Secant varieties and the complexity of matrix multiplication

TL;DR

This survey explores the role of secant varieties in understanding the border rank of tensors, which is crucial for analyzing the computational complexity of matrix multiplication.

Contribution

It provides an overview of algebraic geometry techniques used to determine tensor border rank, linking classical and recent results to computational complexity.

Findings

Highlights the importance of secant varieties in tensor rank analysis

Connects algebraic geometry methods to matrix multiplication complexity

Summarizes recent advances in equations for secant varieties

Abstract

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical computer science, and on the other hand touches on many beautiful topics in algebraic geometry such as classical and recent results on equations for secant varieties (e.g., via vector bundle and representation-theoretic methods) and the geometry and deformation theory of zero dimensional schemes.