Felix Klein

Felix Christian Klein (25 April 1849, Düsseldorf - 22 June 1925, Göttingen) was a German mathematician who showed that metric geometries, Euclidean or non-Euclidean, constitute special cases of projective geometry. In 1871 he presented a remarkable classification of geometry, the "Erlangen program", which ended the split between pure geometry and analytic geometry. In this classification, the concept of group plays a fundamental role, since the object of each geometry becomes the study of the group of transformations that characterizes it.^{[citation required]}

Like Bernhard Riemann, Klein considered the theory of functions of complex variables as a geometric theory and transferred the concept directly to physics. His study of modular functions remains essential for researchers.^{[citation needed]}

Professor at the University of Göttingen (1886), he was the founder of the Great Encyclopedia of Mathematics (1895) and one of the advocates and architects of the renewal of mathematics teaching in secondary studies. Klein was also an important organizer of scientific groups and team teaching activities. He is considered one of the main contributors to Göttingen becoming an important center for the development of mathematics in Europe. ^{[citation needed ]}

The famous Klein bottle, surface with a single face, bears his name.^{[citation needed]}

Biographical data of academic career

Klein studied in Bonn, being a disciple of Rudolf Lipschitz and Julius Plücker, who later became his assistant. After Plücker's death, Alfred Clebsch took over the editorial work of his unfinished work and partly delegated this work to the talented Klein. Klein obtained a doctorate from him in 1868 under Lipschitz's tutelage with a subject of geometry applied to mechanics.

Sepulture in Gotinga Cemetery

In 1869 he was at the University of Berlin and attended Leopold Kronecker's lecture there on quadratic forms. He participated in the seminars of Ernst Kummer and Karl Weierstrass, where he also met Sophus Lie, with whom he befriended and was in Paris in 1870 on a study tour. Due to the Franco-German war he returned to Germany. He obtained the degree and habilitation as a professor in 1871 with Clebsch in Göttingen and remained there between 1871 and 1872 as a private teacher.

Due to Clebsch's efforts, he obtained an appointment as a professor in Erlangen in 1872. His career trajectory took him in 1875 to the Technical University of Munich. In that same year he married Anna Hegel, a granddaughter of Georg Wilhelm Friedrich Hegel.

In the year 1880 Klein received a request to go to Leipzig as a professor of geometry. His most scientifically productive creative stage took place in this Leipzig period. Thus, he maintained correspondence with Henri Poincaré and simultaneously dedicated himself to the organization of teaching. This double workload eventually led to a bodily collapse. In 1886 he accepted an appointment to Göttingen, where he remained until his death. Here he dedicated himself mainly and intensively to the tasks of scientific organization, while David Hilbert, who had been called to Göttingen thanks to his management in 1895, continued to expand Göttingen's fame as one of the world centers of mathematics of that time. so. In 1914 he was awarded the Ackermann Teubner Prize. From 1908 he represented the University of Göttingen in the Prussian House of Parliament. In 1924 Klein was made an honorary member of the DMV, serving as its president in 1897, 1903 and 1908. He was buried in the City Cemetery on Kasseler Landstraße street in Göttingen.

Scheme of a bottle of Klein

Work

Klein's thesis, on linear geometry and its applications to mechanics, classified linear complexes of the second degree using Weierstrass's theory of elementary divisors.

Klein's first important mathematical discoveries were made during the 1870s. In collaboration with Sophus Lie, he discovered the fundamental properties of asymptotic lines on the Kummer surface. Later they investigated W-curves, invariant curves under a group of projective transformations. It was Lie who introduced Klein to the concept of the group, which would play a pivotal role in his later work. Klein also learned about groups from Camille Jordan.

A Klein bottle blown by hand.

Klein devised the "Klein bottle" named after him, a one-sided closed surface that cannot be embedded in three-dimensional Euclidean space, but can plunge like a rolled-up cylinder to join its other end from the "inside";. It can be immersed in Euclidean space of dimensions 4 and higher. The Klein Bottle concept was devised as a three-dimensional Möbius strip, one construction method being the joining of the edges of two Möbius strips.

During the 1890s, Klein began to study mathematical physics more intensively, writing about the gyroscope with Arnold Sommerfeld. During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften. This company, which lasted until 1935, provided an important standard reference of enduring value.

Erlangen Program

In 1871, while in Göttingen, Klein made important discoveries in geometry. He published two papers On so-called non-Euclidean geometry in which he showed that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley-Klein metric. This idea had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was consistent, giving Euclidean and non-Euclidean geometries equal status, and putting an end to all controversy over non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, considering it circular.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given set of transformations, known as the Erlangen program (1872), profoundly influenced the evolution of mathematics. This program began with Klein's inaugural lecture as a professor at Erlangen, although it was not the actual speech he delivered on that occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus, the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.

Today, the importance of Klein's contributions to geometry is evident. They have become so ingrained in mathematical thought that it is difficult to appreciate the novelty of it when they were first introduced, and to understand the fact that they were not immediately accepted by all of his contemporaries.

Complex analysis

Klein considered his work on complex analysis to be his greatest contribution to mathematics, specifically his work on:

• The link between certain ideas of Riemann and the theory of invariants,
• Theory of numbers and abstract algebra;
• Group theory;
• Geometry in more than 3 dimensions and differential equations, especially equations invented by it, satisfied by elliptical modular function and automotive function.

Klein showed that the modular group moves the fundamental region of the complex plane to tile the plane. In 1879, he examined the action of PSL (2,7), considered as an image of the modular group, and obtained an explicit representation of a Riemann surface now called the Klein quartic. He showed that it was a complex curve in projective space, that its equation was x³y + y ³z + z³x = 0, and that its group of symmetry was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats complex analysis geometrically, connecting potential theory and [ [conformal mapping] ] s. This work was based on notions of fluid dynamics.

Klein considered equations of degree > 4 and was especially interested in using transcendental methods to solve the general quadratic equation. Drawing on the methods of Charles Hermite and Leopold Kronecker, he produced similar results to Brioschi's and then completely solved the problem by means of the icosahedral group. This work allowed him to write a series of articles on elliptic modular function.

In his 1884 book on the icosahedron, Klein established a theory of automorphic functions, associating algebra and geometry. Poincaré had published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men. Both attempted to state and prove a grand uniformization theorem that would establish the new theory more completely. Klein managed to formulate this theorem and describe a strategy to prove it. He came up with the demonstration for himself during an asthma attack at 2:30 a.m. on March 23, 1882.

Klein summarized his work on automorphic and elliptic modular functions in a four-volume treatise, written with Robert Fricke over the course of some 20 years.

Eponymy

• The asteroid (12045) Klein bears this name in his memory.