# Lectures on compact Riemann surfaces

Abstract : This is an introduction to the geometry of compact Riemann surfaces. We largely follow the books [8, 9, 10]. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann-Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, and also with the uniformization theorem that maps Riemann surfaces to hyperbolic surfaces. 6) An application of Riemann surfaces to integrable systems, more precisely finding sections of an eigenvector bundle over a Riemann surface, which is known as the "algebraic reconstruction" method in integrable systems, and we mention how it is related to Baker-Akhiezer functions and Tau functions.
Type de document :
Cours
Doctoral. Paris-Saclay, France. 2018

https://hal-cea.archives-ouvertes.fr/cel-02013559
Contributeur : Bruno Savelli <>
Soumis le : lundi 11 février 2019 - 10:22:52
Dernière modification le : mercredi 13 février 2019 - 01:17:06

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Eynard.pdf
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### Identifiants

• HAL Id : cel-02013559, version 1
• ARXIV : 1805.06405

### Citation

B. Eynard. Lectures on compact Riemann surfaces. Doctoral. Paris-Saclay, France. 2018. 〈cel-02013559〉

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