  Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and
related problems. A high level overview of this research plan and the results
obtained so far was presented in a series of three lectures in the Institute of
Advanced study, Princeton, Feb 9-11, 2009. This article contains the material
covered in those lectures after some revision, and gives a mathematical
overview of GCT. No background in algebraic geometry, representation theory or
quantum groups is assumed.
