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- Riemann's theory of Abelian integrals was far ahead of its time in dealing with the fundamental properties of Riemann surfaces and in introducing theta functions. His viewpoint was to consider plane curves as one-dimensional complex manifolds, namely, Riemann surfaces, and it led to very important results in algebraic geometry as well. One of his basic discoveries was that a Riemann surface is uniquely determined by the set of all rational functions on it (called the field of rational functions). This was the birth of birational geometry.