Ketan Mulmuley

Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems

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.

Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496-526], henceforth referred to as Part I, we suggested an approach to the

A fast planar partition algorithm, I

P

On levels in arrangements and Voronoi diagrams

vs.

A fast planar partition algorithm. I

NP

Randomized multidimensional search trees: lazy balancing and dynamic shuffling

and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the

Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma

NP \not \subseteq P

Lower Bounds in a Parallel Model without Bit Operations

conjecture to the problem of showing that a variety associated with the complexity class

On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna

NP

Geometric Complexity Theory, P vs. NP and Explicit Obstructions

cannot be embedded in a variety associated with the complexity class