Leipzig
Who did Felix Klein work for in Jan 1884?
Felix Klein Christian Felix Klein ( ; 25 April 1849 – 22 June 1925 ) was a German mathematician and mathematics educator , known for his work with group theory , complex analysis , non-Euclidean geometry , and on the associations between geometry and group theory . His 1872 Erlangen program , classifying geometries by their basic symmetry groups , was an influential synthesis of much of the mathematics of the time . Life . Felix Klein was born on 25 April 1849 in Düsseldorf , to Prussian parents . His father , Caspar Klein ( 1809–1889 ) , was a Prussian government officials secretary stationed in the Rhine Province . His mother was Sophie Elise Klein ( 1819–1890 , née Kayser ) . He attended the Gymnasium in Düsseldorf , then studied mathematics and physics at the University of Bonn , 1865–1866 , intending to become a physicist . At that time , Julius Plücker had Bonns professorship of mathematics and experimental physics , but by the time Klein became his assistant , in 1866 , Plückers interest was mainly geometry . Klein received his doctorate , supervised by Plücker , from the University of Bonn in 1868 . Plücker died in 1868 , leaving his book concerning the basis of line geometry incomplete . Klein was the obvious person to complete the second part of Plückers Neue Geometrie des Raumes , and thus became acquainted with Alfred Clebsch , who had relocated to Göttingen in 1868 . Klein visited Clebsch the next year , along with visits to Berlin and Paris . In July 1870 , at the beginning of the Franco-Prussian War , he was in Paris and had to leave the country . For a brief time he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen in early 1871 . Erlangen appointed Klein professor in 1872 , when he was only 23 years old . For this , he was endorsed by Clebsch , who regarded him as likely to become the best mathematician of his time . Klein did not wish to remain in Erlangen , where there were very few students , and was pleased to be offered a professorship at the Technische Hochschule München in 1875 . There he and Alexander von Brill taught advanced courses to many excellent students , including Adolf Hurwitz , Walther von Dyck , Karl Rohn , Carl Runge , Max Planck , Luigi Bianchi , and Gregorio Ricci-Curbastro . In 1875 Klein married Anne Hegel , granddaughter of the philosopher Georg Wilhelm Friedrich Hegel . After spending five years at the Technische Hochschule , Klein was appointed to a chair of geometry at Leipzig . There his colleagues included Walther von Dyck , Rohn , Eduard Study and Friedrich Engel . Kleins years at Leipzig , 1880 to 1886 , fundamentally changed his life . In 1882 , his health collapsed ; in 1883–1884 , he was afflicted with depression . Nevertheless , his research continued ; his seminal work on hyperelliptic sigma functions , published between 1886 and 1888 , dates from around this period . Klein accepted a professorship at the University of Göttingen in 1886 . From then on , until his 1913 retirement , he sought to re-establish Göttingen as the worlds prime center fo mathematics research . However , he never managed to transfer from Leipzig to Göttingen his own leading role as developer of geometry . He taught a variety of courses at Göttingen , mainly concerning the interface between mathematics and physics , in particular , mechanics and potential theory . The research facility Klein established at Göttingen served as model for the best such facilities throughout the world . He introduced weekly discussion meetings , and created a mathematical reading room and library . In 1895 , Klein recruited David Hilbert from the University of Königsberg . This appointment proved of great importance ; Hilbert continued to enhance Göttingens primacy in mathematics until his own retirement in 1932 . Under Kleins editorship , Mathematische Annalen became one of the best mathematical journals in the world . Founded by Clebsch , it grew under Kleins management , to rival , and eventually surpass Crelles Journal , based at the University of Berlin . Klein established a small team of editors who met regularly , making decisions in a democratic spirit . The journal first specialized in complex analysis , algebraic geometry , and invariant theory . It also provided an important outlet for real analysis and the new group theory . In 1893 , Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the Worlds Columbian Exposition . Due partly to Kleins efforts , Göttingen began admitting women in 1893 . He supervised the first Ph.D . thesis in mathematics written at Göttingen by a woman , by Grace Chisholm Young , an English student of Arthur Cayleys , whom Klein admired . In 1897 Klein became a foreign member of the Royal Netherlands Academy of Arts and Sciences . Around 1900 , Klein began to become interested in mathematical instruction in schools . In 1905 , he was instrumental in formulating a plan recommending that analytic geometry , the rudiments of differential and integral calculus , and the function concept be taught in secondary schools . This recommendation was gradually implemented in many countries around the world . In 1908 , Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians . Under his guidance , the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany . The London Mathematical Society awarded Klein its De Morgan Medal in 1893 . He was elected a member of the Royal Society in 1885 , and was awarded its Copley Medal in 1912 . He retired the following year due to ill health , but continued to teach mathematics at his home for several further years . Klein was one of ninety-three signatories of the Manifesto of the Ninety-Three , a document penned in support of the German invasion of Belgium in the early stages of World War I . He died in Göttingen in 1925 . Work . Kleins dissertation , on line geometry and its applications to mechanics , classified second degree line complexes using Weierstrasss theory of elementary divisors . Kleins first important mathematical discoveries were made during 1870 . In collaboration with Sophus Lie , he discovered the fundamental properties of the asymptotic lines on the Kummer surface . They later investigated W-curves , curves invariant under a group of projective transformations . It was Lie who introduced Klein to the concept of group , which was to have a major role in his later work . Klein also learned about groups from Camille Jordan . Klein devised the Klein bottle named after him , a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space , but it may be immersed as a cylinder looped back through itself to join with its other end from the inside . It may be embedded in the Euclidean space of dimensions 4 and higher . The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip , with one method of construction being the attachment of the edges of two Möbius strips . During the 1890s , Klein began studying mathematical physics more intensively , writing on 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 enterprise , which endured until 1935 , provided an important standard reference of enduring value . Erlangen program . In 1871 , while at Göttingen , Klein made major discoveries in geometry . He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric . This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was , giving the same status to geometries Euclidean and non-Euclidean , and ending all controversy about non-Euclidean geometry . Arthur Cayley never accepted Kleins argument , believing it to be circular . Kleins synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations , known as the Erlangen program ( 1872 ) , profoundly influenced the evolution of mathematics . This program was initiated by Kleins inaugural lecture as professor at Erlangen , although it was not the actual speech he gave on the 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 programs definition of geometry encompassed both Euclidean and non-Euclidean geometry . Currently , the significance of Kleins contributions to geometry is evident . They have become so much part of mathematical thinking that it is difficult to appreciate their novelty when first presented , and understand the fact that they were not immediately accepted by all his contemporaries . Complex analysis . Klein saw his work on complex analysis as his major contribution to mathematics , specifically his work on : - The link between certain ideas of Riemann and invariant theory , - Number theory and abstract algebra ; - Group theory ; - Geometry in more than 3 dimensions and differential equations , especially equations he invented , satisfied by elliptic modular functions and automorphic functions . Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate 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 termed the Klein quartic . He showed that it was a complex curve in projective space , that its equation was and that its group of symmetries was PSL ( 2,7 ) of order 168 . His Ueber Riemanns Theorie der algebraischen Funktionen und ihre Integrale ( 1882 ) treats complex analysis in a geometric way , connecting potential theory and conformal mappings . This work drew on notions from fluid dynamics . Klein considered equations of degree > 4 , and was especially interested in using transcendental methods to solve the general equation of the fifth degree . Building on methods of Charles Hermite and Leopold Kronecker , he produced similar results to those of Brioschi and later completely solved the problem by means of the icosahedral group . This work enabled him to write a series of papers on elliptic modular functions . 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 resulted in a friendly rivalry between the two men . Both sought to state and prove a grand uniformization theorem that would establish the new theory more completely . Klein succeeded in formulating such a theorem and in describing 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 . Selected works . - 1882 : Über Riemanns Theorie der Algebraischen Functionen und ihre Integrale - , also available from Cornell - 1884:Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade - English translation by G . G . Morrice ( 1888 ) Lectures on the Ikosahedron ; and the Solution of Equations of the Fifth Degree via Internet Archive - 1886 : Über hyperelliptische Sigmafunktionen Erster Aufsatz p . 323–356 , Mathematische Annalen Bd . 27 , - 1888 : Über hyperelliptische Sigmafunktionen Zweiter Aufsatz p . 357–387 , Math . Annalen , Bd . 32 , - 1894 : Über die hypergeometrische Funktion - 1894 : Über lineare Differentialgleichungen der 2 . Ordnung - 1897 : ( with Arnold Sommerfeld ) Theorie des Kreisels ( later volumes : 1898 , 1903 , 1910 ) - 1890 : ( with Robert Fricke ) Vorlesungen über die Theorie der elliptischen Modulfunktionen ( 2 volumes ) and 1892 ) - 1894 : Evanston Colloquium ( 1893 ) reported and published by Ziwet ( New York , 1894 ) - Zweiter Band . 1901 . - 1901 : - 1897 : Mathematical Theory of the Top ( Princeton address , New York ) - 1895 : Vorträge über ausgewählte Fragen der Elementargeometrie - 1897 : English translation by W . W . Beman and D . E . Smith Famous Problems of Elementary Geometry via Internet Archive - 1908 : Elementarmathematik vom höheren Standpunkte aus ( Leipzig ) - 1926 : Vorlesungen über die Entwicklung der Mathematik im 19 . Jahrhundert ( 2 Bände ) , Julius Springer Verlag , Berlin & 1927 . S . Felix Klein Vorlesungen über die Entwicklung der Mathematik im 19 . Jahrhundert - 1928 : Vorlesungen über nichteuklidische Geometrie , Grundlehren der mathematischen Wissenschaften , Springer Verlag - 1933 : Vorlesungen über die hypergeometrische Funktion , Grundlehren der mathematischen Wissenschaften , Springer Verlag Bibliography . - 1887 . The arithmetizing of mathematics in Ewald , William B. , ed. , 1996 . From Kant to Hilbert : A Source Book in the Foundations of Mathematics , 2 vols . Oxford Uni . Press : 965–71 . - 1921 . Felix Klein gesammelte mathematische Abhandlungen R . Fricke and A . Ostrowski ( eds. ) Berlin , Springer . 3 volumes . ( online copy at GDZ ) - 1890 . Nicht-Euklidische Geometrie