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March 2018

Mr Michael Foster

Honours graduate


Honours thesis

Complex tori

This thesis is an introduction to Riemann surfaces with an emphasis on tori. First we characterise the isomorphisms between tori, and then look at the moduli space of isomorphism classes. After this we explore some of the characteristics of Weierstrass' F-function, and the ?eld of meromorphic functions on a torus. We then shift our focus to the concept of genus, by studying sheaf cohomology and differential forms. We explore the famous Riemann-Roch and Serre Duality theorems and use them to prove the Riemann-Hurwitz Formula. With these results we can find the genus of a torus, and prove the equivalence of our definition of genus to the various topological definitions, in particular, the Euler characteristic.