A small topology problem is also neatly solved. However, Junxin suddenly reacted, and many theories and reasoning have not yet been circulated in this era. When I solve the problem, I habitually omit some inferences, and I don't know if this large number of professors can understand it. So Junxin didn't care too much about these, what he cared about was some thoughts brought to him by the last two additional questions in this exam.

Since its birth, mathematics has rapidly developed into an important tool for the progress of human civilization, whether it is the temple calculation in the war, or the taxation and population in the cultural governance, etc., are inseparable from the development of mathematics.

From the perspective of the history of the development of mathematics in the West, the three great mathematicians of ancient times, such as Euclid, Archimedes and Diophantus, established the classical theory of ancient mathematics, laid the foundation of theoretical mathematics, and extracted mathematics from the concreteness of life to form the initial systematic mathematical theory.

Since modern times, Ischak Newton and Leibniz of Germany have established calculus respectively, and in terms of the development of science, the theory of the former has a greater impact on physics, while the theory of the latter has a greater impact on mathematics. Regardless of the great fights between the two countries that followed them over the invention of calculus, the establishment of calculus by the two men really pushed the development of mathematics forward. The development of mathematics began to enter a period of rapid progress, and began to intersect with other disciplines and advance each other.

Later mathematicians, such as Gauss, Euler, Hardy, Lagrange, Hilbert, Grothendieck, etc., each led their own era and promoted the progress of mathematics. and made unimaginable achievements in their respective fields of study, just like the result of Gauss's lifetime, that is, there are hundreds of theorems and formulas named after him.

However, even these mathematicians still have problems that cannot be solved or have not been solved, and these problems have changed over time, either in the process of research, or in the process of research, they are still at a loss, and they have been plaguing the mathematical community.

One of the most famous is the famous 23 problems of Hilbert proposed by the famous German mathematician David Hilbert at the World Congress of Mathematicians in August 1900.

Hilbert's 23 questions are divided into four sections: questions 1 to 6 are about the foundations of mathematics, questions 7 to 12 are about number theory, 13 to 18 are about algebra and geometry, and 19 to 23 are about mathematical analysis. As of the beginning of the 21st century, Hilbert's 23 questions remain unresolved. Become a problem that mathematicians are eager to solve.

Hilbert's 23 problems have had a great impact on the history of mathematics, which has lasted for more than 100 years, and has not been eliminated, which is certainly related to Hilbert's great achievements in mathematics, but these problems are originally difficult problems in the field of mathematics, and the ideas and ideas obtained by solving these problems are the most important values to promote the development of mathematics.

As a simple example, the proof of Fermat's theorem is the best example of how famous mathematical conjectures are driving mathematical progress. After the death of the Frenchman Fermat, the notes he wrote in a book of arithmetic did not disappear with it. His eldest son, realizing that the scribbled handwriting might have some value, spent five years sorting it out and printing a special edition of Arithmetic, containing side notes made by his father, which contained a series of theorems.

In the margins near question 8, Fermat writes:

"It is impossible to write a cubic number as the sum of two cubic numbers; Or write a power of 4 as the sum of two powers of 4; Or, in general, it is impossible to write a power higher than 2 as the sum of two powers of the same power. ”

This genius, who loves to play pranks, wrote an additional commentary at the end:

"I have a very wonderful proof of this proposition, and the space here is too small to write."

Fermat wrote these lines around 1637, and these lucky discoveries became the misfortune of all subsequent mathematicians. A theorem that could be understood by a high school student became the biggest mystery in mathematics, tormenting the world's brightest minds for 358 years. Generations of mathematical geniuses have challenged this conjecture.

Fermat's theorem is inextricably linked to the history of mathematics and touches on all the important issues of number theory in the 20th century. The proof process involved the group theory developed by Galois, a genius French mathematician in the 19th century, in order to find the solution of the fifth equation, and the Japanese mathematicians Taniyama and Shimura in the 20th century proposed the Taniyama-Shimura conjecture to analyze the Iwasawa theory and the Koliwakin-Fletcher theory for elliptic equations. Most importantly, it connects several unrelated areas of mathematics and develops entirely new mathematical techniques.

The application of these theories to the proof of Fermat's theorem once again shows the great role of mathematical conjecture in promoting the development of mathematics. Therefore, every proof of mathematical conjecture, whether right or wrong, will cause a huge sensation in the mathematical community, and it can make mathematicians in different fields put down the important work at hand and then exchange their ideas in a common place, thus further promoting the development of mathematics. At the same time, this is an important reason why the study of mathematics is the least confidential scientific research in the industry.

The last two questions on the paper, one from number theory, are from number theory, and in the field of number theory, nothing is more legendary than Fermat's theorem, although the Goldbach conjecture is a little more widely known in China. Again, in the field of topology, there is nothing more fascinating than the Poincaré conjecture.

The Poincaré conjecture is a conjecture about manifolds put forward by the famous French mathematician Poincaré in 1904, that is, "any single-connected, closed three-dimensional manifold must be the same as a three-dimensional sphere." "To put it simply, a closed three-dimensional manifold is a three-dimensional space without boundaries; Single connectivity means that every closed curve in this space can be continuously contracted into a point, or in a closed three-dimensional space, if each closed curve can be contracted into a point, this space must be a three-dimensional sphere. Later, this conjecture was extended to more than three dimensions, and it was called the "high-dimensional Poincaré conjecture".

This is a fundamental proposition in topology, which will help mankind to better study three-dimensional space, and its results will deepen people's understanding of the nature of manifolds. In 2000, it was even set up by the Clay Institute in the United States as one of the seven mathematical problems of the millennium. Together with the P-pair NP complete problem, the Hodge conjecture, the Riemann hypothesis, the Young-Mills existence and mass gap, the existence and smoothness of the Navier-Stokes equation, and the Bayhe and Swinaton-Dale conjectures, it is also a mathematical problem that needs to be solved urgently.

As a senior researcher at the Princeton Institute, Junxin naturally knows Andrew Wiles's proof process well, and because of the foundation of physics he has previously studied, he is also familiar with the proof of Poincaré's conjecture. This is also the reason why he studied topology so well, all as a result of a lot of study in order to understand Perelman's proof of Poincaré's conjecture.

Thinking of this, Junxin found a remote place, sat down, and began his own reasoning and proof process. The first to write is a famous conjecture put forward by the British mathematician Moder in 1922, which is an important conjecture to prove Fermat's theorem.