The course will deal primarily with Brill-Noether theory on compact Riemann surfaces, the question of the number of meromorphic functions with at most some specified singularities. That has been discussed extensively on generic (or primitive) Riemann surfaces, as in the book Geometry of Algebraic Curves I by Arbarello, Cornalba, Griffiths and Harris.

I will focus more on special curves, in particular on relations between discrete invariants of the analytic structures of curves such as the maximal sequence and the Luroth sequence, and mostly in terms of line bundles. If there is time and interest perhaps some discussion of the less complete case of vector bundles, or some possibilities in higher dimensions. I will try to make the course fairly self-contained, so initially not assuming much detailed knowledge of function theory on compact Riemann surfaces; but I will not attempt to prove the basic background results.