To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

(LO1) Understand the most basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

(LO2) Understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

(LO3) Learn different techniques to calculate the genus and the dimensions of spaces of meromorphic functions, and they will have acquired some understanding of uniformisation.

(S1) Problem solving skills

-A brief introduction to topology and topological spaces.
-Some topological properties: connected, path-connected, sequentially compact and Hausdorff.
-The definition of a Riemann surface. Charts and atlases. Examples.
-Plane algebraic curves that are also Riemann surfaces. A weak form of Bezout's theorem.
-More advanced complex analysis material. Identity principle for holomorpic maps. Open mapping theorem. The holomorphic implicit function theorem in 2 variables.
-Maps of Riemann surfaces. Isomorphisms of Riemann surfaces. Canonical domains and their automorphisms.
-Holomorphic and meromorphic differentials on Riemann surfaces.
-Residues and integrals of meromorphic differentials. The residue theorem for compact Riemann surfaces.
-Finite triangulations, Euler characteristic and genus. The topological classification of compact Riemann surfaces.
-The Riemann-Hurwitz formula and its consequences.
-The Riemann-Roch theorem.
- Uniformisation of Riemann surfaces.