Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program

We determine set-theoretic defining equations for the variety Dualk,d,N ⊂ P(S d C N ) of hypersurfaces of degree d in C that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety GLn2 · [detn] ⊂ P(S n C n2), showing it is an irreducible component of the variety of hypersurfaces of degree n in C 2 with dual of dimension at most 2n − 2. We establish add...

Geometric Complexity Theory: Introduction

Foreword These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the...

Gauge Theory and the Geometric Langlands Program

The Langlands program of number theory, or what we might call Langlands duality, was proposed in more or less its present form by Robert Langlands, in the late 1960s. It is a kind of unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as Andrew Wiles’ proof of Fermat’s last theorem, which involved a sort of ...

The Geometric Complexity Theory Approach to Algebraic Complexity

Geometric complexity theory and matrix powering

Valiant’s famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a proc...