraic clusters.
3. 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 you imagine the same rubber belt stretching in the proper direction on the tire tread, there is no way to shrink it to a point without damaging the rubber belt or tire tread.
The surface of apple is "simply connected", but the 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.
4. Riemann hypothesis
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. The distribution of this prime number in all natural numbers does not follow any law; However, German mathematician Riemann (1826~ 1866) observed.
The frequency of prime numbers is closely related to the behavior of the well-constructed so-called Riemannian zeta function zeta (S). The famous Riemann hypothesis asserts that all meaningful solutions of the equation ζ(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.
5. The existence and quality gap of poplar paper 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 high-energy experiments in laboratories all over the world.
Brockhaven, Stanford, CERN and Standing Wave. Mathematically strict equations describing heavy particles have no known solutions. The "mass gap" hypothesis, which was confirmed by most physicists and applied to explain quark invisibility, has never been proved mathematically satisfactorily. The progress of the problem needs to introduce basic new ideas into physics and mathematics.
6. Existence and smoothness of 6.Navier Stoke 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 conjecture
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 has no solution.
There is no universal method to determine whether such an equation has an integer solution. When the solution is a point of the Abelian cluster, Behe and Sveneton Dale suspect that the size of the rational point set is related to the behavior of a related Zeta function z(s) near the point s= 1. This interesting conjecture holds that if z( 1) is equal to 0, there are infinite rational points (solutions). If z( 1) is not equal to 0, then there are only a limited number of such points.