Problem description:
The specific contents of the seven difficult problems in mathematics are as follows
Analysis:
Seven mathematical problems in 2 1 century
Recently, the Clay Institute of Mathematics in Massachusetts, USA, announced on May 24th, 2000 at the Institut de France in Paris.
A big event, which was heated by the media, was put out: a reward of/kloc-0.0 million dollars was given to each of the seven "Millennium Mathematical Problems". along with
Here is a brief introduction to these seven difficult problems.
One of the Millennium Problems: P (polynomial algorithm) versus NP (non-polynomial algorithm)
On a Saturday night, you attended a grand party. Embarrassed, I just want to know about this hall.
Is there anyone you already know? Your host hinted to you that you must know the woman in the corner near the dessert plate.
Ross. You don't need a second to glance there and find that your master is right. However, without this.
In this way, you have to look around the hall and look at everyone one by one to see if there is anyone you know. Generative problem
Solving a problem usually takes more time than verifying a given solution. This is an example of this common phenomenon. and
Similarly, if someone tells you that the number 13, 7 17, 42 1 can be written as the product of two smaller numbers, you
Maybe I don't know if I should trust him, but if he tells you that it can be broken down into 3607 times 3803,
Then you can easily verify this with a pocket calculator. Are we proficient in writing a program and judging a
The answer is that it can be quickly verified by internal knowledge, or it will take a lot of time to solve without such hints.
As one of the most prominent problems in logic and computer science. It's StephenCook
) stated 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 how to
To some extent, we can increase the size by gluing simple geometric building blocks, thus gluing the shapes of given objects together.
Form. This technology has become so useful that it can be popularized in many different ways; Eventually lead to some strong
With the help of force, mathematicians have made great progress in classifying objects of various shapes encountered in their research.
Unfortunately, in this generalization, the geometric starting point of the program becomes blurred. In a sense, it must be increased.
Parts without any geometric explanation. Hodge conjecture asserts that for the so-called projective algebra family, a particularly perfect space type,
In other words, the 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 wrap the rubber band around the surface of the apple, we will neither break it nor let it leave the watch.
Face, let it move slowly and shrink into a point. On the other hand, if we imagine that the same rubber band is stretched in a suitable direction.
If you shrink on the tire tread, there is no way to shrink it to a point without destroying the rubber belt or tire tread. We say
Apple surface is "single connected", but tire tread is not. About a hundred years ago, Poincare knew about two-dimensional balls.
Face can be characterized by simple connectivity in essence. He proposed a three-dimensional sphere (the sum of points in four-dimensional space at a unit distance from the origin)
) the corresponding problem. 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. The distribution of this prime number among all natural numbers.
A pattern that does not follow any rules; But German mathematician Riemann (1826~ 1866) observed that the frequency of prime numbers is close.
It is related to the behavior of the so-called Riemannian zeta function z(s$). The famous Riemann hypothesis asserts that the equation z(s)=0.
All meaningful solutions are on a straight line. This has been verified in the original 1, 500,000,000 solutions. Prove it
The establishment of every meaningful solution will bring light to 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. big
About half a century ago, Yang Zhenning and Mills discovered that quantum physics revealed the basic particle physics and the mathematics of geometric objects.
The amazing relationship between them. The prediction based on Young-Mills equation has been completed in the following laboratories around the world.
It has been confirmed in high-energy experiments in Brookhaven, Stanford, CERN and Tsukuba. Although such as
Therefore, they describe heavy particles and the mathematically strict equation has no known solution. In particular, most physicists
The hypothesis of "poor quality" has been confirmed by economists and applied to explain the invisibility of quarks, but it has never been confirmed.
There is no satisfactory mathematical proof. It is necessary to introduce the progress of this problem from both physical and mathematical aspects.
Become a basic new concept.
The Sixth Millennium Problem: Existence and Smoothness of Navier-Stokes Equation
The undulating waves follow our boat across the lake, and the turbulent airflow follows our modern jet.
The flight of the plane. Mathematicians and physicists are convinced that both breeze and turbulence can be understood by Naville-Stokes.
Tox equation to explain and predict them. Although these equations were written in19th century, we are very interested in them.
I still know very little. The challenge is to make substantial progress in mathematical theory, so that we can solve the problems hidden in Naville-Stowe.
The secret in cox 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 Zeng
After the complete solution of this equation is given, it becomes extremely difficult for more complicated equations. In fact, it is.
Ruyu. V.Matiyasevich pointed out that Hilbert's tenth problem is unsolvable, that is, there is no one.
A general method to determine whether this method has an integer solution. When the solution is a point of Abelian cluster, Behr and Svenna
According to Tong-Dale conjecture, the size of rational point group is related to the behavior of zeta function z(s) near point s= 1. special
In particular, this interesting conjecture holds that if z( 1) is equal to 0, there are infinitely many rational points (solutions). On the contrary, if z (
1) is not equal to 0, so there are only a few such points.