Titlebook: Algebraic Curves and Riemann Surfaces for Undergraduates; The Theory of the Do Anil Nerode,Noam Greenberg Textbook 2022 The Editor(s) (if a

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Connectedness, Smooth and SimpleDefining the notion of a connected space, path-connected space, path homotopy, and simply connected spaces. The winding number of a path. A review of multivariable calculus, including the inverse function theorem. Path-connected manifolds are differentiably path-connected.

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Path IntegralsIntegration of generalised 1-forms. The length of a path. Vector fields; conservative and symmetric vector fields.

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Riemann SurfacesDefining holomorphic surfaces; the open mapping theorem; compact Riemann surfaces and degrees of holomorphic maps. The Riemann surfaces for the complex logarithm and roots. Analytic continuation. Differential forms on Riemann surfaces.

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Curves Are SurfacesThe implicit function theorem allows us to define an atlas for a nonsingular complex curve which makes it a Riemann surface. This structure is used to give a topological meaning to intersections with lines.

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