Seven mathematical problems in the world 1, Poincare conjecture
2.NP complete problem
3. The existence and quality gap of Poplar Mill.
4. Hodge conjecture
5. Existence and smoothness of Naville-Stoke equation.
6.BSD conjecture
7. Riemann hypothesis
1, 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, what is the surface of an apple? Simple connectivity? , and 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.
Between 2002 10 (1 1) and July 2003, Russian mathematician grigori? Perelman published three papers in preprint, and claimed to prove the geometric conjecture.
After perelman, two groups of researchers published papers to fill in the missing details in perelman's proof. Including Bruce of the University of Michigan? Kleiner and John? Lott; John from Columbia University? Morgan and Tian Gang from MIT.
In August 2006, the 25th International Congress of Mathematicians awarded perelman Fields Prize. Mathematics finally confirmed that perelman's proof solved Poincare's conjecture.
2.NP complete problem
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. The host of the party hinted to you that you must know Ms. Ross sitting in the corner near the dessert plate. You don't need a second to glance over there and find that the host of the party 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 number 137 1742 1 can be written as the product of two smaller numbers, you may not know whether to believe him, but if he tells you that this number can be decomposed into 3607 times 3803, then you can easily verify that this is correct with a pocket calculator.
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, and they can directly calculate or search for the correct answers in polynomial time. This is the famous NP=P? Guess. 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. Is that Steven? The cock is stated in 197 1.
3. The existence and quality gap of Poplar 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: Brockhavan, Stanford, CERN and standing wave. However, they describe heavy particles and mathematically strict equations have no known solutions. In particular, it has been recognized by most physicists, and they are not satisfied with it. Quarks? In the invisible interpretation? Quality gap? The hypothesis has never been proved mathematically satisfactory. The progress on this issue needs to introduce basic new concepts into physics and mathematics.
4. 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, 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 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.
5. Existence and smoothness of Naville-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.
6.BSD conjecture
Mathematicians are always fascinated by the characteristics of all integer solutions of this kind of algebraic equations. Euclid once gave a complete solution to this equation, but for more complex equations, it became extremely difficult. In fact, as Matthias Sevic pointed out, Hilbert's tenth problem has no solution, that is, 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 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) equals 0, there are infinite rational points (solutions). On the other hand, if z( 1) is not equal to 0. Then there are only a few such points.
7. 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. 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 a well-constructed so-called Riemann zeta function. Nature. The famous Riemann hypothesis assertion equation? All meaningful solutions of (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.
Negative Riemann hypothesis;
In fact, although the number of factors is distributed, it is wrong, because the general formula of pseudo prime numbers and prime numbers tells us that prime numbers and pseudo prime numbers are determined by their variable sets. See pseudo prime number and prime number entries for details.
Since the seven major mathematical problems in the world are recognized, it means that there will be solutions. If it is solved, it will have a great impact on our lives.