A short inotrduction to sheaf cohomology

Classification of isomorphism classes of line bundles on a Riemann surfaces: H^1(M,O*)=Pic(M), Pic_0(M)=Jac(M)

Classification of spin bundles by means of Riemann's vector of constants and theta characteristics.

Exercise: genus 1 algebro-geometric solution of KdV (travelling waves) and their degeneration (solitons). Necessary and sufficient conditions for reality and smoothness.