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• W2186533991 abstract "0.0. Riemann and Riemann surfaces. Some of Riemann’s ideas were about a century ahead of his time. Here is a short list of them related to the surfaces bearing Riemann’s name. • The definition of a Riemann surface transformed a vague concept of a multi-valued (analytic) function into the concept of a usual function defined on something spread over the complex line. During the XXth century this something was further transformed into the definition of a topological surface endowed with the sheaf of holomorphic functions. • The definition of genus was one of the first rigorously defined topological concepts, and it turned out to be closely related to algebraic geometry and complex analysis. • The seemingly technical problem of calculating the dimensions of spaces of meromorphic functions with a prescribed pole structure was completely solved by Riemann himself and his student Gustav Roch; it turned out to be the origin of one of the most powerful generalizations of the XXth century – the Riemann-Roch-Hirzebruch-Grothendieck. . . theorem. • The idea of classification of abstract Riemann surfaces up to isomorphism inspired Riemann to introduce one of the most mysterious objects of modern math, the moduli spaces Mg of Riemann surfaces of genus g, and to calculate their dimensions. The complete understanding of moduli spaces (e.g., their homologies ) is still out of the scope of our understanding. • Riemann’s existence theorem identified these newly introduced Riemann surfaces with the well-developed theory of algebraic curves. This list can be easily continued (abelian integrals, theta functions, . . . ). The theory of Riemann surfaces is one of several from Riemann’s legacy that has developed into a deep branch of mathematics and is quite active nowadays. It is, however, unique in one aspect: its objects are as visualizable as, say, the triangles in the elementary geometry (the zeros of the zeta function are not). The present paper, as its title suggests, is mainly devoted to the details of the visualization of Riemann surfaces. There is, however an important difference between the Riemann’s and the Grothendieck’s approach. Riemann thought about his surfaces in the context of continuous mathematics; therefore, the genuine Riemann surfaces are subject to continuous variation of parameters — usually complex numbers. As for the visualization tools introduced by Grothendieck (dessins d’enfants, see below), they are intended to store the complex structure on a surface in the finite amount of graphical information. Therefore, not all the Riemann surfaces are storable by dessins" @default.
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• W2186533991 date "2008-01-01" @default.
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• W2186533991 title "Visualizing algebraic curves: from Riemann to Grothendieck" @default.
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