Geometric Complexity Theory, P vs. NP and Explicit Obstructions

Abstract

Theory of computing has given rise to some fundamental mathematical problems, notably the P ≠ NP conjecture, and the related lower bound problems concerning formula or circuit size. We develop an approach to these problems through geometric invariant theory. The goal of this approach is to reduce the hard nonexistence problems under consideration to tractable existence problems.Accordingly, we reduce the arithmetic (characteristic 0) version of the P ≠ NP conjecture, and other related lower bound problems to proving existence of obstructions. These are representations in the homogeneous coordinate rings of orbit-closures in geometric invariant theory [MFK], of a class of points which are partially stable and whose stabilizers have special representation-theoretic properties. However, the Luna-Vust complexity [LV] of these orbit closures is quite high, in contrast with the well-understood homogeneous or almost-homogeneous-spaces, such as G/P [LLM], toric varieties [F3], and spherical embed-dings [BLV], whose Luna-Vust complexity is zero.We take a step towards explicit construction of obstructions by proving two results regarding these orbit closures. The first is a generalization of the Borel-Weil theorem for G/P to these orbit-closures. Second, we conjecture a nice representation-theoretic set of generators for their ideals, and prove a weaker version of the conjecture. Such a set of generators had earlier been given for the ideal of G/P by Lakshmibai, Seshadri, Littelmann [LS, Li3, LLM] and Kostant (cf. [PK]).Finally, using these results, we reduce, in essence, the arithmetic non-existence problems under consideration to fundmental existence and construction problems in representation theory and algebraic geometry that are conjectured to be in the complexity class P. 2000 Mathematics Subject Classification68Q1514L2414L3520G0514M17Key words and phrasesComputational ComplexityGeometric Invariant TheoryRepresentation TheoryClassical GroupsPlethysmHomogeneous Spaces