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I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting:

Look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.

Poincare makes this observation when comparing two different types of mathematical minds and Klein, according to him, is of the intuitive type.

I do not know much about Riemann surfaces but I am very curious what problem Klein was working on. How could he solve an abstract problem using conductivity and battery? The book gives no reference to this paragraph.

Thanks!

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Mathematics has been known to be related to physics... –  Qiaochu Yuan Dec 20 '12 at 20:41
    
@QiaochuYuan Yes. I just want to see how conductivity can be related to Riemann surfaces. –  Hui Yu Dec 21 '12 at 1:43

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The reference is to Klein's On Riemann's Theory of Algebraic Functions and Their Integrals. From my review of this book: Where a complex function is analytic it satisfies the Cauchy-Riemann equations, which implies that its real and imaginary part satisfy the Laplace equation, which means that the function corresponds to a steady fluid flow. Restricting ourselves to algebraic functions and their integrals, we can understand the corresponding flows by studying the infinities of these functions, and we can imagine electromagnetic set-ups that would produce such flows. "Now it seems possible, ab initio, to reverse the whole order of this discussion; to study the [flows] in the first place and thence to work out the theory of certain analytical functions" (p. 22). So we study this type of flows on general spheres with handles, and then show that this corresponds precisely to the algebraic functions and their integrals on their Riemann surfaces. In this way, without any of the usual analytic machinery, we reveal the qualitative insights of Riemann's theory. In particular, "among the numerous accidental properties of the functions, we distinguish certain essential ones" (p. 56), namely the infinities, the Riemann surface, and the topological types of the cross-cuts.

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This is fantastic! How did you find that? –  Hui Yu Dec 21 '12 at 15:03

I haven't studied the passage of Klein that Poincare is referring to, but here is my (vaguely informed) guess as to what Poincare is describing:

A basic problem in function theory on Riemann surfaces is to construct meromorphic functions with prescribed poles. Using the well-known relationship between holomorphic functions and harmonic functions, this can equally well be described in terms of harmonic functions with certain singularities.

Now if you have a distribution of point charges in a vacuum, the electric potential will be a harmonic function with singularities at the points where the charges are. Klein's situation is slightly different, namely he is considering a distribution of charges in a metallic plate, when a current is applied at certain points, but the physics, and the PDE it gives rise to, will be similar to the case of electric potential in the presence of point charges.

By choosing the points where the current is applied in an appropriate way, he can show that the singularities of the charge distribution match the singularities of the harmonic function he is trying to construct, and in this way he "proves" the existence of the desired harmonic function. (I put proves in quotes because this argument from physical intuition is not accepted as a rigorous proof; to give a rigorous proof, one must use tools of analysis to show that the relevant version of Laplace's equation actually has a solution of the desired type.)

This kind of argument, which exhibits harmonic functions with certain properties via arguments from physics, goes back (at least) to Riemann's first work on Riemann surfaces.

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Thanks for sharing your insight! I learnt more than I asked for. –  Hui Yu Dec 21 '12 at 15:04

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