Jules Henri Poincaré, a French mathematician, celestial mechanic, mathematical physicist and philosopher of science, is one of the pioneers of dynamic system theory and multiple complex variables theory. 1854 was born in Nancy, France on April 29th, and 19 12 was born on July 29th. Poincare's research involves many fields, such as number theory, algebra, geometry, topology, celestial mechanics, mathematical physics, theory of multiple complex variables, philosophy of science and so on. He is recognized as the leading mathematician in the last quarter of the19th century and the beginning of the 20th century, and he is also the last person who has a comprehensive understanding of mathematics and its applications. Poincare's outstanding work in mathematics has a far-reaching impact on mathematics in the 20th century and today. His research on celestial mechanics is a milestone after Newton, and he is recognized as a theoretical pioneer of relativity because of his research on electronic theory.
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Poincare's research involves many fields, such as number theory, algebra, geometry, topology and so on. The most important work is in function theory. His early main work was to establish the theory of automorphism function (1878). He introduced fuchs group and Klein group, and constructed a more general basic domain. He constructed an automorphism function by using the series named after him later, and found the utility of this function as a univalent function of algebraic function.
In 1883, poincare put forward a general single-valued theorem (in 1907, he and Kirby gave a complete proof of mutual independence). In the same year, he continued to study the general analytic function theory, and studied the genus of the whole function and its relationship with the growth rate of Taylor expansion coefficient or absolute value of the function, which together with piccard's theorem formed the basis for the later development of the whole function and meromorphic function theory. He is also one of the pioneers of the theory of multiple complex variables.
In order to study the stability of planetary orbits and satellite orbits, Poincare established the qualitative theory of differential equations in four papers published in 188 1 ~ 1886. He studied the behavior of solutions of differential equations near four singularities (focus, saddle point, node and center). He suggested that the stability of the solution can be judged according to the relationship between the solution and the limit cycle (a special closed curve he calculated).
1885, King Oscar II of Sweden established the "N-body problem" prize, which aroused Poincare's interest in studying celestial mechanics. He won a prize for his paper on the periodic solutions of three-body when the mass of two three-body problems is much smaller than the other, and proved that the number of periodic solutions of this restricted three-body is as large as the potential of continuum. After that, he did a lot of research on celestial mechanics, introduced the method of gradual expansion, and obtained strict celestial mechanics calculation technology. Poincare's work has brought great and far-reaching influence to the solution of N-body problem and the research of dynamic system. Firstly, Poincare proved that when n is greater than 2, there is no uniform first integral for the N-body problem. That is to say, even for a general three-body, it is impossible to finally reduce the degree of freedom of the problem by looking for various invariants, and simplify the problem into a simpler problem that can be solved, which broke the illusion that many people hoped to find a general explicit solution of the three-body at that time. People who study differential equations a hundred years later know from their teachers in the second week that most differential equations can't find quantitative solutions, but generally they can learn more about the nature of solutions from qualitative theory, and even "see" the shape and behavior of solutions through computers. In Poincare's time, most mathematicians were more interested in finding solutions by algebraic or power function methods. Using qualitative and geometric methods to discuss differential equations originated from Poincare's research on N-body problems, which completely changed people's basic thinking of studying differential equations. Secondly, in order to study the N-body problem, Poincare invented many new mathematical tools. For example, he put forward the concept of invariant integral completely and used it to prove the famous recurrence theorem. For another example, in order to study the behavior of periodic solutions, he introduced the concept of first regression mapping, which was called Poincare mapping in later dynamic system theory. There are also characteristic indices, continuous dependence of solutions on parameters, and so on. These have become the basic concepts in modern differential equations and dynamic system theory. Thirdly, by studying the so-called asymptotic solution, homoclinic orbit and heteroclinic orbit, Poincare found that even in a simple three-body near such homoclinic orbit or heteroclinic orbit, the solution of the equation will be so complicated that it is almost impossible to predict the final fate of this orbit when time approaches infinity under given initial conditions. In fact, half a century later, later mathematicians discovered that this phenomenon is common in general power systems. They call it homoclinic entanglement caused by the normal intersection of stable manifold and unstable manifold. This uncertainty about the long-term behavior of orbit is called chaos by mathematicians and physicists. Poincare's discovery can be said to be the pioneer of chaos theory.
Poincare also initiated the theory of dynamic system, and proved "Poincare regression theorem" in 1895. Another important achievement of his celestial mechanics is that under the action of gravity, there are three kinds of Poincare pears with rotating fluid shapes besides the known rotating ellipsoid, unequal axis ellipsoid and annular body.
Poincare also made contributions to mathematical physics and partial differential equations. He proved the existence of solutions to Dirichlet problem by sweeping method, which promoted the new development of potential theory. He also studied the eigenvalue problem of Laplace operator, and gave a strict proof of the existence of eigenvalue and eigenfunction. He introduced the complex parameter method into the integral equation, which promoted the development of Fredholm's theory.
Poincare's most important influence on modern mathematics is the creation of combinatorial topology. 1892 published the first paper, and 1895 ~ 1904 established combinatorial topology in six papers. He also introduced Betty number, torsion coefficient, basic group and other important concepts, and created tools such as triangulation of manifold, simple complex, redistribution of center of gravity, dual complex and correlation coefficient matrix of complex. Using these tools, Euler's polyhedron theorem becomes Euler-Poincare formula, and the homology duality theorem of manifolds is proved.
Poincare's thought foreshadows Drumm's theorem and Hodge's theory. He also put forward the Poincare conjecture. In "Poincare's last theorem", he attributed the existence of periodic solutions of restricted three-body to the existence of fixed points of plane continuous transformation satisfying certain conditions.
Poincare's work in number theory and algebra is not much, but it is very influential. His book Algebraic Geometry on Rational Number Field initiated the study of understanding Diophantine equation. He defined the rank number of curves and became an important research object of Diophantine geometry. He introduced group algebra into algebra and proved its decomposition theorem. The concepts of left ideal and right ideal in algebra are introduced for the first time. The third fundamental theorem of Lie algebra and Campbell-Hausdorf formula are proved. The envelope algebra of Lie algebra is introduced, its foundation is described, and the Poincare-boekhoff-Witt theorem is proved.
Poincare made in-depth and extensive research on classical physics and contributed to the establishment of special relativity. Earlier than Einstein, Poincare published an article "Relativity of Space" in 1897, which already reflected the shadow of special relativity. 1898, poincare published the article "the measurement of time", and put forward the hypothesis that the speed of light is constant. 1902, poincare expounded the principle of relativity. 1904, poincare named the coordinate transformation relationship between two inertial reference systems given by Lorentz as' Lorentz transformation'. Later, in June of 1905, Poincare published a related paper before Einstein: on electron dynamics. [2] He began to study electronic theory from 1899, and first realized that Lorentz transformation constitutes a group (1904). The next year, Einstein also got the same result in his paper on the establishment of special relativity.
Poincare's philosophical works Science and Hypothesis, The Value of Science and Science and Method also have great influence. He is a representative of conventionalism philosophy, and thinks that scientific axioms are convenient definitions or conventions. You can choose from all possible conventions, but you need to avoid all contradictions based on experimental facts. In mathematics, he disagreed with Russell and Hilbert, opposed the concept of infinite set, agreed with infinite potential, thought that the most basic intuitive concept of mathematics was natural number, and opposed the reduction of natural number to set theory. This made him one of the pioneers of intuitionism.
1905, the Hungarian Academy of Sciences awarded the Baljo Prize with a prize of 10000 kronor. This award is given to mathematicians who have made the greatest contribution to the development of mathematics in the past 25 years. Since Poincare has been engaged in mathematical research since 1879 and has made outstanding contributions in almost the whole field of mathematics, this award belongs to him.
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