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Alfred Clebsch entered the school of mathematics at the University of Königsberg in 1850. In this school, founded by Jacobi, he was influenced by Jacobi through his teachers Hesse and Friedrich Richelot who were both students of Jacobi. In fact although he never met Jacobi, who died one year after Clebsch entered the University of Königsberg, Jacobi was to influence him both through these two teachers and also directly through the fact that Clebsch was to collaborate in the production of the Collected Works of Jacobi. At Königsberg Clebsch was taught mathematical physics by Franz Neumann.
After graduating in 1854 Clebsch went to Berlin where he taught at various schools. His first academic appointment was in 1858 when he was appointed to the University of Berlin. He left after a short spell and, still in 1858, he took up an appointment at the Technische Hochschule in Karlsruhe.
Clebsch had submitted a doctoral dissertation to Königsberg on hydrodynamics. At the start of his research career his topics were mainly concerned with hydrodynamics and elasticity. He remained in Karlsruhe until 1863 but before he left Karlsruhe the direction of his research had changed. The end of his work on applied mathematical topics is perhaps most clearly defined by the publication of Theorie der Elastizität fester Körper in 1862 which was a major work on elasticity.
Pure mathematics became Clebsch's main research topic when he began to study the calculus of variations and partial differential equations. Clebsch moved to Giessen in 1863 and there he collaborated with Paul Gordan. Their joint work culminated in a major work on abelian functions Theorie der Abelschen Funktionen in 1866. Clebsch helped build a school of algebraic geometry and invariant theory at Giessen which included Gordan, Brill, Max Noether, Lindemann and Lueroth.
Hesse had advised Clebsch to investigate the algebraic geometry of Cayley, Sylvester and Salmon and he was particularly attracted to the contributions that Aronhold had made to their theories. Clebsch went back to Abel's approach to algebraic geometry and considered an algebraic approach rather than Riemann who used a geometric approach. His interpretation of the works of Cayley, Sylvester and Salmon in this way led Clebsch to a brilliant new interpretation of Riemann's function theory.
Clebsch's work on algebraic geometry Uber die Anwendung der Abelschen Functionen in der Geometrie is described in [3] as the:
birth cry of modern algebraic geometry.In 1868 Clebsch was appointed to Göttingen. There with Carl Neumann, the son of his former teacher at Königsberg, he cofounded Mathematische Annalen , a mathematics journal of major importance.
Sadly Clebsch's brilliant career came to a sudden end in 1872 when he died of diphtheria. Max Noether and Brill, who were among his students at Giessen, continued his work on curves. Two volumes of his lectures on geometry were published after his death in 1876 and 1891. A second edition of part of one of these volumes, with Clebsch as joint author, was published in three parts in 1906, 1910 and 1932.
W Burau, writing in [1], makes the following comments about Clebsch's work:
... Clebsch described the plane representations of various rational surfaces, especially that of the general cubic surface. Clebsch must also be credited with the first birational invariant of an algebraic surface, the geometric genus that he introduced as the maximal number of double integrals of the first kind existing on it.
References (4 books/articles)
References elsewhere in this archive:
Tell me about Clebsch's work on abstract linear spaces