Geometric complexity theory I: An approach to the P vs. NP and related problems

Abstract

We suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size. Specifically, we introduce the notion of a partially stable point in a reductive-group representation, which generalizes the notion of stability in geometric invariant theory due to Mumford[Geometric Invariant Theory, Springer-Verlag, Berlin, 1965]. Then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points. We also suggest an approach to tackle the latter class of problems via construction of explicit obstructions.