Bernhard Riemann's legacy of 1859

摘要

The German version of Riemann's Collected Works is confined to asingle volume of 690 pages~*. Even so, this volume has had an abiding andprofound impact on modern mathematics and physics, as we shall see. Infifteen years of activity, from 1851, when he gained his doctorate at theUniversity of Gottingen, to his death in 1866, two months short of hisfortieth birthday, Riemann contributed to almost all areas of mathematics.He perceived mathematics from the analytic point of view and used analysisto illuminate subjects as diverse as number theory and geometry. Althoughregarded principally as a mathematician Riemann had an abiding interest inphysics and researched significantly in the methods of mathematicalphysics, particularly in the area of partial differential equations. There wasa healthy interplay between physics and mathematics in his thinkingthroughout his creative life. As for Riemann's style, it was characterised bythe conceptual rather than the algorithmic; mathematical objects wereregarded as vehicles of conceptual properties rather than as given primarilyas formulas. Indeed, formulas were used sparingly in his papers.