Geometry and the complexity of matrix multiplication

TL;DR

This paper explores the deep connections between algebraic geometry, representation theory, and the complexity of matrix multiplication, highlighting open problems and their broader implications in various fields.

Contribution

It demonstrates how geometric and algebraic approaches can address fundamental questions in matrix multiplication complexity and related computational problems.

Findings

Secant varieties of Segre varieties are central to understanding matrix multiplication complexity.

Open problems in algebraic complexity are naturally expressed through geometric and representation-theoretic questions.

Connections are established between these geometric objects and applications in statistics, phylogenetics, and quantum computing.

Abstract

We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.) to motivate researchers to work on these questions, and (iii.) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.