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See searches and references containing ERLANGEN PROGRAM!ERLANGEN PROGRAM
Research program on the symmetries of geometry
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix
Erlangen_program
Public research university in Bavaria, Germany
of Erlangen–Nuremberg (German: Friedrich-Alexander-Universität Erlangen-Nürnberg, FAU) is a public research university in the cities of Erlangen and
University of Erlangen–Nuremberg
University_of_Erlangen–Nuremberg
Study of angle-preserving transformations
transformation geometry soon appreciates the significance of Felix Klein's Erlangen program, an outgrowth of certain models of hyperbolic geometry. The combination
Inversive_geometry
Conjectures connecting number theory and geometry
required in the proof of some essential conjectures in the Langlands program. Erlangen program Geometric Langlands correspondence Jacquet–Langlands correspondence
Langlands_program
German mathematician (1849–1925)
and the associations between geometry and group theory. His 1872 Erlangen program classified geometries by their basic symmetry groups and was an influential
Felix_Klein
Branch of mathematics that studies abstract algebraic structures
via invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program. There are many approaches
Representation_theory
City in Bavaria, Germany
Erlangen (German pronunciation: [ˈɛʁlaŋən] ) is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt
Erlangen
Mathematical model of the physical space
coordinates (analytic geometry), the use of transformation groups (Erlangen program), or the use of calculus (differential geometry). The Elements is mainly
Euclidean_geometry
Type of geometry
is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive
Klein_geometry
Geometry without using coordinates
geometries may be created by discarding or modifying them. Following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection
Synthetic_geometry
Fundamental space of geometry
Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries
Euclidean_space
Topological space in group theory
1) or Galilean and Carrollian spaces. From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry
Homogeneous_space
Branch of mathematics
Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly
Differential_geometry
Geometric transformation that preserves lines but not angles nor the origin
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. Snapper, Ernst; Troyer, Robert
Affine_transformation
Property that is not changed by mathematical transformations
invariant: ICount % 3 == 1 || ICount % 3 == 2 } Curvature invariant Erlangen program Graph invariant Invariant differential operator Invariant estimator
Invariant_(mathematics)
historical and recent, include Felix Klein's Erlangen program, Hilbert's problems, Langlands program, and the Millennium Prize Problems. In the Mathematics
Future_of_mathematics
Geometry
approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries. The
Projective differential geometry
Projective_differential_geometry
Mathematics of smooth surfaces
in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and
Differential geometry of surfaces
Differential_geometry_of_surfaces
Italian mathematician (1863–1924)
cradle of some of the most interesting studies on such issues." The Erlangen program of Felix Klein appealed early on to Segre, and he became a promulgator
Corrado_Segre
Branch of mathematics
between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the
Geometry
Belgian scientist and Catholic priest (1894–1966)
theory of quaternions from first principles, in the spirit of the Erlangen program. Lemaître also worked on the three-body problem, introducing a new
Georges_Lemaître
Inclusion of one mathematical structure in another, preserving properties of interest
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9.. Spivak, Michael (1999)
Embedding
Mathematical space with a notion of closeness
homeomorphic or not." The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation
Topological_space
Branch of mathematics that studies the properties of groups
projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry
Group_theory
Process of extracting the underlying essence of a mathematical concept
geometry, affine geometry and finite geometry. Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining
Abstraction_(mathematics)
Fiber bundle whose fibers are group torsors
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. page 37 Lawson, H. Blaine;
Principal_bundle
Polish–American mathematician (1901–1983)
Silva, anticipated Tarski in applying the Erlangen Program to logic.[citation needed] The Erlangen program classified the various types of geometry (Euclidean
Alfred_Tarski
Mathematical condition
(1997). Differential geometry : Cartan's generalization of Klein's Erlangen program. New York: Springer. ISBN 0-387-94732-9. OCLC 34356972. Conlon 2001
Poincaré_lemma
Subset of a manifold that is a manifold itself; an injective immersion into a manifold
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. Warner, Frank W. (1983). Foundations
Submanifold
Isometric automorphisms of a hyperbolic space
Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed
Hyperbolic_motion
French mathematician, physicist and engineer (1854–1912)
representations. The subject is clearly defined by Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation
Henri_Poincaré
Bijection of a set using properties of shapes in space
Lorentz transformations in special relativity. Coordinate transformation Erlangen program Symmetry (geometry) Motion Reflection Rigid transformation Rotation
Geometric_transformation
Norwegian mathematician (1842–1899)
That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program. In 1872, Lie spent eight months together with Peter
Sophus_Lie
Branch of mathematics
general concepts of cyclic groups and abelian groups. Klein's 1872 Erlangen program studied geometry and led to symmetry groups such as the Euclidean group
Abstract_algebra
Generalization of affine connections
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. Slovák, Jan (1997)
Cartan_connection
Group of flat spacetime symmetries
quantum mechanics (see Wigner's classification). In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group:
Poincaré_group
Fiber bundle induced by a map of its base space
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Graduate Texts in Mathematics. Vol. 166. New York: Springer-Verlag
Pullback_bundle
Study of computable functions and Turing degrees
on the natural numbers (this suggestion draws on the ideas of the Erlangen program in geometry). The idea is that a computable bijection merely renames
Computability_theory
Concept in differential geometry
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, ISBN 978-0-387-94732-7, MR 1453120 Schwachhöfer,
Holonomy
Algebraic object with geometric applications
(2000). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer. p. 194. ISBN 978-0-387-94732-7. Schouten, Jan Arnoldus (1954)
Tensor
Scientific principles enabling the use of the calculus of variations
in the 1926 discovery of Schrodinger's equation. Felix Klein's 1872 Erlangen program attempted to identify invariants under a group of transformations.
Variational_principle
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
Klein saw screw theory as an application of elliptic geometry and his Erlangen program. He also worked out elliptic geometry, and a fresh view of Euclidean
Screw_theory
Set with associative invertible operation
systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry
Group_(mathematics)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
projective geometry and related geometries in the sense of Felix Klein's Erlangen program. It emerged in the course of classification of simple Lie groups that
Simple_Lie_group
Heinz Kunert: Defogger for automobiles. Felix Klein: Invented the Erlangen Program, classifying geometries by their underlying symmetry groups, was a
List of German inventors and discoverers
List_of_German_inventors_and_discoverers
History of a branch of mathematics
the guise of symmetry groups, was initiated by Felix Klein's 1872 Erlangen program. The study of what are now called Lie groups started systematically
History_of_group_theory
German polymath, linguist and mathematician (1809–1877)
developing higher-dimensional geometry. Meanwhile, Klein was advancing his Erlangen program, which also expanded the scope of geometry. Comprehension of Grassmann
Hermann_Grassmann
Soviet mathematician (1921–1988)
symmetry in the 19th century". In his chapter on "Felix Klein and his Erlangen Program", Yaglom says that "finding a general description of all geometric
Isaak_Yaglom
Overview of and topical guide to geometry
non-Euclidean geometry History of topology History of algebraic geometry Erlangen program Noncommutative geometry Topology Mathematics and fiber arts Van Hiele
Outline_of_geometry
Measure of the curvature of a pseudo-Riemannian manifold
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. Singer, I.M.; Thorpe
Weyl_tensor
German-born American mathematician (1924–2021)
"Among today's treatises, the best one from the point of view of the Erlangen Program is Differential Geometry by H. Guggenheimer, Dover Publications, 1977
Heinrich_Guggenheimer
Mathematical concept
(1996). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, Berlin. ISBN 0-387-94732-9. Shlomo Sternberg (1964)
Maurer–Cartan_form
Euclidean geometry without distance and angles
in his Der barycentrische Calcul (chapter 3). After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry
Affine_geometry
Group that is also a differentiable manifold with group operations that are smooth
modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate
Lie_group
Geometry without the parallel postulate
Euclidean geometry. However the converse is not true. Affine geometry Erlangen program Foundations of geometry Incidence geometry Non-Euclidean geometry Faber
Absolute_geometry
Geometric theorem
restricted form. It influenced Felix Klein in the development of the Erlangen program. Since its original conception, it was generalized by many mathematicians
Hesse's_principle_of_transfer
2020 book by Matilde Marcolli
coordinate geometry, and topology, fractals, tessellations, and the Erlangen program of understanding geometries through their symmetries. Two more chapters
Lumen_Naturae
Rational function of the form (az + b)/(cz + d)
the Möbius group, is a geometric structure (in the sense of Klein's Erlangen program) called Möbius geometry. An isomorphism of the Möbius group with the
Möbius_transformation
2002 book on fractal geometry
the connections between group theory, symmetry and geometry - see Erlangen program. The contents of Indra's Pearls are as follows: Chapter 1. The language
Indra's_Pearls_(book)
Type of geometry
like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants
Projective_geometry
German journalist and television presenter
Erlangen) is a German journalist and television presenter of the ZDF program heute. Barbara Hahlweg is the daughter of the former mayor of Erlangen,
Barbara_Hahlweg
surface Sharpe, R.W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9.
Development (differential geometry)
Development_(differential_geometry)
German nuclear physicist (1903–2002)
1903 – 3 February 2002) was a German experimental nuclear physicist from Erlangen, Bavaria. He worked for Walther Bothe at the Physics Institute of Heidelberg
Rudolf_Fleischmann
Constructs a fiber bundle from a base space, fiber and a set of transition functions
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. Steenrod, Norman (1951). The
Fiber bundle construction theorem
Fiber_bundle_construction_theorem
Function in mathematics
techniques of Pfaffian systems to the geometries of Felix Klein's Erlangen program. In these investigations, he found that a certain infinitesimal notion
Connection_(mathematics)
German mathematician (1899–1934)
Wilhelm Blaschke (Thomsen's doctoral advisor) to apply Felix Klein's Erlangen Program on differential geometry. He also edited and organized Blaschke's lectures
Gerhard_Thomsen
Form of geometry without distances
relation on lines. Absolute geometry Affine geometry Cyclic order Erlangen program Euclidean geometry Hilbert's axioms Tarski's axioms Incidence geometry
Ordered_geometry
Type of manifold in differential geometry
all symplectomorphisms is a symplectic invariant. In the spirit of Erlangen program, symplectic geometry is the study of symplectic invariants. Let { v
Symplectic_manifold
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. Dillen, F.J.E.; Verstraelen
Web_(differential_geometry)
Mathematical idealization of the surface of a body
in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and
Surface_(mathematics)
Italian philosopher
L. Magnani and R. Dossena (eds.) (2004), Felix Klein: the Erlangen Program, Pristem/Storia, Springer, Milan (in Italian). Book series editor SAPERE
Lorenzo_Magnani
1920s books on mathematical history by Felix Klein
Felix Klein (1849–1925) was a German mathematician best known for his Erlangen program, which emphasised the use of groups in geometry. From 1886 to 1913
Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert
Vorlesungen_über_die_Entwicklung_der_Mathematik_im_19._Jahrhundert
Fiber bundle
ISBN 978-0-387-94087-8. Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9.
Associated_bundle
German mathematician (1843 to 1905)
developments. Rowe indicates that Klein was most interested in developing his Erlangen program. In 1875 Schlegel countered with the second part of his System der
Victor_Schlegel
Generalization of an ordered basis of a vector space
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94732-7. Spivak
Moving_frame
Geometrical property
reflections such as circle reflection on the plane. In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects
Symmetry_(geometry)
Worldwide competitive programming contest for university students
meets FAU". icpc.informatik.uni-erlangen.de. Archived from the original on 2016-09-14. Retrieved 2016-07-01. "Programming Environment". Archived from the
International Collegiate Programming Contest
International_Collegiate_Programming_Contest
(1996). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, Berlin. ISBN 0-387-94732-9. Shlomo Sternberg (1964)
Darboux_derivative
Study of angle-preserving transformations of a geometric space
Conformal geometric algebra Conformal gravity Conformal Killing equation Erlangen program Möbius plane Paul Ginsparg (1989), Applied Conformal Field Theory.
Conformal_geometry
Transformation of a geometric space preserving structure
their "groups of motions". He proposed using symmetry groups in his Erlangen program, a suggestion that was widely adopted. He noted that every Euclidean
Motion_(geometry)
Construct allowing differentiation of tangent vector fields of manifolds
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. This fills in some
Affine_connection
Mathematical metric in geometry
In Ji, L.; Papadopoulos, A. (eds.). Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. pp. 91–136. arXiv:1406.7309
Cayley–Klein_metric
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. Section 6.1 discusses
Ricci_decomposition
Set of lines described by homogeneous polynomial equations
\omega } , then it is a symplectic transformation. In the spirit of Erlangen program, symplectic geometry studies invariants of symplectic transformations
Line_complex
Series of mathematics textbooks
ISBN 978-0-387-94655-9) Differential Geometry — Cartan's Generalization of Klein's Erlangen Program, R. W. Sharpe (1997, ISBN 978-0-387-94732-7) Field and Galois Theory
Graduate_Texts_in_Mathematics
Concept in mathematical group theory
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. Peter Scherk (1960)
Conformal_group
(1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, p. 146, ISBN 0-387-94732-9 Katok, Anatole;
Linear_flow_on_the_torus
Austrian mathematician (1911–1999)
geometrical studies of kinematics, in particular in relation to the Erlangen program and the works of Wilhelm Blaschke. He also authored some textbooks
Hans_Robert_Müller
cubic form. The name affine differential geometry reflects Klein's Erlangen program, in which geometries are studied through the invariants of transformation
Affine_differential_geometry
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. Bishop, Richard L.; Goldberg
Affine_manifold
German mathematician and philosopher (1915–1994)
October 1, 1994) was a German philosopher and mathematician, founder of the Erlangen School (with Wilhelm Kamlah) and inventor of game semantics (with Kuno
Paul_Lorenzen
Type of transport in differential geometry
(1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9. Ü. Lumiste (2001) [1994]
Projective_connection
Maths textbook
elliptic geometry, spherical geometry, and (in line with Felix Klein's Erlangen program) the transformation groups of these geometries as subgroups of Möbious
Geometry_of_Complex_Numbers
Mathematics timeline
concept is developed, using local formulae. 1872 Felix Klein Klein's Erlangen program puts an emphasis on the homogeneous spaces for the classical groups
Timeline_of_manifolds
irrationale Zahlen, a theory of irrational numbers Felix Klein produces the Erlangen program on geometries February 15 – George Huntington makes the first detailed
1872_in_science
Fighter aircraft family developed from 1958
PHANTOMs Part 3 - The MDD RF-4E Phantom II in German Air Force Service. Erlangen, Germany: AirDOC Aircraft Documentations. pp. 3–7. ISBN 3-935687-08-7.
McDonnell Douglas F-4 Phantom II
McDonnell_Douglas_F-4_Phantom_II
City in Bavaria, Germany
forms a continuous conurbation with the neighbouring cities of Fürth, Erlangen and Schwabach, with the built-up area comprising around 1.4 million inhabitants
Nuremberg
German nuclear physicist
latter years of World War II, he became a professor at the University of Erlangen–Nuremberg. He declined to take a position offered to him in the United
Helmut_Volz
German artist and university professor (born 1972)
Susanne Kriemann (born 1972 in Erlangen) is a German artist, photographer and professor at Karlsruhe University of Arts and Design. Susanne Kriemann graduated
Susanne_Kriemann
Mathematical transformation
updated and edited by Blaschke in 1926.) Kunle H.; Fladt K. (1926). "Erlangen program and higher geometry – Laguerre geometry". In Heinrich Behnke (ed.)
Spherical_wave_transformation
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