Explore the groundbreaking mathematical concepts of Bernhard Riemann in this 32-minute lecture on the complex sphere. Delve into Riemann's geometrical thinking, beginning with an abstract discussion on the nature of mathematical space and its relationship to arithmetical systems. Examine the Riemann sphere, both in its planar version and as it relates to complex numbers. Learn about stereographic projection, the number circle, and complex functions from a geometric perspective. Gain insights into Riemann's contributions to higher-dimensional spaces, complex analysis, and the famous Riemann Conjecture, while considering the ongoing debates surrounding these ideas in modern mathematics.