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世界七大数学难题
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http://hi.baidu.com/jrckkyy/blog/item/46c7f0a7790c9090d043584c.html世界七大数学难题
难题的提出
20世纪是数学大发展的一个世纪。数学的许多重大难题得到完满解决， 如费马大定理的证明，有限单群分类工作的完成等， 从而使数学的基本理论得到空前发展。
计算机的出现是20世纪数学发展的重大成就，同时极大推动了数学理论的深化和数学在社会和生产力第一线的直接应用。回首20世纪数学的发展， 数学家们深切感谢20世纪最伟大的数学大师大卫·希尔伯特。希尔伯特在1900年8月8日于巴黎召开的第二届世界数学家大会上的著名演讲中提出了23个数学难题。希尔伯特问题在过去百年中激发数学家的智慧，指引数学前进的方向，其对数学发展的影响和推动是巨大的，无法估量的。
效法希尔伯特， 许多当代世界著名的数学家在过去几年中整理和提出新的数学难题，希冀为新世纪数学的发展指明方向。 这些数学家知名度是高的， 但他们的这项行动并没有引起世界数学界的共同关注。
2000年初美国克雷数学研究所的科学顾问委员会选定了七个“千年大奖问题”，克雷数学研究所的董事会决定建立七百万美元的大奖基金，每个“千年大奖问题”的解决都可获得百万美元的奖励。克雷数学研究所“千年大奖问题”的选定，其目的不是为了形成新世纪数学发展的新方向， 而是集中在对数学发展具有中心意义、数学家们梦寐以求而期待解决的重大难题。
2000年5月24日，千年数学会议在著名的法兰西学院举行。会上，98年费尔兹奖获得者伽沃斯以“数学的重要性”为题作了演讲，其后，塔特和阿啼亚公布和介绍了这七个“千年大奖问题”。克雷数学研究所还邀请有关研究领域的专家对每一个问题进行了较详细的阐述。克雷数学研究所对“千年大奖问题”的解决与获奖作了严格规定。每一个“千年大奖问题”获得解决并不能立即得奖。任何解决答案必须在具有世界声誉的数学杂志上发表两年后且得到数学界的认可，才有可能由克雷数学研究所的科学顾问委员会审查决定是否值得获得百万美元大奖.
世界七大数学难题
这七个“千年大奖问题”是： NP完全问题、霍奇猜想、庞加莱猜想、黎曼假设、杨－米尔斯理论、纳卫尔－斯托可方程、BSD猜想。
美国麻州的克雷（Clay）数学研究所于2000年5月24日在巴黎法兰西学院宣
布了一件被媒体炒得火热的大事：对七个“千年数学难题”的每一个悬赏一百万美元。
其中有一个已被解决(庞加莱猜想),还剩六个.（庞加莱猜想，已被我国中山大学朱熹平教授和旅美数学家、清华大学兼职教授曹怀东破解了。）
整个计算机科学的大厦就建立在图灵机可计算理论和计算复杂性理论的基础上,
一旦证明P=NP,将是计算机科学的一场决定性的突破,在软件工程实践中,将革命性的提高效率.从工业,农业,军事,医疗到生活,软件在它的各个应用域,都将是一个飞跃.
P=NP吗? 这个问题是著名计算机科学家(1982年图灵奖得主)斯蒂文·考克（StephenCook ）于1971年发现并提出的.
“千年大奖问题”公布以来， 在世界数学界产生了强烈反响。这些问题都是关于数学基本理论的，但这些问题的解决将对数学理论的发展和应用的深化产生巨大推动。认识和研究“千年大奖问题”已成为世界数学界的热点。不少国家的数学家正在组织联合攻关。 可以预期， “千年大奖问题” 将会改变新世纪数学发展的历史进程。
“千年难题”之一：Ｐ（多项式算法）问题对ＮＰ（非多项式算法）问题
在一个周六的晚上，你参加了一个盛大的晚会。由于感到局促不安，你想知道这一大厅中是否有你已经认识的人。你的主人向你提议说，你一定认识那位正在甜点盘附近角落的女士罗丝。不费一秒钟，你就能向那里扫视，并且发现你的主人是正确的。然而，如果没有这样的暗示，你就必须环顾整个大厅，一个个地审视每一个人，看是否有你认识的人。生成问题的一个解通常比验证一个给定的解时间花费要多得多。这是这种一般现象的一个例子。与此类似的是，如果某人告诉你，数１３，７１７，４２１可以写成两个较小的数的乘积，你可能不知道是否应该相信他，但是如果他告诉你它可以因式分解为３６０７乘上３８０３，那么你就可以用一个袖珍计算器容易验证这是对的。不管我们编写程序是否灵巧，判定一个答案是可以很快利用内部知识来验证，还是没有这样的提示而需要花费大量时间来求解，被看作逻辑和计算机科学中最突出的问题之一。它是斯蒂文·考克于１９７１年陈述的。
“千年难题”之二：霍奇(Hodge)猜想
二十世纪的数学家们发现了研究复杂对象的形状的强有力的办法。基本想法是问在怎样的程度上，我们可以把给定对象的形状通过把维数不断增加的简单几何营造块粘合在一起来形成。这种技巧是变得如此有用，使得它可以用许多不同的方式来推广；最终导致一些强有力的工具，使数学家在对他们研究中所遇到的形形色色的对象进行分类时取得巨大的进展。不幸的是，在这一推广中，程序的几何出发点变得模糊起来。在某种意义下，必须加上某些没有任何几何解释的部件。霍奇猜想断言，对于所谓射影代数簇这种特别完美的空间类型来说，称作霍奇闭链的部件实际上是称作代数闭链的几何部件的(有理线性)组合。
“千年难题”之三：庞加莱(Poincare)猜想
如果我们伸缩围绕一个苹果表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点。另一方面，如果我们想象同样的橡皮带以适当的方向被伸缩在一个轮胎面上，那么不扯断橡皮带或者轮胎面，是没有办法把它收缩到一点的。我们说，苹果表面是“单连通的”，而轮胎面不是。大约在一百年以前，庞加莱已经知道，二维球面本质上可由单连通性来刻画，他提出三维球面(四维空间中与原点有单位距离的点的全体)的对应问题。这个问题立即变得无比困难，从那时起，数学家们就在为此奋斗。
6月3日，新华社报道，中山大学朱熹平教授和旅美数学家、清华大学兼职教授曹怀东破解了国际数学界关注上百年的重大难题——庞加莱猜想。
“千年难题”之四：黎曼(Riemann)假设
有些数具有不能表示为两个更小的数的乘积的特殊性质，例如，2、3、5、7……等等。这样的数称为素数；它们在纯数学及其应用中都起着重要作用。在所有自然数中，这种素数的分布并不遵循任何有规则的模式；然而，德国数学家黎曼(1826~1866)观察到，素数的频率紧密相关于一个精心构造的所谓黎曼蔡塔函数z(s$的性态。著名的黎曼假设断言，方程z(s)=0的所有有意义的解都在一条直线上。这点已经对于开始的1,500,000,000个解验证过。证明它对于每一个有意义的解都成立将为围绕素数分布的许多奥秘带来光明。
“千年难题”之五：杨－米尔斯(Yang-Mills)存在性和质量缺口
量子物理的定律是以经典力学的牛顿定律对宏观世界的方式对基本粒子世界成立的。大约半个世纪以前，杨振宁和米尔斯发现，量子物理揭示了在基本粒子物理与几何对象的数学之间的令人注目的关系。基于杨－米尔斯方程的预言已经在如下的全世界范围内的实验室中所履行的高能实验中得到证实：布罗克哈文、斯坦福、欧洲粒子物理研究所和筑波。尽管如此，他们的既描述重粒子、又在数学上严格的方程没有已知的解。特别是，被大多数物理学家所确认、并且在他们的对于“夸克”的不可见性的解释中应用的“质量缺口”假设，从来没有得到一个数学上令人满意的证实。在这一问题上的进展需要在物理上和数学上两方面引进根本上的新观念。
“千年难题”之六：纳维叶－斯托克斯(Navier-Stokes)方程的存在性与光滑性
起伏的波浪跟随着我们的正在湖中蜿蜒穿梭的小船，湍急的气流跟随着我们的现代喷气式飞机的飞行。数学家和物理学家深信，无论是微风还是湍流，都可以通过理解纳维叶－斯托克斯方程的解，来对它们进行解释和预言。虽然这些方程是19世纪写下的，我们对它们的理解仍然极少。挑战在于对数学理论作出实质性的进展，使我们能解开隐藏在纳维叶－斯托克斯方程中的奥秘。
“千年难题”之七：贝赫(Birch)和斯维讷通－戴尔(Swinnerton-Dyer)猜想
数学家总是被诸如x2+y2=z2那样的代数方程的所有整数解的刻画问题着迷。欧几里德曾经对这一方程给出完全的解答，但是对于更为复杂的方程，这就变得极为困难。事实上，正如马蒂雅谢维奇指出，希尔伯特第十问题是不可解的，即，不存在一般的方法来确定这样的方法是否有一个整数解。当解是一个阿贝尔簇的点时，贝赫和斯维讷通－戴尔猜想认为，有理点的群的大小与一个有关的蔡塔函数z(s)在点s=1附近的性态。特别是，这个有趣的猜想认为，如果z(1)等于0,那么存在无限多个有理点(解)，相反，如果z(1)不等于0,那么只存在有限多个这样的点。
### The Seven Millennium Problems in Mathematics
#### Proposal of the Problems
The 20th century was a century of great development in mathematics. Many major mathematical problems were perfectly solved, such as the proof of Fermat's Last Theorem and the completion of the classification of finite simple groups, which led to the unprecedented development of basic mathematical theories.
The emergence of computers was a significant achievement in the development of mathematics in the 20th century, and it also greatly promoted the deepening of mathematical theories and the direct application of mathematics in the front line of society and productivity. Looking back on the development of mathematics in the 20th century, mathematicians deeply thanked David Hilbert, the greatest mathematician of the 20th century. In a famous speech at the Second World Congress of Mathematicians held in Paris on August 8, 1900, Hilbert proposed 23 mathematical problems. Hilbert's problems inspired the wisdom of mathematicians in the past hundred years and guided the direction of mathematical progress. Their influence and promotion on the development of mathematics are enormous and immeasurable.
Following Hilbert's example, many famous contemporary mathematicians sorted out and proposed new mathematical problems in the past few years, hoping to point out the direction for the development of mathematics in the new century. These mathematicians are well-known, but their action did not arouse the common attention of the world's mathematical community.
In early 2000, the Scientific Advisory Board of the Clay Mathematics Institute in the United States selected seven "Millennium Prize Problems". The board of the Clay Mathematics Institute decided to establish a prize fund of $7 million, and each "Millennium Prize Problem" solved could receive a prize of $1 million. The selection of the "Millennium Prize Problems" by the Clay Mathematics Institute was not aimed at forming new directions for the development of mathematics in the new century, but at focusing on major problems that are central to the development of mathematics and that mathematicians have long dreamed of and are eager to solve.
On May 24, 2000, the Millennium Mathematics Conference was held at the famous Collège de France. At the conference, Gowers, the 1998 Fields Medalist, gave a speech titled "The Importance of Mathematics". Subsequently, Tate and Atiyah announced and introduced these seven "Millennium Prize Problems". The Clay Mathematics Institute also invited experts in relevant research fields to elaborate on each problem in detail. The Clay Mathematics Institute has made strict regulations on the solution and award of the "Millennium Prize Problems". The solution of each "Millennium Prize Problem" cannot immediately win the prize. Any solution must be published in a world-renowned mathematical journal for two years and recognized by the mathematical community before it may be reviewed by the Scientific Advisory Board of the Clay Mathematics Institute to determine whether it is worthy of the $1 million prize.
### The Seven Millennium Problems in Mathematics
These seven "Millennium Prize Problems" are: NP Completeness Problem, Hodge Conjecture, Poincaré Conjecture, Riemann Hypothesis, Yang-Mills Theory, Navier-Stokes Equations, and BSD Conjecture.
The Clay Mathematics Institute in Massachusetts, USA, announced on May 24, 2000, at the Collège de France in Paris a matter that was hyped by the media: a reward of $1 million for each of the seven "Millennium Mathematical Problems".
One of them has been solved (Poincaré Conjecture), and six are left. (The Poincaré Conjecture has been solved by Professor Zhu Xiping from Sun Yat-sen University in China and Professor Cao Huaidong, a visiting mathematician at Tsinghua University.)
The entire edifice of computer science is built on the basis of the Turing machine computability theory and computational complexity theory. Once P=NP is proven, it will be a decisive breakthrough in computer science. In software engineering practice, it will revolutionarily improve efficiency. In all its application fields, from industry, agriculture, military, medical to life, software will be a leap forward.
Is P=NP? This problem was discovered and proposed by the famous computer scientist Stephen Cook (the 1982 Turing Award winner) in 1971.
Since the announcement of the "Millennium Prize Problems", they have had a strong impact in the world of mathematics. These problems are all about basic mathematical theories, but the solution of these problems will greatly promote the development of mathematical theories and the deepening of applications. Understanding and researching the "Millennium Prize Problems" has become a hot spot in the world of mathematics. Mathematicians in many countries are organizing joint research efforts. It can be expected that the "Millennium Prize Problems" will change the historical process of the development of mathematics in the new century.
#### "Millennium Problem" One: P (Polynomial Algorithm) Problem vs. NP (Non-Polynomial Algorithm) Problem
One Saturday night, you attended a grand party. Feeling uneasy, you wanted to know if there was someone you already knew in this large hall. Your host suggested to you that you must know the lady Rose near the corner of the dessert plate. Without a second, you could scan there and find that your host was correct. However, if there was no such hint, you would have to look around the entire hall and examine each person one by one to see if there was someone you knew. Generating a solution to a problem usually takes much more time than verifying a given solution. 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 if you should believe him, but if he tells you that it can be factorized into 3607 multiplied by 3803, then you can easily verify this with a pocket calculator. No matter how clever we write the program, determining whether an answer can be quickly verified using internal knowledge or needs to spend a lot of time to solve without such a hint is regarded as one of the most prominent problems in logic and computer science. It was stated by Stephen Cook in 1971.
#### "Millennium Problem" Two: Hodge Conjecture
Mathematicians in the 20th century discovered powerful ways to study the shapes of complex objects. The basic idea is to ask to what extent we can form the shape of a given object by gluing simple geometric building blocks of increasing dimensions together. This technique has become so useful that it can be generalized in many different ways; eventually leading to some powerful tools that enable mathematicians to make great progress in classifying the various objects they encounter in their research. Unfortunately, in this generalization, the geometric starting point of the program has become vague. In a sense, some components without any geometric interpretation must be added. The Hodge Conjecture asserts that for such a particularly perfect space type called a projective algebraic variety, the components called Hodge cycles are actually (rational linear) combinations of the geometric components called algebraic cycles.
#### "Millennium Problem" Three: Poincaré Conjecture
If we stretch an elastic band around the surface of an apple, then we can either not break it or not let it leave the surface, and slowly move and shrink it to a point. On the other hand, if we imagine that the same elastic band is stretched on a tire surface in an appropriate direction, then without breaking the elastic band or the tire surface, there is no way to shrink it to a point. We say that the apple surface is "simply connected", while the tire surface is not. About a hundred years ago, Poincaré already knew that the two-dimensional sphere can essentially be characterized by simple connectivity, and he proposed the corresponding problem of the three-dimensional sphere (the set of points at a unit distance from the origin in four-dimensional space). This problem immediately became extremely difficult, and since then, mathematicians have been working hard on it.
On June 3, Xinhua News Agency reported that Professor Zhu Xiping from Sun Yat-sen University in China and Professor Cao Huaidong, a visiting mathematician at Tsinghua University, solved the major problem that the international mathematical community has paid attention to for more than a hundred years - the Poincaré Conjecture.
#### "Millennium Problem" Four: Riemann Hypothesis
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, for example, 2, 3, 5, 7... and so on. Such numbers are called prime numbers; they play an important role in pure mathematics and its applications. Among all natural numbers, the distribution of such 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 the behavior of a carefully constructed so-called Riemann Zeta function ζ(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 for the first 1,500,000,000 solutions. Proving that it holds for every meaningful solution will bring light to many mysteries surrounding the distribution of prime numbers.
#### "Millennium Problem" Five: Existence and Mass Gap of Yang-Mills
The laws of quantum physics hold for the world of elementary particles in the way that Newton's laws of classical mechanics hold for the macroscopic world. About half a century ago, Yang Zhenning and Mills discovered that quantum physics reveals a remarkable relationship between elementary particle physics and the mathematics of geometric objects. The predictions based on Yang-Mills equations have been confirmed in high-energy experiments performed in laboratories around the world such as Brookhaven, Stanford, the European Organization for Nuclear Research, and Tsukuba. However, their equations that describe both heavy particles and are strictly mathematical have no known solutions. In particular, the "mass gap" hypothesis, which is recognized by most physicists and applied in their explanation of the invisibility of "quarks", has never been satisfactorily confirmed mathematically. Progress in this problem requires fundamentally new concepts in both physics and mathematics.
#### "Millennium Problem" Six: Existence and Smoothness of Navier-Stokes Equations
The undulating waves follow our small boat meandering through the lake, and the turbulent airflow follows the flight of our modern jet aircraft. Mathematicians and physicists are convinced that whether it is a breeze or a turbulent flow, they can be explained and predicted by understanding the solutions of the Navier-Stokes equations. Although these equations were written in the 19th century, our understanding of them is still very little. The challenge is to make substantial progress in mathematical theory so that we can unlock the mysteries hidden in the Navier-Stokes equations.
#### "Millennium Problem" Seven: Birch and Swinnerton-Dyer Conjecture
Mathematicians are always fascinated by the problem of characterizing 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, this becomes extremely difficult. In fact, as Matijasevich 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 a point of an Abelian variety, the Birch and Swinnerton-Dyer Conjecture holds that the size of the group of rational points is related to the behavior of a related Zeta function ζ(s) near the point s=1. In particular, this interesting conjecture holds that if ζ(1) equals 0, then there are infinitely many rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there are only finitely many such points.
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