### 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.
#### 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.
