Green Family Research Lecture Series: Permanent & Determinant: Non-identical Twins Avi Wigderson Institute for Advanced Study The determinant is undoubtedly the most important polynomial function in mathematics. Its lesser known sibling, the permanent, plays very important roles in enumerative combinatorics, statistical and quantum physics, and the theory of computation. In this lecture I plan to survey some of the remarkable properties of the permanent, its applications and impact on fundamental computational problems, its similarities to and apparent differences from the determinant, and how these relate to the P vs. NP problem. This lecture is intended for a general Math and CS audience.