A linear series on an algebraic curve defines a map from the curve to a projective space. The classical Brill-Noether theory studies the moduli space of linear series of given rank and degree on a general smooth and projective curve. In this class, I will start with some backgrounds; I will then describe the main results and techniques of Brill-Noether theory and, if time permits, discuss a few applications in the geometry of the moduli space of curves.

Basic knowledge of algebraic geometry

Geometry of Algebraic Curves I,E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris

Moduli of Curves,J. Harris, I. Morrison