Friedrich Schottky

Friedrich Schottky
Name	Friedrich Schottky
Birth date	1851
Death date	1935
Nationality	German
Fields	Mathematics
Workplaces	University of Berlin; University of Zürich
Alma mater	University of Königsberg; University of Göttingen
Known for	Schottky problem; Schottky groups; work on theta functions; algebraic curves

Friedrich Schottky

Friedrich Schottky was a German mathematician noted for foundational work on complex analysis, algebraic curves, and the theory of functions of several complex variables. His research on theta functions, Riemann surfaces, and automorphic functions influenced contemporaries and later developments in algebraic geometry and mathematical physics. Schottky interacted with mathematicians across Germany, France, and Switzerland and held academic posts that connected him to the mathematical centers of Göttingen and Berlin.

Life and education

Schottky was born in Breslau (then part of the Kingdom of Prussia) and pursued higher education at the University of Königsberg and the University of Göttingen, where he studied under figures associated with the mathematical traditions of Carl Gustav Jacob Jacobi and Bernhard Riemann. During his formative years he encountered the work of Karl Weierstrass, Leopold Kronecker, and members of the Berlin Academy circle. Schottky completed his doctoral studies in a period when German mathematics was shaped by exchanges between the institutes at Königsberg, Göttingen, and the University of Berlin, and he later moved within the academic networks of Zurich and Munich.

Mathematical career and positions

Schottky held academic positions at universities including the University of Zürich and the University of Berlin, places that connected him with scholars from the École Normale Supérieure tradition and with research groups influenced by Felix Klein and Hermann Amandus Schwarz. His career overlapped with the generation of David Hilbert, Emmy Noether, and Hermann Weyl, and he contributed to seminars and correspondence that circulated among the mathematical societies of Germany and Switzerland. Schottky served as an examiner and advisor in doctoral defenses at institutions that included the Prussian Academy of Sciences and participated in meetings of the Deutsche Mathematiker-Vereinigung. Through these roles he influenced younger mathematicians associated with the schools of Göttingen and Zürich.

Contributions and the Schottky problem

Schottky is best known for posing and beginning to clarify the problem that bears his name, the Schottky problem, concerning the characterization of Jacobian varieties among principally polarized abelian varieties. His investigations used tools developed by Bernhard Riemann, Ferdinand Georg Frobenius, and Adolf Hurwitz and connected with the theory of theta functions introduced by Carl Gustav Jacob Jacobi and furthered by Friedrich Prym and Max Noether. Schottky introduced classes of discontinuous groups now called Schottky groups, which generalized earlier ideas of Poincaré on Kleinian and Fuchsian groups, and these constructions connected the study of uniformization of Riemann surfaces to the structure of moduli spaces analyzed later by Bernard Teichmüller and Alexander Grothendieck.

His work on theta characteristics and period matrices led to explicit conditions—Schottky relations—intended to single out period matrices of Jacobian varieties among those of arbitrary principally polarized abelian varieties, a question later developed by Igor Shafarevich, David Mumford, and Shiing-Shen Chern. Schottky's analytical techniques invoked the theory of abelian integrals as in Riemann and the function-theoretic methods of Weierstrass; his approach anticipated modern treatments using moduli of curves and singular cohomology pursued by Pierre Deligne and John Milnor.

Schottky also worked on the uniformization of algebraic curves, building on contributions by Henri Poincaré, Felix Klein, and Paul Koebe, and his constructions were later incorporated into geometric function theory and the study of Kleinian groups by Lipman Bers and Ahlfors. The Schottky groups he introduced remain central in modern connections among Teichmüller theory, low-dimensional topology studied by William Thurston, and string-theoretic investigations by Edward Witten and Michael Green.

Selected publications

- Papers on theta functions and period matrices, published in journals associated with the Prussian Academy of Sciences and proceedings of the Deutsche Mathematiker-Vereinigung, which presented the original Schottky relations. - Articles on discontinuous groups and uniformization, developing examples of Schottky groups and explicating their action on the Riemann sphere; these engaged with earlier expositions by Henri Poincaré and Felix Klein. - Monographs and lecture notes circulated in the mathematical centers of Göttingen and Zurich that treated abelian integrals in the tradition of Bernhard Riemann and Karl Weierstrass and that influenced expositions by Hermann Weyl and David Hilbert.

Legacy and influence on later research

Schottky's legacy is evident in several enduring research directions. The Schottky problem motivated a program of characterizing Jacobians that was advanced by David Mumford, Igor Shafarevich, and Enrico Arbarello; later work by Carel Faber and Eduard Looijenga connected Schottky's ideas with the geometry of moduli spaces of curves. The notion of Schottky groups influenced the development of Kleinian group theory pursued by Lipman Bers, Ahlfors, and Dennis Sullivan and informed the geometric perspectives used by William Thurston in three-manifold theory.

In algebraic geometry and mathematical physics, Schottky relations reappear in the study of integrable systems and in string perturbation theory, where researchers such as Edward Witten, Michael Green, and Nathan Seiberg have used period matrix characterizations in path-integral formulations. Histories of 19th- and 20th-century mathematics recount Schottky among the contributors who bridged the analytic traditions of Riemann and Weierstrass with the geometric moduli approaches later crystallized by Grothendieck and Deligne.

Category:German mathematicians Category:Complex analysts