A Physicist's Proof of the Atiyah-Singer Index Theorem

In this talk we will review the proof of the Atiyah-Singer index theorem in flat space using Fujikawa's method for calculating the chiral anomaly. We will also discuss some applications of the theorem in physics, from calculating gauge anomalies using the method of descent to the Neilsen-Ninomiya theorem which prohibits discretizing chiral fermions on a lattice.