1September, 826 17, Riemann (1826-1866) was born in a village priest's home in Bre Slentz, Hanover, and was the second of six children.
Riemann loved mathematics since he was a child. At the age of six, he began to learn arithmetic, which showed his mathematical genius. He can not only solve all the math problems left to him, but also often ask some questions to play tricks on his brothers and sisters. /kloc-When I was 0/0 years old, I studied advanced arithmetic and geometry with a professional teacher, and soon surpassed the teacher and often gave better answers to some questions.
Riemann 14 years old went to middle school in Hanover. Due to financial constraints, he always walks back and forth between Hanover and small rural villages. Of course, he has no money to buy reference books. Fortunately, the headmaster of the middle school discovered his talent in mathematics in time. Considering his financial difficulties, the headmaster authorized Riemann to borrow math books from his private library. On the recommendation of the headmaster, Riemann borrowed a mathematician Legendre's Number Theory, which is a four-page masterpiece with 859 pages. Riemann cherished this opportunity to study very much, and he taught himself eagerly. Six days later, Li Man finished learning and returned the book. The headmaster asked him, "How much have you read?" Riemann said, "This is an amazing book, and I have mastered it." A few months later, the headmaster tested him on the content of the book. Riemann answers questions like water and answers comprehensively. Using the principal's library, Riemann also took the time to teach himself the works of Euler, a great mathematician, and thus mastered calculus and its branches. Riemann not only learned mathematics knowledge from Euler's works, but also learned Euler's skills in studying mathematics.
College career
19 years old, Riemann entered the University of G? ttingen. In order to help his family financially and find a paid job as soon as possible, he first studied philosophy and theology, but in addition to these two courses, he also took part in mathematics and physics courses. He listened to Stern's lecture on equation theory and definite integral, Gauss's lecture on least square method and Gordes Mitter's lecture on geomagnetism, and he became interested in mathematics.
Riemann told his father all this and asked for permission to change his major in mathematics. Father heartily agreed to his request. Riemann was very happy and deeply grateful to his father.
1847, in order to learn from more masters, Riemann transferred to Berlin University, where he studied under the great mathematicians jacoby, Dirichlet, Steiner and Eisenstein. He learned advanced mechanics and algebra from jacoby, number theory and analysis from Dirichlet, modern geometry from Stayner and elliptic function theory from Weinstein.
During this period, he was so diligent that he didn't even take a holiday. 1847 in the autumn vacation, Riemann discovered several Journals of the Paris Academy of Sciences, which contained a new paper on analytic functions of simple complex variables published by mathematician Cauchy. He saw at a glance that this was a new mathematical theory, so he stayed indoors for several weeks, devoted himself to studying Cauchy's paper and brewing his new views on this subject, laying the foundation for writing his doctoral thesis "General Theory of Functions of Simple and Complex Variables" four years later.
Riemann not only carefully studied the master's academic monographs, but also humbly asked the master for advice. Once, Dirichlet came to G? ttingen for a holiday. Riemann took this opportunity to ask him questions about mathematics and handed him his unfinished paper for advice. Dirichlet was fascinated by Riemann's modesty, sincerity and genius. He had a long talk with Riemann for two hours, made a lot of comments on Riemann's paper, and gave many directions to the topic that Riemann was studying. Riemann benefited a lot. He said that without Dirichlet's guidance, he would have to study hard in the library for a few days.
Although he lived in poverty, he studied very hard, which made Riemann achieve fruitful results when he graduated from college. 185 1 At the end, Riemann submitted his doctoral thesis to the great mathematician Gauss for review. Gauss was very excited after reading the paper and spoke highly of Riemann's paper, which is rare for Gauss. Gauss commented: "The paper submitted by Mr. Riemann provides convincing evidence that the author has conducted a comprehensive and in-depth study on this issue discussed in this paper, indicating that the author has a creative, active and true mathematical mind and brilliant creativity."
Strive for progress in poverty
At the beginning of 1852, Riemann obtained his doctorate with excellent academic performance and stayed at the University of G? ttingen. In Germany in the middle of19th century, science had almost nothing to do with the national economy. Universities are only established to train lawyers, doctors, teachers and missionaries, and provide places for aristocratic children and rich children to spend attractive and respected years. Only full professors can get government subsidies, and they can teach formal standard courses. These courses are all basic subjects, and there are many students in the class, and the professors charge a lot of tuition. This is also the reason why the course standard was low at that time, because if the course was too difficult, it would be impossible to accept many students, which would affect the professor's income. After all, the purpose of aristocratic children and rich children going to college is not to really study. Lecturers, on the other hand, have no government subsidies and no chance to teach basic formal courses. They live entirely on the tuition fees of students who come to attend classes. There are not many students attending classes at ordinary times, and their income is quite meager, so life is very difficult. Being a lecturer is the only way to become a full professor. But when the lecturer can be promoted to professor, there is no explicit provision. In order to take care of a particularly valuable scholar and there is no vacancy for a full professor, the government can appoint him as a "visiting professor" to make him qualified to teach basic formal courses and increase his income, but this appointment has additional conditions, stipulating that the government will not pay any allowance. Therefore, during his tenure as a lecturer, Riemann did not have any independent source of living expenses, and his life was still poor.
However, despite living in poverty, Riemann still devoted all his energy to mathematics. He feels that as long as he can barely make a living and let him learn mathematics, he will be satisfied. He has never been depressed by financial evidence. On the one hand, he actively prepared the inaugural speech paper of "unpaid lecturer", on the other hand, he seriously engaged in the research work of mathematical physics. His inaugural thesis is quite difficult. At first, in order to determine the topic of the paper, he submitted three topics to Gauss, who asked him to choose one. Among them, the third topic is related to the geometric basis. Riemann didn't have much desk preparation at that time, so Riemann sincerely hoped that Gauss wouldn't choose. However, Gauss studied the third topic deeply, and he thought about it for 60 years. In order to see what kind of creative work Riemann will do on this profound issue, Gauss designated the third topic as the title of Riemann's inaugural speech.
Afterwards, Riemann told his father about it, "so I fell into a desperate situation again" and "I had to create this topic".
Riemann also has infinite enthusiasm for the study of mathematical physics. At that time, he once said to people, "I am fascinated by mathematical research that combines everything with the laws of physics." "Through the general study of the relationship between electricity, light and magnetism, I found an explanation for this phenomenon. This is very important to me, because this is the first time I can apply my work to unknown phenomena. " These two studies were of high level at that time, so they were extremely difficult. Despite poverty and malnutrition, Riemann forgot his work and thought nervously for a long time, so that he was often exhausted and even fell ill. Once he has recovered a little, he will continue his research. Where there is a will, there is a way. 1854, 10 In June, Riemann delivered his inaugural speech with his paper "Assumptions on Geometric Basis", which was recognized and praised by mathematicians attending the meeting. Gauss was surprised after hearing this. He felt that the young man had handled this difficult problem well and was full of praise. Riemann's paper is regarded as one of the masterpieces in the history of mathematics in19th century.
From 65438 to 0855, the University of G? ttingen began to pay Riemann's salary, but it was quite low. A year is only equivalent to $200. In this year, Mann was 29 years old, and his family suffered great misfortune. His father and one sister died one after another, and the three sisters who once relied on their father lost their source of livelihood. So riemann sum and his brothers took on the burden of taking care of the lives of the three sisters. Riemann is always worried about the life of his family. 1857, Riemann's annual salary increased to the equivalent of $300. Because of the low income and the heavy burden of taking care of three sisters, Riemann didn't even dare to consider his marriage. However, in this year, unfortunately, it fell from the sky again, and Riemann's brother died again. This is like adding insult to injury to Riemann, and the burden of taking care of the lives of the three sisters falls on his shoulders alone. During the five years from 1855 to 1859, Riemann was always trapped by economic difficulties and poverty, and sometimes his family even fell into a situation where they needed to calculate their own rations. It is in this situation that Riemann, regardless of the poverty of material life, still devoted himself to mathematical research, struggled hard on the rugged scientific road and made amazing achievements. Many of his important achievements in mathematics were completed during this period. His research on Abel integral and Abel function initiated modern algebraic geometry. He pioneered the study of number theory with complex analytic functions and the analytic number theory in the modern sense; His research on hypergeometric series promoted the development of mathematical physics and differential equation theory. With the advent of research results, Riemann's academic reputation in the field of mathematics has rapidly increased. He was praised by many world-famous mathematicians and won the highest honor that a scientist can usually get.
The death of the master
1859 When Riemann was 33 years old, Gauss died. He was appointed as a full professor at the University of G? ttingen and became the second successor of Gauss after Dirichlet. At this time, Riemann's life began to improve, and he began to think about personal marriage. At the age of 36, he married a friend's sister. A year later, his daughter was born in Pisa.
However, the long-term poverty, overwork and vigorous research made Riemann weak and tired. 1862, Riemann suffered from pleurisy, lung disease soon, and jaundice a year later. Despite his illness, Riemann persisted in his mathematical research as long as he had a little strength. Although Riemann actively sought medical treatment during this period, he was blind due to illness and ultimately had no effect. 1866 On July 20th, Riemann's pure and noble heart stopped beating. He died prematurely, and he also left mathematics prematurely at the age of 40.
Riemann is one of the most original mathematicians in the history of mathematics. He has done a lot of basic and creative research work in many fields of mathematics: he initiated the theory of complex variable function from the geometric direction; Is the founder of analytic number theory in the modern sense; He personally established Riemannian geometry and was a pioneer of combinatorial topology. He made an important contribution to the strict treatment of calculus; It has also achieved fruitful results in the fields of mathematical physics, differential equations and so on. 1859, riemann was elected as an academician of the Berlin institute of communication; 1866, he was elected as an academician of the Paris institute of communication and a member of the royal society abroad.
Riemann's untimely death is a pity for German mathematics and even the whole world! However, there are too many rich concepts in his few published papers, which have not been fully studied by later mathematicians.
1September, 826 17, Riemann was born in Breselenz village in Hanover, northern Germany. His father is a poor country priest. He started school at the age of six, 14 entered the pre-university study, and 19 entered the University of G? ttingen, studying philosophy and theology according to his father's wishes, so as to follow his father's wishes and become a priest in the future.
Because he loved mathematics since he was a child, Riemann listened to some math classes while studying philosophy and theology. At that time, the University of G? ttingen was one of the mathematical centers in the world, and some famous mathematicians such as Gauss, Weber and Steyr had taught in the school. Riemann was infected by the atmosphere of mathematics teaching and research here and decided to give up theology and specialize in mathematics.
From 65438 to 0847, Riemann transferred to Berlin University and became a student of Jacoby, Dirichlet, Steiner and Eisenstein. 1849, he returned to Golding University to study for a doctorate and became a student of Gauss in his later years.
L85 1 year, Riemann received a doctorate in mathematics; 1854, he was hired as a part-time lecturer at the University of G? ttingen. 1857 promoted to associate professor; 1859, Dirichlet was hired as a professor to replace his death.
Due to years of poverty and fatigue, Riemann began to suffer from pleurisy and tuberculosis less than a month after she got married in 1862, and spent most of the next four years in Italy for treatment and rest. 1866 died in Italy on July 20th at the age of 39.
Riemann is one of the most original mathematicians in the history of world mathematics. Riemann's works are few, but extremely profound, full of creation and imagination of concepts. In his short life, Riemann has done a lot of basic and creative work in many fields of mathematics and made great contributions to world mathematics.
The founder of complex variable function theory
/kloc-the most unique creation of mathematics in the 0/9th century is the creation of the theory of complex variable functions, which is the continuation of people's research on complex numbers and complex variable functions in the 0/8th century. Before 1850, Cauchy, jacoby, Gauss, Abel, Wilstrass and others had systematically studied the theory of single-valued analytic functions, but for multi-valued functions, only Cauchy and Pisser had some isolated conclusions.
185 1 year, under the guidance of gauss, riemann completed his doctoral thesis entitled "general theoretical basis of simple complex variable function", and later published four important articles in the journal of mathematics, further expounding the ideas in his doctoral thesis. On the one hand, he summarized the previous achievements about single-valued analytic functions, processed them with new tools, and established the theoretical basis of multi-valued analytic functions.
Cauchy and riemann sum Wilstrass are recognized as the main founders of the theory of complex variable functions, and it was later proved that Riemann method is essential in dealing with the theory of complex variable functions. The thoughts of Cauchy and Riemann are integrated, and the thoughts of Wilstrass can be deduced from Cauchy-Riemann's point of view.
In Riemann's treatment of multivalued functions, the most important thing is that he introduced the concept of "Riemann surface". Multi-valued functions are geometrically intuitive through Riemannian surfaces, and the multi-valued functions represented on Riemannian surfaces are single-valued. He introduced fulcrum, section line, defined connectivity on Riemannian surface, and studied the properties of functions, and obtained a series of results.
The complex function handled by Riemann, single-valued function is an example of multi-valued function. He extended some known conclusions of single-valued functions to multi-valued functions, especially his method of classifying functions according to connectivity, which greatly promoted the initial development of topology. He studied Abel function, Abel integral and the inversion of Abel integral, and got the famous Riemann-Roche theorem. The first double rational transformation constitutes the main content of algebraic geometry developed in the late19th century.
In order to perfect his doctoral thesis, Riemann gave several applications of his function theory in conformal mapping at the end of the paper, extended Gauss's conclusion about conformal mapping from plane to plane in 1825 to arbitrary Riemann surfaces, and gave the famous Riemann mapping theorem at the end of the paper.
The founder of Riemannian geometry
Riemann's most important contribution to mathematics lies in geometry. The study of high-dimensional abstract geometry initiated by him and the methods and means to deal with geometric problems are a profound revolution in the history of geometry. He established a brand-new geometric system named after himself, which had a great influence on the development of modern geometry and even the branches of mathematics and science.
1854, Riemann gave a speech to all the faculty and staff in order to obtain additional lecturer qualification at the University of G? ttingen. This speech was published two years after his death (1868), entitled "Hypothesis as the Basis of Geometry". In his speech, he briefly summarized all known geometries, including hyperbolic geometry, one of the newly born non-Euclidean geometries, and proposed a new geometric system, which was later called Riemannian geometry.
In order to compete for the prize of Paris Academy of Sciences, Riemann wrote an article on heat conduction in 186 1, which was later called his "work in Paris". In this paper, his article 1854 is treated technically to further clarify his geometric thought. This article was included in his anthology 1876 after his death.
Riemann mainly studies the local properties of geometric space, and he adopts differential geometry, which is opposite to the Euclidean geometry or non-Euclidean geometry of Gauss, Bolyai and Lobachevsky, which regards space as a whole. Riemann got rid of the shackles of Gauss and other predecessors who limited geometric objects to curves and surfaces in three-dimensional Euclidean space, and established a more general abstract geometric space from the perspective of dimensions.
Riemann introduced the concepts of manifold and differential manifold, and called dimensional space manifold. A point in a dimensional manifold can be represented by a set of specific values of variable parameters, and all these points constitute the manifold itself. This variable parameter is called the coordinate of manifold and is differentiable. When the coordinates change continuously, the corresponding points traverse the manifold.
Riemann takes the traditional differential geometry as the model, and defines the distance between two points on the manifold, the curves on the manifold and the included angle between the curves. Based on these concepts, the geometric properties of dimensional manifolds are studied. On the dimensional manifold, he also defined the curvature similar to Gaussian when studying the general surface. He proved that when his dimension on the dimensional manifold is equal to 3, the situation of Euclidean space is consistent with the results obtained by Gauss and others, so Riemann geometry is a generalization of traditional differential geometry.
Riemann developed Gauss's geometric thought that the surface itself is a space, and studied the intrinsic properties of dimensional manifolds. Riemann's research led to the birth of another non-Euclidean geometry-elliptic geometry.
In Riemann's view, there are three different geometries. The difference between them lies in the number of parallel lines made by a given point around a fixed straight line. If only one parallel line can be made, it is called Euclidean geometry; If you don't know any of them, it's elliptic geometry; If there is a set of parallel lines, we can get the third geometry, that is, Luo Barczewski geometry. Riemann therefore developed the space theory after Lobachevsky, ending the discussion about Euclid's parallel axiom for more than 1000 years. He asserted that objective space is a special kind of manifold, and foresaw the existence of manifolds with certain properties. These were gradually confirmed by later generations.
Because Riemann considers the geometric space of any dimension, it has deeper practical value to the complex objective space. Therefore, in high-dimensional geometry, due to the complexity of multivariate differential, Riemann adopted some methods different from those of his predecessors, which made the expression more concise, and finally led to the birth of modern geometric tools such as tensor, external differential and connection. Einstein successfully used Riemann geometry as a tool to explain general relativity. Now, Riemannian geometry has become the necessary mathematical basis of modern theoretical physics.
Creative contribution of calculus theory
In addition to his pioneering work in geometry and complex variable functions, Riemann is famous in history for his outstanding contribution to the perfection of calculus theory that rose at the beginning of19th century.
From the end of 18 to the beginning of 19 century, the mathematical community began to care about the imprecision of the concept and proof of calculus, the largest branch of mathematics. Porzano, Cauchy, Abel, Dirichlet and later Wells all devoted themselves to rigorous analysis. Riemann studied mathematics from Dirichlet in Berlin University, and had a deep understanding of Cauchy and Abel's work, so he had his unique views on calculus theory.
1854, Riemann needed to submit a paper reflecting his academic level in order to obtain additional lecturer qualification at the University of G? ttingen. What he handed in was an article about the possibility of expressing functions by trigonometric series. This is a masterpiece with rich contents and profound thoughts, which has far-reaching influence on perfecting analytical theory.
Cauchy once proved that continuous functions must be integrable, and Riemann pointed out that integrable functions are not necessarily continuous. Cauchy and almost all mathematicians of his time believed in the relationship between continuity and differentiability. In the next 50 years, many textbooks "proved" that continuous functions must be differentiable. Riemann gave a famous counterexample of continuity and differentiability, and finally clarified the relationship between continuity and differentiability.
Riemann established the concept of Riemann integral described in calculus textbooks, and gave the necessary and sufficient conditions for the existence of this integral.
Riemann studied Fourier series in his own unique way, and extended Dirichlet condition, that is, Riemann condition on convergence of trigonometric series, and obtained a series of theorems on convergence and integrability of trigonometric series. He also proved that the terms of any conditionally convergent series can be rearranged appropriately so that the new series converges to any specified sum or divergence.
Cross-century achievements of analytic number theory
/kloc-An important development of number theory in the 9th century is the introduction of analytical methods and results initiated by Dirichlet, while Riemann pioneered the study of number theory with complex analytic functions and achieved cross-century results.
1859, riemann published the paper "the number of prime numbers under a given size". This is an extremely in-depth paper, less than ten pages. He attributed the distribution of prime numbers to the problem of function, which is now called Riemann function. Riemann proved some important properties of functions, and simply asserted other properties without proof.
In the more than one hundred years after Riemann's death, many of the best mathematicians in the world have been trying to prove his assertion, and in the process of these efforts, they have created new branches with rich contents for analysis. Now, except for one of his assertions, the rest have been solved as Riemann expected.
That unsolved problem is now called "Riemann conjecture", that is, all zeros in the belt region are on a straight line (the eighth of Hilbert's 23 problems), which has not been proved so far. For some other fields, members of the Bourbaki school have proved the corresponding Riemann conjecture. The solution of many problems in number theory depends on the solution of this conjecture. Riemann's work is not only a contribution to analytic number theory, but also greatly enriches the content of complex variable function theory.
Pioneer of combinatorial topology
Before the publication of Dr. Riemann's paper, there were some scattered results in combinatorial topology, among which Euler euler theorem was the most famous one about the relationship between vertices, edges and faces of closed convex polyhedron. There are still some seemingly simple problems that have not been solved for a long time, such as the problem of the seven bridges in Konigsberg and the four-color problem, which prompted people to study combinatorial topology (then called position geometry or position analysis). However, the biggest motivation of topology research comes from Riemann's theory of complex variables.
In the doctoral thesis of 185 1, and in the study of Abelian functions, Riemann emphasized some theorems that the study of functions necessarily requires position analysis. According to modern topological terminology, Riemann has actually classified closed surfaces according to genus. It is worth mentioning that he said in his dissertation that the idea that all functions are composed of connected closed regions (at spatial points) is the earliest functional idea.