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Title: An introduction to Riemann surfaces
Author: Ruiz Bonilla, Sergio
Director/Tutor: Mundet i Riera, Ignasi
Keywords: Superfícies de Riemann
Treballs de fi de grau
Superfícies (Matemàtica)
Espais de Hilbert
Corbes algebraiques
Riemann surfaces
Bachelor's theses
Surfaces (Mathematics)
Hilbert space
Algebraic curves
Issue Date: 29-Jun-2015
Abstract: In this document we give a first view to Riemann surface theory. Starting from definition and examples in chapter I, in the next chapter one sees the relation between the well known oriented smooth surfaces and this new object, with the result that any oriented smooth surfaces is equivalent to a Riemann surface. In order to prove this result, almost-complex structures and isothermal coordinates (between others) are explained, and the key point is the existence of these isothermal coordinates for a smooth surface as we will see. Finally in Chapter III we stablish the relation between Riemann surfaces and algebraic curves. First we construct a Riemann surface from a polynomial, which is relatively easy, and then we give and prove the Main Theorem for Riemann surfaces using Hilbert space techniques and some tools like the Riesz Representation Theorem. This Main Theorem is the key to prove the existence of meromorphic functions on a Riemann surface and the fact that any compact Riemann surface arises from a polynomial.
Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Ignasi Mundet i Riera
Appears in Collections:Treballs Finals de Grau (TFG) - Matemàtiques

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