TITLE: Elementary Proof of Riemann-Roch for Compact Riemann Surfaces
QUESTION [28 upvotes]: I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps doing the version for projective curves.
I assume the crowd knows little algebraic geometry. I assume however, knowledge of elementary differential geometry, sheaves, cohomology and complex analysis. 
I am looking for an elementary proof (especially because I want time to be able to work out some examples). Any references?

REPLY [2 votes]: Here is one that is not elementary.  Joe Polchinski mentions a derivation of Riemann-Roch via Feynman Path integrals.  Unfortunately most people don't know what "ghosts" are and try to do String Theory without them. 
https://physics.stackexchange.com/questions/110322/number-of-zero-modes-on-the-sphere
It is worked out in Chapter 5 of Polchinski's String Theory Vol 1 - he is trying to talk about the Moduli space of Riemann Surfaces