Wilhelm Karl Joseph Killing | |
|---|---|
 | |
| Born | 10 May 1847 |
| Died | 11 February 1923 (aged 75) |
| Citizenship | German |
| Known for | Lie algebras, Lie groups, and non-Euclidean geometry |
| Spouse | Anna Commer |
| Awards | Lobachevsky Prize (1900) |
| Scientific career | |
| Fields | Mathematics |
| Doctoral advisor | Karl Weierstrass Ernst Kummer |
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. In 1875, he married Anna Commer, who was the daughter of a music lecturer. He became a professor at the seminary college Collegium Hosianum in Braunsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town council. As a professor and administrator, Killing was widely liked and respected. Finally, in 1892 he became a professor at the University of Münster. [1]
In 1886, Killing and his wife entered the Third Order of Franciscans. [1]
In 1878 Killing wrote on space forms in terms of non-Euclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885. [2] Recounting lectures of Weierstrass, he there introduced the hyperboloid model of hyperbolic geometry described by Weierstrass coordinates. [3] He is also credited with formulating transformations mathematically equivalent to Lorentz transformations in n dimensions in 1885. [4]
Killing invented Lie algebras independently of Sophus Lie around 1880. Killing's university library did not contain the Scandinavian journal in which Lie's article appeared. (Lie later was scornful of Killing, perhaps out of competitive spirit and claimed that all that was valid had already been proven by Lie and all that was invalid was added by Killing.) In fact Killing's work was less rigorous logically than Lie's, but Killing had much grander goals in terms of classification of groups, and made a number of unproven conjectures that turned out to be true. Because Killing's goals were so high, he was excessively modest about his own achievement.[ citation needed ]
From 1888 to 1890, Killing essentially classified the complex finite-dimensional simple Lie algebras, as a requisite step of classifying Lie groups, inventing the notions of a Cartan subalgebra and the Cartan matrix. He thus arrived at the conclusion that, basically, the only simple Lie algebras were those associated to the linear, orthogonal, and symplectic groups, apart from a small number of isolated exceptions. Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper. Killing also introduced the notion of a root system. He discovered the exceptional Lie algebra g2 in 1887; his root system classification showed up all the exceptional cases, but concrete constructions came later.
As A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born." [5]
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.
Christian Hugo Eduard Study was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.
Ludwig Otto Blumenthal was a German mathematician and professor at RWTH Aachen University.
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi, states that any finite-dimensional real{Change real Lie algebra to a Lie algebra over a field of characteristic 0} Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra.
Gerhard Hessenberg was a German mathematician who worked in projective geometry, differential geometry, and set theory.
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space.
In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982.
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.
Friedrich Heinrich Schur was a German mathematician who studied geometry.
In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Jürgen Neukirch showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Kôji Uchida strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Florian Pop extended the result to infinite fields that are finitely generated over prime fields.
Ernst Kötter (1859-1922) was a German mathematician.
Ioannis "John" N. Hazzidakis was a Greek mathematician, physicist, author, and professor. He is one of the most important mathematicians of the modern Greek scientific era. His professor was world renowned Greek mathematician Vassilios Lakon. He also studied with famous German mathematicians Ernst Kummer, Leopold Kronecker, Karl Weierstrass. He systematically worked in the field of research and education. He wrote textbooks in the field of algebra, geometry, and calculus. Hazzidakis essentially adopted some elements of Lacon's Geometry. He introduced the Hazzidakis transform in differential geometry. The Hazzidakis formula for the Hazzidakis transform can be applied in proving Hilbert's theorem on negative curvature, stating that hyperbolic geometry does not have a model in 3-dimensional Euclidean space.
Carl Friedrich Geiser was a Swiss mathematician, specializing in algebraic geometry. He is known for the Geiser involution and Geiser's minimal surface.
Friedrich Bachmann was a German mathematician who specialised in geometry and group theory.
Frank-Olaf Schreyer is a German mathematician, specializing in algebraic geometry and algorithmic algebraic geometry.
Marius Nicolae Crainic is a Romanian mathematician working in the Netherlands.
The Lotschnittaxiom is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann. It states:
Perpendiculars raised on each side of a right angle intersect.
Wolfgang Gaschütz was a German mathematician, known for his research in group theory, especially the theory of finite groups.
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