The Weierstrass ``analytical configurations'': Alternatives to the Riemann ``surface'' and forerunners of complex spaces (Q1566141)

A remarkable feature of the history of complex-variable analysis is that during the second half of the 19th century two radically different general theories developed: Weierstrass's, rooted in analytic methods often as similar to those formulated in real-variable analysis; and the `geometric fantasies' (K. Weierstrass) of Riemann, with prominent places given to the weird and wonderful surfaces and their cuts and connectivities. This paper deals with this contrast, by explaining Weierstrass's development of the notion of analytic configuration, his fantasy-free way of handling multi-valuedness of functions by taking them as containing points specified by the pairs of values \((x, y)\) which satisfy some given algebraic equation \(f(x, y) = 0\). He seems to have introduced it directly in reaction to Riemann's manner of handling this by means of the surfaces, as publicised in a paper on 1857. Later Weierstrass also applied his theory to the theory of several complex variables. In 1918 Hermann Weyl formulated both surfaces and configurations in such a way as to claim their (logical) equivalence. The author bases much of his account on several of Weierstrass's lecture courses that were published either long ago in his collected work or more recently, and also a manuscript set of notes of a lecture course taken by Weierstrass's fan Gustav Mittag-Leffler.