Bernhard Riemann

German mathematician

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synonym Riemann (Riemann) generally refers to Bornhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 July 20, 1866), a famous German mathematician mathematical analysis and differential geometry He has made important contributions to Riemannian Geometry And to Einstein later General relativity Provides Fundamentals of Mathematics 。^[1]

On July 20, 1866, he went there for the third time Italy Reasons on the way to recuperation pulmonary tuberculosis He died in Silaska.^[2]

Chinese name: Bernhard Riemann
Foreign name: Georg Friedrich Bernhard Riemann
Alias: Riemann
Nationality: Germany
date of birth: September 17, 1826
Date of death: July 20, 1866

University one is graduated from: University of Gottingen
Occupation: mathematician 、 physical scientist
Key achievements: Riemannian Geometry
one's native heath: Kingdom of Hanover Amern Breschlentz
Representative works: Basis of General Theory of Functions of Single Complex Variable 、 Assumptions Based on Geometry, etc
Gender: male

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Character experience

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Riemann (3 sheets)

He was born on September 17, 1826 Kingdom of Hanover Breselenz (today's Germany). His father, Frederick Bonhard Riemann, was a local Lutheran priest. He is the second of six children. He is a quiet, sickly and shy person who likes to be alone all his life. His colleague DeDekin（ Dedekind ）He is one of the few people who know him. according to Dedekind He said that besides Riemann's really bad physical condition, he is still a Hypochondria patient.

In 1840, Riemann moved to Hanover She lived with her grandmother and entered middle school.

In 1842, after his grandmother died, he moved to Johanneum in L ü neburg.

In 1846, Riemann entered University of Gottingen Study philosophy and theology. During this period, he went to some math lectures, including Gauss's least square method Lectures of. With his father's permission, he changed to mathematics. Two years in college University of Berlin Studied by C.G.J Jacobian And P G. L. Dirichlet's influence.

In the spring of 1847, Riemann transferred University of Berlin And put it under the doors of Jacobi, Dirichlet and Steiner. He returned two years later Gottingen 。

In 1851 University of Gottingen Received a doctor's degree.

In 1851, it was demonstrated that Complex function Derivable necessity sufficient condition (i.e. Cauchy Riemann equation). With the help of Dirichlet principle Elaborated Riemann mapping theorem , which becomes the basis of geometric theory of function.

In 1853, he defined Riemann integral and studied Trigonometric series Convergence criteria.

In 1854, Gauss's differential geometry Research and propose manifold Understand the essence of space by using the square of the length of differential arc Positive definite quadratic form Understand metrics, establish the concept of Riemannian space euclid geometry 、 Non Euclidean geometry Included in his system.

Tomb of Li Man

In 1854, he took the stage for the first time and made a speech entitled "On Action" Geometric Basis ", which created Riemannian geometry and provided Einstein's general theory of relativity with Fundamentals of Mathematics 。

In 1857, he was awarded the associate professor of Gottingen University

In 1857, he published a report on Abel The paper on function leads to the concept of Riemannian surface Abel integral With the theory of Abel function to a new turning point and do systematic research. Of which Riemann surface From topology, analysis Algebraic geometry All aspects have been studied in depth. Created a series of pairs of Algebraic topology Develop the concept of far-reaching influence, and clarify the Riemann Roch theorem supplemented by G. Roch later. In 1857, he was promoted to a non editorial professor of Gottingen University. In 1859, he succeeded Dirichlet as a professor. He also published a paper on the number of prime numbers less than a given value and proposed the Riemann hypothesis.

In 1862, he married Elise Koch.

On July 20, 1866, he went there for the third time Italy On the way to recuperation pulmonary tuberculosis He died in Selasca.^[2]

Character relationship

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He pursues his teacher Gaussian Our motto would rather be less, but mature.^[8]

Main contributions

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In 1859, he published a report on prime number In the paper on the number of prime numbers less than a given value Riemann ζ function The integral representation of ζ function and the Functional equation He pointed out that there is a deep connection between the distribution of prime numbers and Riemann ζ function. The core of this correlation is the integral of J (x) expression 。

In 1854, Riemann University of Gettingen Published under the title《 On Assumptions as the Basis of Geometry 》The speech of Riemannian geometry 。 Riemann regarded the surface itself as an independent geometric entity, rather than just a geometric entity in Euclidean space. In 1915, A. Einstein used Riemannian Geometry and Tensor analysis Tools create new Gravitational theory —— General relativity 。

Besides, he was right partial differential equation And its application in physics. Even for physics itself, such as heat and electromagnetism Action at a distance and shock wave Theory has also made important contributions.

Riemann's work directly affected the development of mathematics in the second half of the 19th century. Many outstanding mathematicians re demonstrated Riemann's asserted theorems. Under the influence of Riemann's thought, many branches of mathematics made brilliant achievements.

Riemann first proposed to use Complex function theory In particular, the new idea and new method of studying number theory with ζ function created Analytic number theory And has a profound impact on the development of the theory of functions of single complex variable. He is the world History of mathematics Riemann, one of the most original mathematicians in Shanghai, has not many works, but he is extremely profound and rich in the creation and imagination of concepts.

His name appears in Riemann ζ function 、 Riemann integral 、 Riemann lemma 、 Riemannian manifold 、 Riemannian space Riemann mapping theorem Riemann Hilbert problem 、 Cauchy Riemann equation Riemann thought loop matrix.

work

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Riemann's works mainly include: Foundation of General Theory of Functions of Single and Complex Variables, About Assumptions Based on Geometry, Possibility of Representing Functions with Trigonometric Series, Differential Equations of Mathematical Physics (co authored with Weber), Elliptic Function Theory, Gravity, Electricity and Magnetism, Number of Prime Numbers Not Exceeding Known Numbers, etc.

DeDekin published the complete works of Riemann in 1876. Riemann's students collected their lecture notes, which were published in 1902 as a supplement to the complete collection.^[5]

Riemann conjecture

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Riemann conjecture

One of the difficult problems left by Riemann is the famous Riemann conjecture , Yes Hilbert (Hilbert) proposed in 1900 tricosa- The eighth of the seven problems in the millennium has been listed as one of the seven major problems in the millennium. It requires that the nontrivial zero points of Riemann zeta function ζ (s) are located on the line Re (s)=1/2 of the complex plane. Mathematicians call this line the critical line. Using this term, Riemann conjecture can be expressed as: all nontrivial zeros of Riemann ζ (s) function are located on the critical line.^[6]

In September 2018, Michael Atiyah The statement proves that Riemann conjecture will be Heidelberg Winners Forum Publicity. On September 24, Michael Atia posted his certificate Riemann hypothesis (conjecture).^[3-4]

Character evaluation

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Sir Eddington said: "A geometer like Riemann can almost foresee the more important features of the real world."

Gaussian He said, "Riemann... has a creative, active, real mathematician's mind and brilliant and rich creativity."

Modern mathematics Bell, a historian, said: "As a mathematician, Riemann's greatness lies in that he gave pure mathematics and applied mathematics Powerful in revealing methods and new ideas universality And unlimited scope. "

German mathematician Klein Riemann said: "Riemann has extraordinary intuitive ability, and his genius for understanding is better than all mathematicians of the same generation."^[7]

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