Alexander Nevsky str., 14, Kaliningrad, Russia, 236041
Gordan was student of Carl Jacobi at the University of Königsberg (now Immanuel Kant Baltic Federal University).
plac Uniwersytecki 1, 50-137 Wrocław, Poland
Gordan obtained his Ph.D. at the University of Breslau (now University of Wrocław) in 1862.
Gordan's early interest in mathematics was encouraged by the private tutoring he received from N. H. Schellbach, a professor at the Friedrich Wilhelm Gymnasium. He was a student of Carl Jacobi at the University of Königsberg (now Immanuel Kant Baltic Federal University) before obtaining his Ph.D. at the University of Breslau (now University of Wrocław) in 1862.
Gordan’s interest in function theory led him to visit F. B. Riemann in Göttingen in 1862, but Riemann was ailing and their association was brief. The following year, Gordan was invited to Giessen by A. Clebsch, with whom he worked on the theory of Abelian functions. Together they wrote an exposition of the theory. In 1874 Gordan became a professor at Erlangen, where he remained until his retirement in 1910.
In 1868 Clebsch introduced Gordan to the theory of invariants, which originated in observation of George Boole’s in 1841 and was further developed by Arthur Cayley in 1846. Following the work of these two Englishmen, a German branch of the theory was developed by S. H. Aronhold and Clebsch, the latter elaborating the former’s symbolic methods of characterizing algebraic forms and their invariants. Invariant theory was Gordan’s main interest for the rest of his mathematical career; he became known as the greatest expert in the field, developing many techniques for representing and generating forms and their invariants. Correcting an error made by Cayley in 1856, Gordan in 1868 proved by constructive methods that the invariants of systems of binary forms possess a finite base. Known as the Gordan finite basis theorem, this instigated a twenty-year search for proof in case of higher-order systems of forms. Making use of the Aronhold-Clebsch symbolic calculus and other elaborate computational techniques, Gordan spent much of his time seeking a general proof of finiteness. The solution to the problem came in 1888, when David Hilbert proved the existence of finite bases for the invariants of systems of forms of arbitrary order. Hilbert’s proof, however, provided no method for actually finding the basis in a given case. Although Gordan was said to have objected to Hilbert’s existential procedures, in 1892 he wrote a paper simplifying them. His version of Hilbert’s theorem is the one presented in many textbooks.
The second major area of Gordan’s contributions to mathematics is in solutions of algebraic equations and their associated groups of substitutions. Working jointly with Felix Klein in 1874-1875 on the relationship of icosahedral groups to fifth-degree equations, Gordan went on to consider seventh-degree equations with the group of order 168; and toward the end of his career, equations of the sixth degree with the group of order 360. His work was algebraic and computational and utilized the techniques of invariant theory. Typical of Gordan’s many contributions to these subjects are papers in 1882 and 1885 in which, following Klein’s exposition of the general problem, he carries out the explicit reduction of the seventh-degree equation to the setting of the substitution group of order 168.
The overall style of Gordan’s mathematical work was algorithmic. He shied away from presenting his ideas in informal literary forms. He derived his results computationally, working directly toward the desired goal without offering explanations of the concepts that motivated his work.
Gordan was a member of the German Academy of Sciences Leopoldina and the Göttingen Academy of Sciences.
Gordan married Sophie Deuer, the daughter of a Giessen professor of Roman law, in 1869.