Goldbach's conjecture and Poincaré's conjecture
Poincaré's conjecture, like the Riemann hypothesis, Hodge's conjecture, Yang-Mill theory, etc., is listed as one of the seven major problems of the century in mathematics. one.
One of the "Thousand-Xi Difficulties": P (Polynomial Algorithm) Problem vs. NP (Non-Polynomial Algorithm) Problem
The second "Thousand-Xi Dilemma": Hodge Conjecture< /p>
The third "Thousand-Xi Dilemma": Poincare conjecture
The fourth "Thousand-Xi Dilemma": Riemann hypothesis
" "Thousand Xi Dilemma" No. 5: Yang-Mills Existence and Quality Gap
"Thousand Xi Dilemma" No. 6: Navier-Stokes Equation The existence and smoothness of "One: P (polynomial arithmetic) problem vs. NP (non-polynomial arithmetic) problem
On a Saturday night, you attended a grand party. Feeling uneasy, you wanted to know this Is there anyone in the hall that you already know? Your host proposes to you that you must know the lady Rose in the corner near the dessert plate. It won't take you a second to glance there and spot yours. The host is correct. However, if there is no such hint, you have to look around the hall and look at everyone one by one to see if there is anyone you know. It usually takes longer to generate a solution to the problem than to verify a given solution. It costs much more. 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 smaller numbers, you may not know whether. You should believe him, but if he tells you that it factors as 3607 times 3803, then you can easily verify that this is true with a pocket calculator. Regardless of our programming dexterity, determining an answer is quickly exploitable. Whether there is no such hint and it takes a lot of time to solve is regarded as one of the most outstanding problems in logic and computer science. It was stated by Stephen Cook in 1971.
"Thousand-Xi Problem" Part 2: Hodge's Conjecture
Mathematicians in the 20th century discovered the basic idea of ??a powerful way to study the shape of complex objects. is asking to what extent we can form 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 used in many different ways. Unfortunately, in this generalization, the program's geometric starting point was used. Becomes vague. In some sense, some components without any geometric explanation must be added. The Hodge conjecture asserts that for a particularly perfect type of space called projective algebraic varieties, it is called a Hodge chain. The components are actually (rational linear) combinations of geometric components called algebraically closed chains.
Part 3 of the "Thousand-Xi Problem": Poincare's Conjecture
If we stretch a rubber band around the surface of an apple, then we can do it without tearing it. Without letting it leave the surface, let it slowly move and shrink to a point. On the other hand, if we imagine the same rubber band being stretched and contracted on a tire tread in the appropriate direction, there is no way to shrink it to a point without tearing the rubber belt or tire tread. We say that the surface of an apple is "simply connected", but the surface of a tire is not. About a hundred years ago, Poincaré already knew that the two-dimensional sphere could essentially be characterized by simple connectivity, and he proposed the correspondence problem of the three-dimensional sphere (the set of points unit distance from the origin in four-dimensional space). The problem immediately became incredibly difficult, and mathematicians have struggled with it ever since.
"Thousand-Xi Dilemma" No. 4: Riemann Hypothesis
Some numbers have special properties that cannot be expressed as the product of two smaller numbers, for example, 2, 3,5,7, etc. Such numbers are called prime numbers; they play an important role in both pure mathematics and its applications. Among all natural numbers, this distribution of prime numbers does not follow any regular pattern; however, the German mathematician Riemann (1826~1866) observed that the frequency of prime numbers is closely related to a carefully constructed so-called Riemann Zeitah function. The behavior of z(s$. The famous Riemann hypothesis asserts that all meaningful solutions to the equation z(s)=0 are on a straight line. This has been verified for the first 1,500,000,000 solutions. Prove that it is true for every The establishment of a meaningful solution will bring light to many mysteries surrounding the distribution of prime numbers.
"Thousand-Xi Problem" Part 5: Yang-Mills Existence and Mass Gap
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The laws of quantum physics are true for the world of elementary particles in the same way that Newton’s laws of classical mechanics are true for the macroscopic world.
About half a century ago, Chen Ning Yang and Mills discovered that quantum physics revealed a remarkable relationship between the physics of elementary particles and the mathematics of geometric objects. Predictions based on the Yang-Mills equation have been confirmed in high-energy experiments performed at laboratories around the world: Brockhaven, Stanford, CERN, and Tsukuba. Still, their mathematically rigorous equations, which both describe heavy particles, have no known solution. In particular, the "mass gap" hypothesis, recognized by most physicists and used in their explanation of the invisibility of "quarks", has never been confirmed to a mathematically satisfactory degree. Progress on this problem requires the introduction of fundamentally new ideas, both in physics and mathematics.
"Thousand-Xi Problem" No. 6: The Existence and Smoothness of the Navier-Stokes Equation
The undulating waves follow us. Small boats meander in the lake, and turbulent air currents follow the flight of our modern jets. Mathematicians and physicists are convinced that both breezes and turbulence can be explained and predicted by understanding the solutions to the Navier-Stokes equations. Although these equations were written down in the 19th century, our understanding of them is still minimal. The challenge is to make substantial advances in mathematical theory that allow us to unlock the mysteries hidden in the Navier-Stokes equations.
"Thousand Xi Problems" No. 7: Birch and Swinnerton-Dyer Conjecture
Mathematicians are always puzzled by questions such as x^2+ I am fascinated by the problem of characterizing all integer solutions to algebraic equations like y^2=z^2. Euclid once gave a complete solution to this equation, but this becomes extremely difficult for more complex equations. In fact, as Yu.V. Matiyasevich pointed out, Hilbert's tenth problem is unsolvable, that is, there is no general way to determine whether such a method has an integer solution. When the solution is a point of an Abelian variety, the Behe ??and Svenetorn-Dyer conjectures hold that the size of the group of rational points is related to the behavior of the Zeitah 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.