Algebraic Geometry and Representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science

TL;DR

This paper explores how algebraic geometry and representation theory can be applied to fundamental problems in theoretical computer science, especially focusing on the complexity of matrix multiplication and orbit closure problems.

Contribution

It introduces a novel perspective connecting algebraic geometry, representation theory, and computational complexity, particularly for matrix multiplication.

Findings

Reformulation of complexity questions as orbit closure problems

Connections between invariant theory and computational complexity

New insights into matrix multiplication complexity

Abstract

Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let $G$ be a complex reductive group, let $V$ be a $G$-module, and let $v,w$ be elements of $V$. Determine if $w$ is in the $G$-orbit closure of $v$. I explain the computer science problems, the questions in representation theory and algebraic geometry that they give rise to, and the new perspectives on old areas such as invariant theory that have arisen in light of these questions. I focus primarily on the complexity of matrix multiplication.