B. Riemann and the complex sphere

Insights into Mathematics

100K subscribers

Bernhard Riemann was a major pioneer in several important areas of mathematics, and in particular he helped develop a theory of higher dimensional spaces and how to view them metrically, made important advances in complex analysis and what became known as Riemann surfaces, and of course he gave rise to the most important modern open problem in pure mathematics the Riemann Conjecture.

Here we give an introduction to his geometrical thinking with regard to the Riemann or complex sphere, starting with some abstract discussion of the "nature of mathematical space" and its relation to arithmetical systems. These ideas are still the subject of discussion and indeed debate in mathematical circles, especially as we move towards critiquing the modern notion of "manifold", which is largely built from ideas of Riemann, some of which he inherited from his teacher C. F. Gauss.