... that this work (on conformal mappings of multiply connected domains) is one of the best he ever examined ...However, in his letter to Kovalevskaya, Weierstrass goes on to make a number of interesting comments about Schottky's personality and about his time as a research student in Berlin (see ):-
The author is of a clumsy appearance, unprepossessing, a dreamer, but, if I'm not completely wrong, he possesses an important mathematical talent. On Christmas Eve he was suddenly arrested and led away to the barracks to serve a 3-year term as a common soldier, for 'he had forgotten' to register in time for a 1-year term as a volunteer (as every student does). Fortunately, he proved to be so useless as a soldier that he was discharged as unsuitable after 6 weeks. Thus, he could return to his dissertation. He then signed up for the examination without presenting the requisite certificates and without knowing anything about the necessary formalities. As rector I had to cancel his name from the register because neither had he attended lectures nor were his whereabouts in Berlin known.Certainly the quality of Schottky's thesis was outstanding. Reinhard Bölling writes :-
In 1875, Schottky discovered a type of function which was later intensively studied, in full generality, by Poincaré and Klein (automorphic functions). He was the first to study systematically conformal mappings of multiply connected domains. In the case of a domain bounded by p closed curves (p ≥ 2), Schottky discovered that there exist 3p - 3 real constants that characterize the conformal class to which this region belongs. These are now called the moduli of the class.After obtaining his doctorate, Schottky remained at the University of Berlin for a while but submitted his habilitation thesis to the University of Breslau in 1878. His school friend Eberhard Gothein also habilitated at the University of Breslau in the same year having undertaken research in the history of economics. Schottky taught as a docent at Breslau until 1882 when he was appointed professor of mathematics in the Eidgenössische Technische Hochschule Zürich. During his four years lecturing at Breslau, Schottky published two papers: Abriss einer Theorie der Abel'schen Functionen von drei Variabeln Ⓣ (1880) and Über eindeutige Functionen mit linearen Transformationen in sich Ⓣ (1882). He held the chair in Zürich for ten years before moving to another chair at the University of Marburg in 1892. Keeping up his move every ten years he went to a chair at the University of Berlin in 1902 but remained there for twenty years until he retired in 1922. It was Lazarus Fuchs's chair at Berlin to which Schottky was appointed and many felt that his personal friendship with Frobenius was a factor in his appointment. Certainly Schottky had made highly significant mathematical contributions (which we will comment on below) but it was clear to most people that his best years as a mathematician were behind him by 1902. However, he still produced significant results although his teaching was considered to be below par :-
Schottky's teaching qualities were not impressive ("As a teacher he is unsuitable", Kummer once said). Perhaps, for this reason, he almost never taught beginners' classes in Berlin.When Schottky retired from his Berlin chair it allowed Issai Schur, who had been an extraordinary professor since 1916, to become a full professor.
Most of Schottky's work concerns elliptic, abelian and theta functions. We have already mentioned above his doctoral thesis which made an important contribution to conformal mappings of multiply connected plane domains. This was the origin of the mapping of a domain bounded by three disjoint circles which provides an example of an automorphic function with a Cantor set boundary. Schottky's thesis also discusses conformal mappings of domains bounded by circular and conic arcs. This thesis was published in 1877 as a fifty-page paper. Schottky's paper Über eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unverändert bleibt Ⓣ (1887), advanced the theory of Poincaré series considerably. The Schottky problem is the problem of finding characterizations of Jacobians among all principally polarized abelian varieties. He studied this problem, already posed by Bernhard Riemann, in his 1888 paper Zur Theorie der Abelschen Funktionen von vier Variabeln Ⓣ, showing that principally-polarized Abelian varieties (of dimension g) do not coincide with the Jacobian varieties (of an algebraic curve of genus g) for g = 4. The varieties do coincide for g ≤ 3. Schottky's Theorem (1904) is related to Picard's Theorem and has become a classical result in the theory of functions of a complex variable.
Schottky published 55 papers and, in 1880, a book Abriss einer Theorie der Abel'schen Functionen von drei Variablen Ⓣ. In  Hans Freudenthal writes:-
His work is difficult to read. Although he was a student of Weierstrass, his approach to function theory was Riemannian in spirit, combined with Weierstrassian rigour.Among Schottky's doctoral students we should mention Heinrich Jung who was his student at Marburg. Schottky and Jung jointly authored the important 1909 paper Neue Sätze über Symmetralfunktionen und die Abel'schen Funktionen der Riemann'schen Theorie which made a significant contribution to the Schottky problem. They also co-authored Neue Sätze über Symmetralfunktionen und die Abelschen Funktionen der Riemannschen Theorie (1912). Perhaps it is worth noting that these two papers with Jung were the only two joint papers which Schottky published. He was an examiner of Paul Koebe's doctoral thesis at Berlin in 1905, was an advisor for Konrad Knopp's 1907 Berlin thesis and supervised the doctoral work of several other students at Berlin.
Finally we should mention Schottky's family. He married Henriette Hammer (1858-1947) who was the daughter of Heinrich Hammer (1823-1860) from Waldenburg (now Walbrzych) in Silesia, a District Judge from Breslau, and Eveline von Meichsner (1828-1907) from Jauer (now Jawor) in Silesia. Henriette and Friedrich Schottky had one daughter and four sons, including the physicist Walter Schottky who was born on 23 July 1886 in Zürich, Switzerland. Walter Schottky was a doctoral student of Max Planck in Berlin in 1912 at the time when his father was professor there. He made important contributions to the theory of the electron as well as inventing the screen-grid vacuum tube and the pentode. Walter Schottky died on 4 March 1976. Another of Schottky's sons, Hermann Schottky (1885-1974) was a metallurgist, and Ernst Schottky (1888-1915) was a botanist who died in World War I.
Schottky was elected a corresponding member of the Prussian Academy of Sciences (now the Berlin Academy) in 1900 and was made a full member two years later. He died and was buried in the Steglitz area of Berlin after a funeral attended only by close family members.
Article by: J J O'Connor and E F Robertson
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