Lecture 1 : Geometric Complexity Theory Overview

In this exposition, we give a overview of the GCT approach to solve fundamental lower bound problems in algebraic models of computation. 1.1 Lower bound problems Consider a polynomial f(~x) ∈ Z[~x] with integral coefficients. The non-uniform version of the P v/s NP problem, prescribes a specific such f and asks if it can be shown not to have polynomial size arithmetic circuits over F2. The characteristic zero version of the same problem asks whether we can prove that suitable f ’s do not have polynomial sized arithmetic circuits over Z (or equivalently Q). Clearly if an f has a polynomial sized circuit over Z, then it can be transformed into a polynomial sized circuit over F2 to compute f mod 2. Hence the characteristic zero version is an implication of the characteristic 2 version of the problem. Hence forth, we concentrate only on characteristic zero version of the problem. In order to prove that a certain f does not have polynomial size circuits over Z, we hope to prove a (possibly) stronger statement, viz. that it does not have polynomial size circuits over C (since C has more constants than Z). Since C is an algebraically closed field with a well understood topology, we hope that results in Algebraic Geometry, Representation Theory and Geometric Invariant Theory will come to our rescue and help us solve the problem. What we will see is that the current state of knowledge in these areas is not sufficient to help us resolve the lower bound problems we are interested in. However, we will be able to show that knowing answers to certain mathematical questions (which have independent mathematical interest) will help us resolve our lower bound questions. So what is so great about this approach? Lower bound problems are essentially problems of non-existence, and hence they are hard to solve. This approach reduces the hard non-existence problems into tractable existence problems. This is akin to the NP-characterization of primality, where the proof of primality of p is a generator of Zp. These existence problems are in the areas of Representation Theory and Algebraic Geometry. The primality reduction uses basic number theory and group theory, while the GCT reduction uses classical GIT (due to Hilbert and Weyl), modern GIT (due to Mumford, Kempf and others), together with some new results. 1Geometric Complexity Theory 2Geometric Invariant Theory