MathematicianDavid Hilbert The second session was held in Paris on August 8, 1900World Congress of MathematiciansIn the famous speech on, 23Mathematical puzzle。Hilbert's Problems In the past hundred years, it has stimulated the wisdom of mathematicians and guided the direction of mathematics. Its impact and promotion on the development of mathematics is enormous and immeasurable.
The 20th century is a century of great development of mathematics.Many major mathematical problems have been solved satisfactorily, such asFermat's big theoremProof of,Finite simple groupThe completion of classification work, etcfundamental theoryIt has achieved unprecedented development.
United States at the beginning of 2000Cray Institute of MathematicsThe Scientific Advisory Committee of,Cray Institute of MathematicsThe board of directors decided to set up a prize fund of seven million dollars, and each solution to the "Millennium Prize Problem" would receive a reward of one million dollars.
Cray Institute of MathematicsThe purpose of the selection of the "Millennium Prize Problem" is not to form a new direction for the development of mathematics in the new century, but to focus on the major problems that are central to the development of mathematics and that mathematicians are eager to solve.
On May 24, 2000, the Millennium Mathematics ConferenceCollege de France Held.At the conference, 1997Fields Medal The winner Gavos gave a speech on the topic of "the importance of mathematics". Later, Tate and Atia announced and introduced the seven "Millennium Prize questions".Cray Institute of Mathematics also invited experts in relevant research fields to give a more detailed description of each problem.The Cray Institute of Mathematics has made strict regulations on the solution and awarding of the "Millennium Prize Problem".If every "Millennium Prize problem" is solved, it will not win the prize immediately.Any solution must be based on theJournal of MathematicsIt is only two years after the last publication and the recognition of the mathematical community that the Scientific Advisory Committee of Cray Institute of Mathematics may review and decide whether it is worth winning the award of one million dollars.
Since the issue of the "Millennium Prize" was announced, it has generated a strong response in the world's mathematical community.These problems are all about the basic theory of mathematics, but the solution of these problems will greatly promote the development and application of mathematical theory.Understanding and studying the "Millennium Prize" has become a hot topic in the world of mathematics.Mathematicians in many countries are organizing joint research.The "Millennium Prize Problem" will change the historical process of mathematical development in the new century.
Seven difficult problems
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NP complete problem
Ex.: On a Saturday night, you attended a grand party.Feeling embarrassed, you want to know if there are people you already know in this hall.The host of the banquet suggested to you that you must know the lady Rose in the corner near the dessert plate.It doesn't take you a second to glance over there and find that the host of the banquet is right.However, if there is no such hint, you must look around the hall and examine everyone one by one to see if there are people you know.
It usually takes much more time to generate a solution to a problem than to verify a given solution.This is an example of this general phenomenon.Similarly, if someone tells you that the number 13717421 can be written as the product of two smaller numbers, you may not know whether you should trust him or not, but if he tells you that it can be decomposed into 3607 times 3803, then you can easily verify this with a pocket calculator.
It is found that all complete polynomialsUncertaintyProblems can be transformed into a kind of satisfaction problemsLogical operationQuestion.Since all possible answers to such questions can be found inpolynomial time In internal computing, people then wonder whether there is a deterministic algorithm for such problems that can directly calculate or search for the correct answer in polynomial time?This is the famous NP=P?Guess.No matter whether we are clever in writing programs, whether we can quickly use internal knowledge to verify the answer or spend a lot of time to solve without such prompt is regarded as logic andcomputer scienceOne of the most prominent problems in.It was stated by Steven Coker in 1971.
Hodge conjecture
Mathematicians in the 20th century discovered thatComplex objectsThe shape of the powerful approach.The basic idea is to ask to what extent we can change the shape of a given object bydimensionAn increasing number of simple geometric building blocks are glued together to form.This technique has become so useful that it can be promoted in many different ways;Eventually, it led to some powerful tools that enabled mathematicians to 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 Assertions, for the so-calledProjective algebraic varietyFor this particularly good space type, the components called Hodge closed chain are actually calledAlgebraic closed chainA (rational linear) combination of geometric components of.
Poincare conjecture
If we stretch the rubber band around the surface of an apple, we can neither break it nor let it leave the surface, so that it slowly moves and shrinks to a point.On the other hand, if we imagine that the same rubber band is telescoped in a proper directionTire treadIf the rubber band or tire tread is not broken, there is no way to shrink it to a point.We say that the surface of the apple is“simply-connected And the tire tread is not.About a hundred years ago, Poincare had known that the two-dimensional sphere could be characterized by simple connectivity in essence, and he proposedThree-dimensional sphere(Four-dimensional spaceAll points with unit distance from the origin).The problem immediately became extremely difficult. Since then, mathematicians have been struggling for it.
Between November 2002 and July 2003, Russian mathematiciansGregory PerelmanThree preprints of the paper were published and claimed to prove thatGeometrization conjecture 。
Some numbers have special properties that cannot be expressed as the product of two smaller numbers, such as 2, 3, 5, 7, etc.Such numbers are calledprime number；They play an important role in pure mathematics and its application.At allNatural numberThe distribution of such prime numbers does not follow any regular pattern;However, German mathematician Riemann (1826~1866) observed that the frequency of prime numbers is closely related to the behavior of an elaborately constructed so-called Riemann zeta function ζ (s).famousRiemann hypothesisIt is asserted that all meaningful solutions of the equation ζ (s)=0 are on a straight line.This has been verified for the first 150000000 solutions.To prove that it holds true for every meaningful solution will bePrime distributionMany mysteries of bring light.
Denial of Riemann hypothesis:
In fact, although the number of factors is distributed, it is a wrong way becausePseudo prime numberandprime numberThe general formula of C tells us that prime numbers and pseudo prime numbers are determined by their variable sets.SeePseudo prime numberandprime numberEntries.
Young Mills existence and quality gap
The laws of quantum physics are based onclassical mechanicsOfNewton's lawyesMacro worldThe way ofElementary particleThe world was founded.About half a century ago,Yang ZhenningAnd Mills found that,Quantum physicsThe remarkable relationship between elementary particle physics and the mathematics of geometric objects is revealed.The prediction based on Young Mills equation has been confirmed in the following high-energy experiments carried out in laboratories around the world: Brocklehaven, StanfordEuropean Institute for Particle Physicsandstanding wave。However, their equations that describe heavy particles and are mathematically rigorous have no known solutions.In particular, it is confirmed by most physicists, and“quark”The "quality gap" hypothesis applied in the interpretation of invisibility has never been proved mathematically satisfactory.Progress on this issue requires the introduction of fundamental new ideas in physics and mathematics.
Existence and Smoothness of the Navier Stoke Equation
The undulating waves follow our boat winding through the lake, and the turbulent current follows our modernityJet aircraftThe flight of.Mathematicians and physicists are convinced that both breeze and turbulence can be understood bySolution of the equation, to explain and predict them.Although these equations were written in the 19th century, we still have very little understanding of them.The challenge lies in makingSubstantiveWe can solve the mystery hidden in the Navier Stokes equation.
BSD conjecture
Mathematicians are always treated as
That wayalgebraic equationThe problem of characterizing all integer solutions of the.EuclidThe complete solution of this equation was once given, but for more complex equations, it becomes extremely difficult.In fact, as Matyashevich pointed out, Hilbert's tenth problem is unsolvable, that is, there is no general method to determine whether such an equation has an integer solution.When the solution is aAbelian varietyBach and Schweinerton Dale conjectures believe that the size of the group of rational points is related to the behavior of a Zetta 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 an infinite number ofRational point(Solution).On the contrary, if z (1) is not equal to 0, then there are infinitely many such points.
