Abstract
This chapter attempts to give a brief summary of the life and work of Riemann. It tries to address the following issues: the influence of his education and early life on his work, and a summary of his major works and an overview of his work, and the general impact of his work through the concepts and terminology named after him.
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Notes
- 1.
As it is known, the theory of elliptic integrals motivated the theory of Abelian integrals. One basic insight in the theory of elliptic integrals is to consider the inverse of an elliptic integral. The general framework for Abelian integrals on a fixed compact Riemann surface is the map via integration of holomorphic 1-forms from the group of divisors of degree 0 to the Jacobian variety of the Riemann surface. To invert this map, we need to identify its image. One version of the Jacobi inversion problem, as presented in most modern textbooks on Riemann surfaces, asks to prove that this map is surjective. The full version asks, in addition, for a precise description of the inverse images of this map.
- 2.
One explanation was given by Klein [8, p. 167]: “The outward life of Riemann may perhaps appeal to your sympathy; but it was too uneventful to arouse particular interest. Riemann was one of those retiring men of learning who allow their profound thoughts to mature slowly in the seclusion of their study.”
- 3.
It might be interesting to compare this with Gauss’ evaluation on Dedekind’s thesis: “The paper submitted by Mr. Dedekind deals with problems in calculus which are by no means commonplace. The author not only shows very good knowledge in this field but also an independence which indicates favorable promise for his future achievements. As paper for admission to the examination this text is fully sufficient”.
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Ji, L. (2017). Bernhard Riemann and His Work. In: Ji, L., Papadopoulos, A., Yamada, S. (eds) From Riemann to Differential Geometry and Relativity. Springer, Cham. https://doi.org/10.1007/978-3-319-60039-0_20
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