Geometric Complexity Theory: Introduction

TL;DR

This paper provides introductory lecture notes on geometric complexity theory (GCT), explaining its basic structure and invariant theory aspects for graduate students without requiring advanced algebraic geometry background.

Contribution

It offers accessible, comprehensive introductions to GCT and invariant theory, facilitating understanding of complex concepts for newcomers in the field.

Findings

Clarifies the foundational ideas of GCT

Connects invariant theory to computational complexity

Provides educational material for graduate courses

Abstract

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 second author in the spring quarter, 2003. It gives introduction to invariant theory with a view towards GCT. No background in algebraic geometry or representation theory is assumed. These lecture notes in conjunction with the article \cite{GCTflip1}, which describes in detail the basic plan of GCT based on the principle called the flip, should provide a high level picture of GCT assuming familiarity with only basic notions of algebra, such as groups, rings, fields etc.