Talking about Three Mathematical Crises——Fermat's Last Theorem
At the end of the 19th century and the beginning of the 20th century, with the rapid development of fields such as non-Euclidean geometry and infinitesimal analysis, the mathematics community was facing unprecedented challenges. This debate on the basis of mathematics has been called the "mathematical crisis". The mathematical crisis originated from the re-examination of the basic concepts and axiomatic systems of mathematics, involving many aspects such as set theory, logic, and infinitesimal quantities. The three mathematical crises refer to three extremely difficult problems in the 19th and 20th centuries that profoundly affected the development of mathematics. These three problems are: Riemann conjecture (1826-1866), Poincaré conjecture (1854-1912) and the well-known Fermat's last theorem (1607-1995). These questions amaze everyone and challenge the limits of mathematicians. Next, I will focus on introducing the problems related to Fermat's last theorem.
Fermat's last theorem is a mathematical problem with a long history and much attention. At the time, the French mathematician Pierre de Fermat left a brief note on his blog, claiming to have found a proof that was very elegant, but unlikely to fit in the margins. Such a brief note is enough to give rise to noble, fruitless discussions in the field of algebra.
The problem remained unsolved until British mathematician Andrew Wiles found a proof in 1995. The story is famous all over the world because of its magic. In the first half of the 20th century, mathematicians all over the world tried to solve Fermat's last theorem, but failed. When computers became an integral part of mathematical calculations, people started using them to find solutions. However, the proof of Fermat's last theorem could not be found at the technical level at that time. Now Andrew Wiles represents all those who have tried to solve this problem, and by using a series of advanced mathematical techniques including his own genius and inspiration, he finally found the proof.
The content of Fermat's last theorem is: when n is greater than 2, a^n + b^n = c^n cannot be satisfied. (where a, b, and c are all positive integers)
Wiles' method of proving Fermat's Last Theorem is called the "elliptic curve" method. The so-called elliptic curve is simply the (V, λ) point when the proportional (straight line in the real number field) curve is tangent to L×V0. And Wiles' method of proving its correctness has been successfully practiced, which has also gained a very high degree of recognition in the field.
However, the way to get to this level is not just to discover the proof itself, but involves a comprehensive science that combines the deep research of multiple fields of mathematics and intelligent computer technology. Wiles was not just a mathematician, he was one of the pioneers of exploration and one of those who followed him.
Wiles started reading about Fermat's Last Theorem in his 20s. He thinks that solving this problem has become an unbearable burden in his life. He almost gave up on this puzzle, but overcame it and resumed his attempt to prove Fermat's last theorem. He chose a research approach, which is to decompose complex problems into sub-problems that are easier to study, and seek proofs in these relatively simple problems and methods. In the end, he succeeded in using this new type of research to allow people to study hard mathematical problems in greater depth, demonstrating the flexibility and rigor of human intelligence.
He proved the Fermat conjecture on a very strong integer field category that does not contain zero points, and applied it to the "elliptic curve in the modular sense" to speed up the process of proof. Wiles' proof starts from the conclusion, and solves the problem through the in-depth analysis of the judgment condition subset chain and the method of type classification. The specific implementation process also requires related technologies in the branch of advanced algebraic geometry, such as Frobenius perturbation method and Grothendieck-Teichmüller group, etc.
Andrew Wiles' proof is mainly based on the categorical algebra method in the field of algebraic geometry and the theory of elliptic curves, transforming Fermat's theorem into an isomorphism problem, that is, proving whether a special type of ring exists. Specifically:
First, Wiles defined a set of "Moduli spaces" by using tools such as modular groups and elliptic functions, and transformed Fermat's theorem into two A structural simplification problem known as an "ellipse scheme".
Then, a definite judgment condition is given through the relevant techniques of categorical algebra, which can tell us that some part of this family of rings does not exist, and then deduce the result of Fermat's theorem. This judgment condition is the result of an in-depth analysis of the classification of canonical representations of modular groups and the "three point singularities" produced by some operations on Grothendieck-Teichmüller groups.
Then, he used the "model change" developed in the field of algebraic geometry to complete the in-depth analysis of the judgment condition subset chain and the traversal of the typed classification, and finally obtained the conclusion of infinite recursion, proving that there are such representative elements that are indeed different. exist.
While Wiles' efforts received a lifetime honor, they also demonstrated the importance of modern technology in scientific research. Fermat himself was a very gifted mathematician, but the information that created the puzzle seems to have remained unrevealed for millennia, especially when it came time to find its solution. Therefore, this story also demonstrates the importance of collaboration between multiple fields.
The story of Andrew Wiles' proof of Fermat's Last Theorem not only illustrates the need for a high degree of specialization in the field and what modern computers can achieve, but also shows that the road to success is far from a one-man effort.
By appreciating this story, we can realize the value of hard work, courage to try, continuous exploration, friendship across domains, and the value of boldly moving forward to break through difficulties.
Technological advances have brought greater depth to our understanding. Although Fermat's theorem has been proposed since the 17th century, the rapid development of technologies such as high-speed computers, computer graphics and predictive models in recent years has greatly accelerated the field of mathematics, and promoted the frontier and in-depth academic research in the context of big data.
In short, the proof process of Fermat's theorem makes us understand how difficult and admirable it is to persevere in exploring and solving difficult problems. At the same time, it also educates us to be good at innovative thinking and cross the boundaries of knowledge to solve complex problems in multiple fields.