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. My advisor snorted at this question, so you know it is good. Bonus points for pictures.