Hello, I'm here to answer your question about the process of proving Poincare's conjecture in China, which I believe many friends still don't know. Now let's have a look!
1, Poincare conjecture Poincare conjecture is a long-term concern of the international mathematics community, and it is listed as one of the seven "mathematical century problems".
2. Henri Poincare, a French mathematician, put forward this conjecture in a set of papers published in 1904: "A simply connected three-dimensional closed flow is like an embryo on a three-dimensional sphere.
3. Later, it was extended to: "Any closed manifold with N-dimensional homotopy must be homeomorphic on the N-dimensional sphere".
4. "We might as well make a shallow metaphor with the help of two-dimensional examples: a rubber film without holes is topologically equivalent to a two-dimensional closed surface, while an inflatable balloon can be regarded as a two-dimensional spherical surface, and the points between them are one-to-one correspondence, while the adjacent points on the rubber film are still adjacent points on the inflatable balloon, and vice versa.
5. Poincare conjecture, like Riemann hypothesis, Hodge conjecture and Jan-Mill theory, is listed as one of the seven mathematical century problems.
6. In May 2000, the Clay Institute of Mathematics in the United States offered a reward of100000 dollars for each problem solved.
7. 100 years, countless mathematicians have paid attention to and tried to confirm Poincare's conjecture.
8. In the early 1980s, Professor thurston, an American mathematician, won the Fields Prize for partially proving the Poincare conjecture in geometry.
9. Later, American mathematician Hamilton also made important progress in proving this conjecture.
10 and in 2003, Russian mathematician perelman put forward the key to solve this conjecture.
1 1, Qiu Chengtong, a professor at Harvard University, a famous mathematician and a Fields Prize winner in the United States, announced at Morningside Mathematics Research Center of China Academy of Sciences on June 3, 2006 that on the basis of the work of scientists in the United States, Russia and other countries, Professor Zhu Xiping of Sun Yat-sen University and Cao Huaidong, a mathematician living in the United States and an adjunct professor in Tsinghua University, have completely proved this conjecture.
12. Using the theories of Hamilton and perelman, Zhu Xiping and Cao Huaidong successfully solved the "singularity" problem in the conjecture for the first time, published more than 300 pages of papers, and gave a complete proof of Poincare's conjecture.
13, (the above proof process needs to be tested for one or two months and analyzed by mathematicians from other countries.
14) The June issue of Asian Journal of Mathematics published in the United States published a long article with more than 300 pages, entitled "The Complete Proof of Poincare Conjecture and Geometric Conjecture: The Application of Hamilton-perelman Theory".
15, this proof is of great significance, which will help human beings to better study three-dimensional space and have a far-reaching impact on physics and engineering.
16, four-color problem, also known as four-color conjecture, is one of the three major mathematical problems in the modern world.
17, the content of the four-color problem is: "Any map with only four colors can make countries with the same border have different colors.
18 ",expressed in mathematical language, means" subdivide the plane into non-overlapping areas at will, and each area can always be marked with one of the four numbers 1, 2, 3 and 4, without making two adjacent areas get the same number.
19, "(right) The adjacent area mentioned here means that a whole section of the boundary is common.
20. Two regions are not adjacent if they intersect at only one point or a limited number of points.
2 1, because painting them the same color will not cause confusion.
22. The four-color conjecture was put forward by Britain.
23. 1852, when Fernandez guthrie, who graduated from London University, came to a scientific research institute to paint maps, he found an interesting phenomenon: "It seems that every map can be painted in four colors, so countries with the same border will be painted in different colors.
24. "Can this phenomenon be strictly proved by mathematical methods? He and his younger brother, Grace, who is in college, are determined to give it a try.
25. The manuscripts used by the two brothers to prove this problem have been piled up, but the research work has not progressed.
1852101On October 23rd, his younger brother asked his teacher, the famous mathematician De Morgan, about the proof of this problem. Morgan couldn't find a solution to this problem either, so he wrote to his good friend, Sir Hamilton, a famous mathematician, for advice.
27. After receiving Morgan's letter, Hamilton demonstrated the four-color problem.
28. However, until the death of 1865 Hamilton, this problem remained unsolved.
29, 1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became the concern of the world mathematical community.
30. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture.
During 3 1 and 1878 ~ 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. Everyone thought that the four-color conjecture was solved from now on.
32. Kemp's proof is as follows: First, it is pointed out that if no country surrounds other countries, or no more than three countries intersect at one point, this map is called "regular" (left picture).
33. If it is a regular chart, otherwise it is an irregular chart (right).
34. A map is often linked by a regular map and an informal map, but the number of colors required by an informal map generally does not exceed that required by a regular map. If a map needs five colors, it means that its regular map is five colors. To prove the four-color conjecture, it is enough to prove that there is no regular five-color map.
Kemp proved this by reducing to absurdity, to the effect that if there is a regular five-color map, there will be a "minimal regular five-color map" with the least number of countries. If there is a country with less than six neighboring countries in the minimal regular five-color graph, then there will be a regular graph with fewer countries that is still five-colored, so there will be no country with minimal five-color graph, and there will be no regular five-color graph.
In this way, Kemp thought he had proved the "four-color problem", but later people found him wrong.
37. However, Kemp's proof clarifies two important concepts and provides ideas for solving problems in the future.
The first concept is "configuration".
He proved that in every regular graph, at least one country has two, three, four or five neighbors, and there is no regular graph in which every country has six or more neighbors. In other words, a set of "configurations" consisting of two neighboring countries, three neighboring countries, four or five neighboring countries is inevitable, and each map contains at least one of these four configurations.
Another concept put forward by Kemp is reducibility.
4 1, the use of the word "reducible" comes from Kemp's argument.
42. He proved that as long as there is a country with four neighboring countries in a five-color map, there will be a country with fewer five-color maps.
Since the concepts of "configuration" and "reducibility" were put forward, some standard methods for testing configurations to determine whether they are reducible have been gradually developed, and the inevitable groups of reducible configurations can be found, which is an important basis for proving the "four-color problem".
44. However, to prove that a large configuration is reducible, many details need to be checked, which is quite complicated.
45. After 1 1 year, that is, 1890, Herwood, who was only 29 years old and studied at Oxford University, pointed out the loopholes in Kemp's certificate with his own accurate calculation.
46. He pointed out that Kemp's reason that a country can have five neighboring countries without a minimum five-color map is flawed.
47. Soon Taylor's proof was also denied.
48. People found that they actually proved a weak proposition-the five-color theorem.
49. In other words, five colors are enough to color the map.
50. Later, more and more mathematicians racked their brains for this, but found nothing.
5 1, so people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture.
Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea.
53. 19 13. boekhoff, a famous American mathematician and Harvard University, used Kemp's ideas and combined his new ideas; It is proved that some large configurations are reducible.
54. Later, American mathematician Franklin proved in 1939 that maps below 22 countries can be colored in four colors.
55, 1950, someone pushed from 22 countries to 35 countries.
56, 1960, which proves that maps below 39 countries can be colored with only four colors; And then push it to 50 countries.
57. It seems that this progress is still very slow.
The invention of high-speed digital computers has prompted more mathematicians to study the "four-color problem".
59. Heck, who studied the four-color conjecture from 1936, publicly declared that the four-color conjecture can be proved by finding the necessary group of reducible graphs.
60. His student Toure wrote a calculation program. Heck can not only prove that the configuration is reducible with the data generated by this program, but also describe the reducible configuration by transforming the map into a shape called "duality" in mathematics.
6 1, he marked the capital of each country, and then connected the capitals of neighboring countries with a railway crossing the border. Except for the capital (called vertex) and the railway (called arc or edge), all other lines have been erased, and the rest are called dual graphs of the original graph.
62. In the late 1960s, Heck introduced a method similar to moving charges in an electrical network to find an inevitable set of configurations.
63. The "discharge method", which first appeared in a rather immature form in Heck's research, is a key to the later study of inevitable groups and a central element to prove the four-color theorem.
64. Since the advent of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid increase of calculation speed and the appearance of man-machine dialogue.
65. Harken of the University of Illinois in the United States began to improve the "discharge process" in 1970, and then compiled a good program with Appel.
66.1In June, 976, they spent 1200 hours on two different computers in the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem, which caused a sensation in the world.
67. This is a great event that has attracted many mathematicians and math enthusiasts for more than 0/00 years. When two mathematicians published their research results, the local post office stamped all the mails sent that day with a special postmark of "four colors are enough" to celebrate the solution of this problem.
68. The "four-color problem" proved to be only a solution to a difficult problem that lasted for more than 100 years, and it became the starting point of a series of new ideas in the history of mathematics.
In the process of studying the "four-color problem", many new mathematical theories have emerged and many mathematical calculation skills have been developed.
70. If the problem of map coloring is transformed into a problem of graph theory, the content of graph theory will be enriched.
7 1. Not only that, the "four-color problem" promotes the effective design of airline flight schedules and the design of computer coding programs.
72. However, many mathematicians are not satisfied with the achievements of computers. They think there should be a simple and clear written proof method.
73. Today, many mathematicians and math enthusiasts are still looking for more concise proof methods.