ICM'98
Weierstrass's school of analysis and its influence on Italian
Mathematics
By Umberto Bottazzini
Berlin emerged as the leading centre of mathematical analysis in the
early 1860s when Weierstrass, who had joined Kronecker and Kummer in
1856, began to teach the theory of analytic functions at the
university and founded the mathematical seminar together with
Kummer. In autumn of 1864 the Italian mathematician Casorati visited
Berlin to discuss with Weierstrass, Kronecker, Kummer and their pupils
the most recent progress in mathematics. The notes taken by Casorati
of the talks with the Berlin mathematicians provide a vivid picture of
the questions which were then at the forefront of mathematical
research. In particular, they discussed such topics as continuity,
differentiability, analytic continuation, natural boundaries, and
Riemann's use of the Dirichlet principle. The latter became the main
subject of the correspondence between Casorati and Schwarz, the
student of Weierstrass who mostly contributed to the diffusion of his
teacher's methods in Italy. Indeed, Schwarz was the true trait d'union
between the Berlin mathematicians and their Italian colleagues. In the
early 1870s he had an intensive correspondence with Casaroti and Dini
in particular. In his letters to them Schwarz presented the methods
and the main results as expounded by Weierstrass in his lectures and
seminars. Dini's lectures at the university of Pisa, inspired by the
new methods of Weierstrass (and Cantor), were at the basis of his
celebrated treatise Fondamenti per la teorica delle funzioni di
variabili reali (1878) where Dini acknowledged his debt to
Schwarz. From the mid-1860s on, Weierstrass lectured regularly on the
theory of analytic functions, the theory of elliptic and Abelian
functions and the calculus of variations. This set of lectures, which
Weierstrass repeated and refined for nearly 30 years, was never
published during his lifetime. The first presentation in Italy of
Weierstrass's theory of analytic functions was given in 1880 by
Salvatore Pincherle. A student of Dini and Casorati, Pincherle spent
one year in Berlin before being appointed in 1880 to a chair of
analysis at the University of Bologna. By then Weierstrass's theory of
analytic functions was beginning to be taught in Italy.
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Last modified: June 19, 1998