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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. –  Zhen Lin Sep 15 '11 at 10:04
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Simul-posted to MathOverflow, mathoverflow.net/questions/75504/… –  Gerry Myerson Sep 15 '11 at 12:47
    
@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... –  UGPhysics Sep 16 '11 at 20:22
    
... (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 ... –  UGPhysics Sep 16 '11 at 20:35
    
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.] –  UGPhysics Sep 16 '11 at 20:37
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The theory of Riemann surfaces developed from the theory of elliptic functions. Some milestones:

  1. Cauchy's theory of contour integrals in the complex plane.

  2. Abel's work on elliptic functions; the Abel part of the Abel-Jacobi theorem

  3. Jacobi's inversion theorem

  4. Riemann's bilinear relations

  5. The Weierstrass theory of elliptic functions

  6. Picard's work in complex analysis, which was precursor to the notion of the Picard group/variety.

Starting with Riemann's work is also theory of moduli spaces. The development of topology in the background, with the uniformization theorem in particular was a prominent achievement. Homology and Cohomology theories in topology enabled the existing way of presenting the theory of Riemann surfaces. An important later development is the de Rham theorem, which was later phrased using cohomological ideas.

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What is the existing way of presenting the theory of Riemann surfaces? Thanks. –  Andrey Rekalo Sep 15 '11 at 12:06
    
What I meant was that initially things like the Abel-Jacobi theorem was phrased using elliptic integrals. Nowadays, for example, one embeds the integral homology as a lattice in the dual of the space of differential forms, and take the quotient, etc.. –  George Sep 15 '11 at 12:10
    
Right. IMO for those who are interested in quick applications of the Abel-Jacobi theorem and are not prepared/motivated to study homology and cohomology theories, a classical exposition of the theory might work just fine. –  Andrey Rekalo Sep 15 '11 at 12:30
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You have to understand that the notion of function as it is used nowadays is quite recent. During a long time, analysts were perfectly happy to work with so-called multiform functions. For example, $\log : \mathbb C \setminus \{0\} \to \mathbb C$ was a perfectly fine function to work with, even if you had to be careful : if the argument of this “function” makes a turn around 0, the image of $\log$ changes (something like $2i\pi$ is added).

One of the great things Riemann did is to notice that this creates a true (“uniform”) function on something slightly more complicated. In modern language, there is a Riemann surface $X$, a function $p : X \to \mathbb C \setminus \{0\}$ (actually, $X = \mathbb C$ and $p=\exp$, but allow me to forget that) and a true function $f : X \to \mathbb C$ such that the various “values” of $\log(z)$ are just the $f(\tilde z)$ for $\tilde z \in p^{-1}\{z\}$. And the fact that the “value” of $\log$ changes when you make a turn around 0 is just the fact that your loop around 0 is the image of no loop in $X$ : when you try to lift that loop to $X$, you get a non closed path. The difference in value for $\log$ (the “monodromy”) is just the consequence of the topology of $X$. (Of course, that quickly leads to covering spaces, etc.) Somehow, the main change of focus is here: to understand functions, you have to understand Riemann surfaces.

Now, for what kind of functions do you want to play that game? If I remember correctly, Riemann quotes the example of the logarithm in his article, but he is mainly interested in (inverses of) algebraic functions, which give birth to compact Riemann surfaces and finite (branched) coverings over the Riemann sphere. A goal was to understand elliptic and Abelian integrals, functions that were becoming more and more important in physics, analysis and number theory...

A later source of examples was the theory of differential equations. We now tend to look at these from a real point of view (that is, with a real time $t \in \mathbb R$) but the complex point of view used to be more important. Differential equations, even if they are linear and of order two, also define “multiform functions” that you want to understand better. This was for example the main motivation of Poincaré, whose work (in that subject) culminated with the uniformisation theorem. (Note: ”uniformisation”, because you want uniform functions!)

Of course, everything I say is grossly simplified. For a start, every word here is an anachronism, because a lot of complex analysis, algebraic geometry, even surface topology was developed precisely during that period between 1851 (Riemann's first article) and 1907 (Poincaré and Koebe's first convincing proofs of the uniformisation theorem). But it is certainly true that a huge part of the second half of the mathematical nineteenth century revolved around such notions...

If you want to read more about that, McKean and Moll's Elliptic Curves is a great book to understand the relation between those innocent-looking elliptic integrals and that somehow fearsome theory of Riemann surfaces. And if you read French, there are two relevant historical/mathematical books: Dieudonné's Cours de géométrie algébrique (first volume) and Saint-Gervais's Uniformisation des surfaces de Riemann. Normally, a pdf version of the latter exists somewhere on the internet, but I'm afraid you'll need a good library to find the former. (How can such a wonderful book not to be reedited?)

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