Bernhard Riemann
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Bernhard Riemann | |
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Scientific career | |
Fields | |
Institutions | University of Göttingen |
Thesis | Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe (1851) |
Doctoral advisor | Carl Friedrich Gauss |
Other academic advisors | |
Notable students | Gustav Roch Eduard Selling |
Signature | |
Georg Friedrich Bernhard Riemann (German:
Biography
Early years
Riemann was born on 17 September 1826 in
Education
During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years). 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. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances.
During the spring of 1846, his father, after gathering enough money, sent Riemann to the
Academia
Riemann held his first lectures in 1854, which founded the field of
In 1862 he married Elise Koch; their daughter Ida Schilling was born on 22 December 1862.[9]
Protestant family and death in Italy
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.[10] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore
Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord's Prayer with his wife and died before they finished saying the prayer.[11] Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost.[10]
Riemann's tombstone in
Georg Friedrich Bernhard Riemann
Professor in Göttingen
born in Breselenz, 17 September 1826
died in Selasca, 20 July 1866
Riemannian geometry
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Geometers |
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.
In 1853,
The subject founded by this work is
The Riemann metric is a collection of numbers at every point in space (i.e., a tensor) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a manifold, no matter how distorted it is.
Complex analysis
In his dissertation, he established a geometric foundation for
His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous
Other highlights include his work on abelian functions and
Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface.
According to Detlef Laugwitz,[16] automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces.
Real analysis
In the field of
In his habilitation work on Fourier series, where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n.
Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory.
He also worked with
Number theory
Riemann 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 investigated the zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers. The Riemann hypothesis was one of a series of conjectures he made about the function's properties.
In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for .
Riemann knew of Pafnuty Chebyshev's work on the Prime Number Theorem. He had visited Dirichlet in 1852.
Writings
Riemann's works include:
- 1851 – Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inauguraldissertation, Göttingen, 1851.
- 1857 – Theorie der Abelschen Functionen, Journal für die reine und angewandte Mathematik, Bd. 54. S. 101–155.
- 1859 – Über die Anzahl der Primzahlen unter einer gegebenen Größe, in: Monatsberichte der Preußischen Akademie der Wissenschaften. Berlin, November 1859, S. 671ff. With Riemann's conjecture. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile of the manuscript Archived 2016-03-03 at the Wayback Machine with Clay Mathematics.
- 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted to the Paris Academy for a prize competition
- 1867 – Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
- 1868 – Über die Hypothesen, welche der Geometrie zugrunde liegen. Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation EMIS, pdf On the hypotheses which lie at the foundation of geometry, translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also 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.
- 1876 – Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind, Leipzig, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Max Noether and Wilhelm Wirtinger, Teubner 1902). Later editions The collected Works of Bernhard Riemann: The Complete German Texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017
- 1876 – Schwere, Elektrizität und Magnetismus, Hannover: Karl Hattendorff.
- 1882 – Vorlesungen über Partielle Differentialgleichungen 3. Auflage. Braunschweig 1882.
- 1901 – Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen. PDF on Wikimedia Commons. On archive.org: Riemann, Bernhard (1901). Weber, Heinrich Martin (ed.). "Die partiellen differential-gleichungen der mathematischen physik nach Riemann's Vorlesungen". archive.org. Friedrich Vieweg und Sohn. Retrieved 1 June 2022.
- 2004 – Riemann, Bernhard (2004), Collected papers, Kendrick Press, Heber City, UT, MR 2121437
See also
- List of things named after Bernhard Riemann
- Non-Euclidean geometry
- On the Number of Primes Less Than a Given Magnitude, Riemann's 1859 paper introducing the complex zeta function
References
- ISBN 978-3-411-04067-4.
- ISBN 978-3-11-018202-6.
- ^ Wendorf, Marcia (2020-09-23). "Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity". interestingengineering.com. Retrieved 2023-10-14.
- ^ Ji, Papadopoulos & Yamada 2017, p. 614
- ^ Mccleary, John. Geometry from a Differentiable Viewpoint. Cambridge University Press. p. 282.
- ISBN 978-0-7624-1922-7.
- ^ a b Wendorf, Marcia (2020-09-23). "Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity". interestingengineering.com. Retrieved 2023-04-06.
- ^ Werke, p. 268, edition of 1876, cited in Pierpont, Non-Euclidean Geometry, A Retrospect
- ^ "Ida Schilling". 22 December 1862.
- ^ ISBN 978-0-06-621070-4.
- ^ "Christian Mathematician – Riemann". 24 April 2012. Retrieved 13 October 2014.
- ^ "Riemann's Tomb". 18 September 2009. Retrieved 13 October 2014.
- ^ Riemann, Bernhard: Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868), S. 133-150.
- ^ On the Hypotheses which lie at the Bases of Geometry. Bernhard Riemann. Translated by William Kingdon Clifford [Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.]
- Vorlesungen über theoretische Physik“, Bd.2 (Mechanik deformierbarer Medien), Harri Deutsch, S.124. Sommerfeld heard the story from Aachener Professor of Experimental Physics Adolf Wüllner.
- ISBN 978-3-7643-5189-2
Further reading
- ISBN 0-309-08549-7.
- Monastyrsky, Michael (1999), Riemann, Topology and Physics, Boston, MA: Birkhäuser, ISBN 0-8176-3789-3.
- Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds. (2017). From Riemann to Differential Geometry and Relativity. Springer. ISBN 9783319600390.
External links
- Bernhard Riemann at the Mathematics Genealogy Project
- The Mathematical Papers of Georg Friedrich Bernhard Riemann
- Riemann's publications at emis.de
- O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics Archive, University of St Andrews
- Bernhard Riemann – one of the most important mathematicians
- Bernhard Riemann's inaugural lecture
- ScienceWorld.
- Richard Dedekind (1892), Transcripted by D. R. Wilkins, Riemanns biography.