We will begin by using complex analysis to discuss the field of meromorphic functions on Riemann surfaces of genus 0 and 1, as a motivation for the definition of a divisor and the statement of Riemann-Roch for both Riemann surfaces and algebraic curves, along with a few "applications" of Riemann-Roch. Time permitting, we will discuss the gonality of a curve and the Brill-Noether theorem. Speaker(s): Ben Krakoff (UM)