Bernhard Riemann
Bernhard Riemann, 1863
Born	September 17, 1826 Breselenz, Kingdom of Hanover (modern-day Germany)
Died	July 20, 1866(1866-07-20) (aged 39) Selasca, Kingdom of Italy
Residence	Kingdom of Hanover
Nationality	German
Fields	Mathematics
Institutions	Georg-August University of Göttingen
Alma mater	Georg-August University of Göttingen Berlin University
Doctoral advisor	Carl Friedrich Gauss
Other academic advisors	Gotthold Eisenstein Moritz Abraham Stern
Notable students	Gustav Roch
Known for	See list
Influences	Johann Peter Gustav Lejeune Dirichlet

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

Biography[link]

Early years[link]

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of 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.

Education[link]

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. 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, after gathering enough money, sent Riemann to university at the renowned University of Göttingen, where he planned to study towards a degree in Theology. However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his parents' approval, Riemann transferred to the University of Berlin in 1847.^[1] During his time of study, Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

Academia[link]

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^[2]—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.

Austro-Prussian War[link]

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.^[3] 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.^[3]

Influence[link]

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.

Riemann was the inspiration for mathematician Charles Lutwidge Dodgson (better known by his pen name Lewis Carroll) to write Alice's Adventures in Wonderland and Through the Looking-Glass.^[4]

Euclidean geometry versus Riemannian geometry[link]

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 and delivered his lecture at Göttingen in 1854 entitled Über die Hypothesen welche der Geometrie zu Grunde liegen ("On the hypotheses which underlie geometry"). When it was finally published in 1868, two years after his death, the mathematical public received it with enthusiasm and it is now recognized as one of the most important works in geometry.

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[link]

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[link]

• 1868 “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).
• 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.
• Riemann, Bernhard (2004), Collected papers, Kendrick Press, Heber City, UT, ISBN 978-0-9740427-2-5, MR 2121437 

See also[link]

• List of topics named after Bernhard Riemann
• Riemann's 1859 paper introducing the complex zeta function

Notes[link]

Further reading[link]

• Derbyshire, John (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Washington, DC: John Henry Press, ISBN 0-309-08549-7 .
• Monastyrsky, Michael (1999), Riemann, Topology and Physics, Boston, MA: Birkhäuser, ISBN 0-8176-3789-3 .

External links[link]

Wikimedia Commons has media related to: Bernhard Riemann

Wikisource has the text of the 1905 New International Encyclopedia article Riemann, Georg Friedrich Bernhard.

• 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) from ScienceWorld.
• Bernhard Riemann at Find a Grave

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2008)

George Frederick of Brandenburg-Ansbach
George Frederick, Margrave of Brandenburg-Ansbach
Spouse(s)	Sophie of Brunswick-Lüneburg
Noble family	House of Hohenzollern
Father	George, Margrave of Brandenburg-Ansbach
Mother	Emilie of Saxony
Born	(1539-04-05)5 April 1539 Ansbach
Died	25 April 1603(1603-04-25) (aged 64) Ansbach

George Frederick of Brandenburg-Ansbach (German: Georg Friedrich der Ältere) (April 5, 1539, Ansbach – April 25, 1603) was Margrave of Ansbach and Bayreuth, as well as Regent of Prussia. He was the son of George, Margrave of Brandenburg-Ansbach and a member of the House of Hohenzollern. First (1559) he married Elisabeth of Brandenburg-Küstrin (29 August 1540 – 8 March 1578). He secondly married (1579) Sophie of Brunswick-Lüneburg (October 30, 1563 – 1639), daughter of William of Brunswick-Lüneburg and Dorothea of Denmark.

George Frederick reigned in his native Ansbach, Franconia and Jägerndorf, Upper Silesia since 1556 and, after the death of his cousin Albert Alcibiades in 1557, also in Kulmbach. He took over the administration of the Duchy of Prussia in 1577, when the then-reigning Duke Albert Frederick became ill.

He was the last of the older Franconia line of the House of Hohenzollern. Upon his death Ansbach and Kulmbach were inherited by younger princes of the Brandenburg line according to the House Treaty of Gera of 1598.

George Frederick rebuilt the palace and fortress Plassenburg, which was destroyed after the second margravian war (1552–1554), as one of the most impressive residences of the renaissance in the German empire. He also built the fortress Wülzburg and the old palace in Bayreuth.

During his reign between 1557 and 1603 in the Franconian territories of the Hohenzollern (Brandenburg-Ansbach and Brandenburg-Kulmbach) he kept peace, rebuilt cities and Castles, founded several schools and a University.

Ancestors[link]

This article relies on references to primary sources or sources affiliated with the subject, rather than references from independent authors and third-party publications. Please add citations from reliable sources. (September 2011)

Manuel De Landa, (born 1952 in Mexico City), is a writer, artist and philosopher who has lived in New York since 1975. He is presently the Gilles Deleuze Chair of Contemporary Philosophy and Science at the European Graduate School in Saas-Fee, Switzerland; a visiting professor at the University of Southern California School of Architecture in Los Angeles^{[citation needed]}; a lecturer at the University of Pennsylvania School of Design in Philadelphia, Pennsylvania; and an adjunct professor at the Pratt Institute School of Architecture in Brooklyn, New York.^[1] He was previously an adjunct professor in the Graduate School of Architecture, Planning and Preservation at Columbia University and a lecturer at Canisius College in Buffalo, New York. De Landa has a BFA from New York's School of Visual Arts.

He is the author of War in the Age of Intelligent Machines (1991), A Thousand Years of Nonlinear History (1997), Intensive Science and Virtual Philosophy (2002) and A New Philosophy of Society: Assemblage Theory and Social Complexity (2006). He has published many articles and essays and lectured extensively in Europe and in the United States. His work focuses on the theories of the French philosopher Gilles Deleuze on one hand, and modern science, self-organizing matter, artificial life and intelligence, economics, architecture, chaos theory, history of science, nonlinear dynamics, cellular automata on the other.

Alongside his intellectual work, De Landa made several short Super 8 and 16mm films in the 1970s and early 1980s. He pulled them from circulation after the original negatives were lost. In 2011, Anthology Film Archives restored and reissued them. Cited by filmmaker Nick Zedd in his Cinema of Transgression Manifesto, De Landa associated with many of the experimental and art filmmakers of this New York based movement. Much of De Landa's film work is inspired by his interest in philosophy and critical theory; one of his best known films, Raw Nerves, has been described as a 'Lacanian thriller' by at least one critic.

His latest book (as of March 2011) is Philosophy and Simulation: The Emergence of Synthetic Reason, which explores simulations of emergence in systems of different scales, from the atomic to the social.

Books[link]

• War in the Age of Intelligent Machines (1991)
• A Thousand Years of Nonlinear History (1997)
• Intensive Science and Virtual Philosophy (2002)
• A New Philosophy of Society: Assemblage Theory and Social Complexity (2006)
• Deleuze: History and Science (2010)
• Philosophy & Simulation: The Emergence of Synthetic Reason (2011)

See also[link]

• Gilles Deleuze
• Noosphere
• Realism
• Wargaming

External links[link]

• Manuel de Landa Faculty profile at European Graduate School EGS. Biography, bibliography, photos, quotes and video lectures.
• Manuel DeLanda Annotated Bibliography with links to articles
• Various texts
• http://www.canisius.edu/topos/delanda.asp
• Interview
• Interview on Ctheory
• Other interview by Paul Miller
• "Markets and antimarkets in the world economy" by DeLanda

Notes[link]

1. ^ See: Manuel de Landa Faculty profile at European Graduate School