Bernhard Riemann

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(Georg Friedrich) Bernhard Riemann

Library of Congress

[b. Hanover, Germany, September 17, 1826, d. Selasca, Italy, July 20, 1866]

Riemann's work made mathematics more general or abstract. He put the definite integral on a firmer basis with a generalization now called the Riemann integral. In 1851 Riemann invented a new way to show functions of complex numbers on a plane and developed fundamental ideas of topology to handle such representations. In 1854 Riemann addressed the foundations of geometry in spaces of n dimensions, suggesting a new form of non-Euclidean geometry that later became the geometry of Einstein's general theory of relativity. In 1859 Riemann made a conjecture about a complex function called the zeta function that remains among the main unresolved issues of mathematics.

The German mathematician Georg Friedrich Bernard Riemann (1826-1866) was one of the founders of algebraic geometry. His concept of geometric space cleared the way for the general theory of relativity.

On Sept. 17, 1826, Georg Riemann was born in Breselenz. Shortly afterward, the family moved to Quickborn, Holstein, where his father, a Lutheran minister, assumed the pastorate. Riemann senior quickly recognized his younger son's mathematical talent. When Georg was 10 years old, he was placed under a mathematics tutor who soon found himself outdistanced by his pupil.

Riemann had planned on a career in the Church in accordance with his father's wishes. In 1846 he entered the University of Göttingen as a student of theology and philology. But mathematics called, and he had probably already decided to change his mind, should his father consent. He may have strengthened his argument by a grand attempt to prove Genesis mathematically. The proof was hardly valid, but Riemann senior appreciated the effort and gave his blessing to the mathematical career. In 1847 Georg transferred to the University of Berlin, where such vigorous innovators as K. G. J. Jacobi, P. G. Lejeune-Dirichlet, J. Steiner, and F. G. M. Eisenstein had created a livelier atmosphere for learning. In 1849 Riemann returned to Göttingen to prepare for his doctoral examinations under Wilhelm Weber, the famous electrodynamicist.

Riemann Surfaces

Riemann's doctoral dissertation was, in Karl Friedrich Gauss's words, the product of a "gloriously fertile originality." Its novel ideas were further developed in three papers published in 1857. Here is a crude explanation of the principal novelty:

A complex number may be represented by a point in a plane. A function (single-valued) of a complex variable is a rule which pairs each point in one plane with a unique point in another plane. Imagine a fly wandering about the surface of a plate-glass window. As the fly moves from point to point, its shadow moves from point to point on the floor of the room. Each point which the fly occupies on the window determines a unique point that its shadow occupies on the floor.

Now suppose that the floor is a highly reflective surface. The incoming light strikes the floor and is reflected to the wall, and we see a second image of our wandering fly. Now each position of the fly on the window determines two shadows - one on the floor and one on the wall.

But that is not quite right. There are some positions in which the fly still casts only one shadow. These are the points which throw the shadow on the line of intersection between floor and wall. Let us call these points branch points.

Now suppose that we replace the plate-glass window with two parallel sheets of plate glass (like a double window for insulation against cold). We endow the sheets with the following magical properties: any object on the outside sheet will cast a shadow only on the floor, and any object on the inside sheet will cast a shadow only on the wall. Furthermore, we join the two sheets along the line of branch points, so that they now form a single surface. Our fly may crawl from one sheet to the other, but to each point that he occupies on the glass surface, there once again corresponds one unique location of his shadow.

This is what Riemann did for multiple-valued functions of a complex variable. His surfaces restore single-valuedness to functions and at the same time provide a method of representing these functions geometrically. Moreover, it turns out that the analytic properties of many functions are mirrored by the geometric (topological) properties of their associated Riemann surfaces.

His Göttingen Lecture

After successfully defending his dissertation, Riemann applied for an opening at the Göttingen Observatory but did not get the job. He next set his sights on becoming a privatdozent (unpaid lecturer) at the university. There were two hurdles to surmount before he could obtain the lecture-ship: a probationary essay and a trial lecture before the assembled faculty. The former, a paper on trigonometric series, included the definition of the "Riemann integral" in almost the form that it appears in current textbooks. The essay was submitted in 1853.

For his trial lecture Riemann submitted three possible titles, fully expecting Gauss to abide by tradition and assign one of the first two. But the third topic was one with which Gauss himself had struggled for many years. He was curious to hear what Riemann had to say "On the Hypotheses Which Lie at the Foundations of Geometry."

The lecture that Riemann delivered to the Göttingen faculty on June 10, 1854, is one of the great masterpieces of mathematical creation and exposition. Riemann wove together and generalized three crucial discoveries of the 19th century: the extension of Euclidean geometry to n dimensions; the logical consistency of geometries that are not Euclidean; and the intrinsic geometry of a surface, in terms of its metric and curvature in the neighborhood of a point. In his synthesis Riemann demonstrated the existence of an infinite number of different geometries, each of which could be characterized by its peculiar differential form. Finally, he pointed out that the choice of a particular geometry to represent the structure of real physical space was a matter for physics, not mathematics.

The impact of the lecture was enormous but delayed. Riemann worked out some of the analytical machinery in a memoir of 1861 on the conduction of heat, but the lecture itself was not published until 1868. Twenty years later a respected historian noted simply that the paper "had excited much interest and discussion." By 1908 the same historian was calling it a "celebrated memoir" which had attracted "general attention to the subject of non-Euclidean geometry."

"Riemann Hypothesis"

Riemann spent 3 years as a privatdozent. In 1857 he was appointed assistant professor, and 2 years later, when Dirichlet died, Riemann succeeded him in the chair of Gauss. After 1860 the honors, including international recognition, came thick and fast. He died on July 20, 1866, in Selasca, Italy.

Riemann's special genius was the penetrating vision that enabled him to see through a mass of obscuring detail and perceive the submerged foundations of a theory intuitively. This uncanny talent was most obvious in his geometric work, but the most remarkable instance occurs in analytic number theory. In an 1859 paper on prime numbers, Riemann proved several properties of what came to be called "Riemann's zeta function." Several other properties of the function he simply stated without proof. After his death a note was found, saying that he had deduced these properties "from the expression of it (the function) which, however, I did not succeed in simplifying enough to publish."

To this day no one has the slightest idea of what this "expression" might be. All but one of the properties have since been proved. The last one, now called the "Riemann hypothesis," still awaits its conqueror, despite the efforts of several generations of talented mathematicians.

Further Reading

The best biography of Riemann in English is in Eric T. Bell, Men of Mathematics (1937). He is discussed in the first volume of Ganesh Prasad, Some Great Mathematicians of the Nineteenth Century: Their Lives and Their Works (1933). The place of Riemannian geometry in relativity theory is discussed in Jagjit Singh, Great Ideas of Modern Mathematics: Their Nature and Use (1959). For a nontechnical introduction to non-Euclidean geometries see Richard Courant and Herbert Robbins, What Is Mathematics? (1941).

Additional Sources

Riemann, topology, and physics, Boston: Birkhauser, 1987.

"Riemann" redirects here. For other uses, see Riemann (disambiguation).

Bernhard Riemann
Bernhard Riemann, 1863
Born	September 17, 1826 Breselenz, Germany
Died	July 20, 1866 (aged 39) Selasca, Italy
Residence	Germany
Citizenship	German
Fields	Mathematician
Institutions	Georg-August University of Göttingen
Alma mater	Georg-August University of Göttingen Berlin University
Doctoral advisor	Carl Friedrich Gauss
Other academic advisors	Ferdinand Eisenstein Moritz Abraham Stern
Notable students	Gustav Roch
Known for	See list
Influences	Johann Peter Gustav Lejeune Dirichlet

Georg Friedrich Bernhard Riemann (help·info) (German pronunciation: [ˈʁiːman]; September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.

Contents 1 Biography 1.1 Early life 1.2 Middle life 1.3 Later life 2 Influence 3 Euclidean geometry versus Riemannian geometry 4 Higher dimensions 5 Writings in English 6 See also 7 Notes 8 References 9 External links

Biography

Early life

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is Germany today. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but suffered from timidity and a fear of speaking in public.

Middle life

During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. To this end, he even tried to prove mathematically the correctness of the Book of Genesis. His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with his family's finances.

During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares.

In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

Later life

Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality^{[citation needed]}—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.^[1] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.^[1]

Influence

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.

Riemann made major contributions to real analysis. He defined the Riemann integral by means of Riemann sums, developed a theory of trigonometric series that are not Fourier series—a first step in generalized function theory—and studied the Riemann–Liouville differintegral.

He made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he introduced the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions by consideration only of their singularities.

Euclidean geometry versus Riemannian geometry

In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture at Göttingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important works in geometry. It was titled Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.

The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.

Higher dimensions

Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.

Writings in English

• 1868.“On the hypotheses which lie at the foundation of geometry” in Ewald, William B., ed., 1996. “From Kant to Hilbert: A Source Book in the Foundations of Mathematics” , 2 vols. Oxford Uni. Press: 652–61.

See also

• List of topics named after Bernhard Riemann
• Riemann's 1859 paper introducing the complex zeta function
• Prime Obsession
• The Music of the Primes

Notes

1. ^ ^{a b} Marcus du Sautoy, The Music of the Primes, (HarperCollins 2003)

References

• John Derbyshire, "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics" (John Henry Press, 2003) ISBN 0-309-08549-7

External links

Wikimedia Commons has media related to: Bernhard Riemann

• Bernhard Riemann at the Mathematics Genealogy Project
• The Mathematical Papers of Georg Friedrich Bernhard Riemann
• All publications of Riemann can be found at: http://www.emis.de/classics/Riemann/
• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html .
• Bernhard Riemann – one of the most important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric W., Riemann, Bernhard (1826–1866) at ScienceWorld.

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