TITLE: How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?
QUESTION [32 upvotes]: As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he reminds me of Thurston in the sense he just drew a picture and then the proof popped out. Nowadays most formulations of Riemann-Roch are couched in algebraic language and do not invoke any geometric intuition for the complex plane, so I am asking here. Bonus points for pictures.

REPLY [5 votes]: And a complementary electromagnetic interpretation is given in the last paragraph of "THE THEOREM OF RIEMANN-ROCH AND ABEL’s THEOREM" by Siu:
The theorem of Riemann-Roch and Abel’s theorem could be interpreted as answering the question: for which configuration of charges, dipoles, or multipoles on a compact Riemann surface of genus ≥ 1 would the flux functions (whose level curves are the flux lines and which are the harmonic conjugates of the electrostatic potential functions) in the case of the theorem of Riemann-Roch, or their exponentiation after multiplication by 2πi in the case of Abel’s theorem, be single-valued on the Riemann surface so that the flux lines are closed curves?
(My library, before I lost it, once contained a classic book on Riemann surfaces by one of the older generations' masters on the topicc that was rife with such physical interpretations. Perhaps someone knows the book and author that I have since forgotten.)