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It is written in Riemann Surfaces (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that:

"[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus"

Can someone please provide an articulated commentary on this statement.

Specifically, the statement suggests, [or seems to suggest], that Riemann surfaces were the logical / mathematical outcome of many years of careful development and refinement of traditional calculus. But: (i) what was / were the major milestones(s) in this road? and (ii) when the author uses the word 'culmination' what specifically is it the culmination of, and what problems / issues did the introduction of Riemann surfaces help to solve / clarify / etc.?

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    This is a bit anachronistic, but it should be pointed out that the theory of Riemann surfaces sits at the confluence of complex analysis, differential geometry, and algebraic geometry. The study of Riemann surfaces also motivated a few developments in modern mathematics, e.g. the concept of sheaves. So one could say it's significant in that sense.2011-09-15
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    Simul-posted to MathOverflow, http://mathoverflow.net/questions/75504/historical-basis-and-mathematical-significance-of-riemann-surfaces2011-09-15
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    @Zhen, please elaborate: "the theory of Riemann surfaces sits at the confluence of complex analysis, differential geometry, and algebraic geometry"... From what I understand- a) within Algebra, the problem of finding solutions to quadratic equations led to the introduction of the complex number $i$; progressively, this led to the field of "complex" analysis. .. Now, complex analytic ideas are central (if not per-se the -*only*- available set of ideas) to help prove the Fundamental Theorem of Algebra; b) the realization that there may exist geometries other than that of Euclid...2011-09-16
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    ... (i.e. the realization that the Parallel Postulate is not absolutely essential) - leading to non-Euclidean geometries of various sorts, and the use of the concept of "manifold" (which, - if I'm not mistaken again - came about via the notion of Riemann surface) as a fundamental object to defining and studying these various geometries; and c) the study of analysis - from real analysis to complex analysis to whatever the field of Analysis has developed to in its current stage - all has some link with the Riemann surface concept. --- I want to to get some idea of the mathematical threads ...2011-09-16
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    which link all these together [Algebra-Geometry-Analysis (and also Number Theory, but not sure what / how Riemann surfaces apply to here / are applied here.]2011-09-16
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    [Correction: "complex analytic ideas are central (if not per-se the -*only*- available set of ideas)" --> "complex analytic ideas are central (if not the -*only*- available set of ideas)"]2011-09-16
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    [Correction 2: "all has some link with the Riemann surface concept." --> "all have some link with the Riemann surface concept."]2011-09-16

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