Lectures notes on compact Riemann surfaces

TL;DR

This paper provides an introductory overview of the geometry of compact Riemann surfaces, covering their definitions, function spaces, key theorems, moduli space, and applications to integrable systems.

Contribution

It compiles fundamental concepts and recent insights into the structure and moduli of Riemann surfaces, emphasizing their role in integrable systems and algebraic geometry.

Findings

Classification of meromorphic functions using Newton polygons

Description of the moduli space via Strebel graphs and uniformization

Application of Riemann surfaces to algebraic reconstruction in integrable systems

Abstract

This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 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.