(Christian) Felix Klein - Life, Work, Bibliography

Mathematician, born in Düsseldorf, W Germany. He studied at the University of Bonn, and held chairs at Erlangen (1872–5), Munich (1875–80), Leipzig (1880–6), and Göttingen (1886–1913). He worked on geometry, including non-Euclidean geometry, function theory (in which he developed Bernhard Riemann's ideas), and elliptic modular and automorphic functions. His Erlanger Programm showed how different geometries could be classified in terms of group theory. He also wrote on the history of mathematics, encouraged links between pure and applied mathematics and engineering, and promoted general mathematical education.

Felix Klein
Felix Christian Klein
Born	April 25, 1849 Düsseldorf, Germany
Died	June 22, 1925 Göttingen, Germany

Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

Life

Klein's parents were Prussian; At that time, Julius Plücker held Bonn's chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868,

Julius Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Clebsch, who had moved to Göttingen in 1868. Klein visited Clebsch the following year, along with visits to Berlin and Paris. Klein was in Paris when (July 1870) Bismarck published a message intended to provoke France into declaring war on Prussia. Klein promptly left Paris when France did so.

Erlangen appointed Klein professor in 1872, when he was only 23. Klein did not build a school at Erlangen where there were few students, and so he was pleased to be offered a chair at Munich's Technische Hochschule in 1875.

In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Hegel.

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life.

His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886.

The research center Klein established at Göttingen served as a model for the best such centers throughout the world. In 1895, Klein hired Hilbert away from Königsberg;

Under Klein's editorship, Mathematische Annalen became one of the very best mathematics journals in the world. Founded by Clebsch, only under Klein's management did it first rival then surpass Crelle's journal based out of the University of Berlin. Klein set up a small team of editors who met regularly, making democratic decisions.

Thanks in part to Klein's efforts, Göttingen began admitting women in 1893. she was an English student of Arthur Cayley's, whom Klein admired.

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress.

The London Mathematical Society awarded Klein its De Morgan Medal in 1893.

Work

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

Klein's first important mathematical discoveries were made in 1870. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan.

Klein devised the bottle named after him, a one-sided closed surface which cannot be constructed in Euclidean space. It is possible, however, to construct a Klein bottle in non-Euclidean space.

In the 1890s, Klein turned to mathematical physics, a subject from which he had never strayed far, writing on the gyroscope with Arnold Sommerfeld.

Erlangen Program

In 1871, while at Göttingen, Klein made major discoveries in geometry. This had the remarkable corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, putting Euclidian and non-Euclidian geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry. Cayley never accepted Klein's argument, believing it to be circular.

Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlangen Program (1872), profoundly influenced the evolution of mathematics. This program was set out in Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. 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 significance of Klein's contributions to geometry are less than evident, but not because those contributions are now seen as strange or wrong.

Function theory

Klein saw his work on function theory as his major contribution to mathematics, specifically his work on:

The link between certain ideas of Riemann's and invariant theory, Number theory and abstract algebra;

Klein showed that that the modular group moves the fundamental region of the complex plane so as to tessellate that plane.

Klein considered equations of degree >

In his 1884 book on the icosahedron, Klein set out a theory of automorphic functions, connecting algebra and geometry. Klein succeeded in formulating such a theorem and in sketching a strategy for proving it.

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

Bibliography

Primary:

1887.