Lectures on topological field theories

These lecture notes cover my talks at the June 2009 “Geometry of Quantum Fields and Strings” summer school at the University of Pennsylvania. I begin by giving a very brief introduction to quantum field theory, quickly moving to nonlinear sigma models and the associated A and B model topological field theories, and the physical realization of derived categories. Later lectures will cover Landau-Ginzburg models, matrix factorization, gauged linear sigma models, and mirror symmetry.