Matrix multiplication exponent equals 2

Determine whether the matrix multiplication exponent ω—defined as the smallest real number for which multiplying two n×n matrices can be done in O(n^{ω+ε}) time for all ε>0—equals 2.

Background

The paper reviews the long-standing effort to reduce the exponent ω of square matrix multiplication, currently bounded above by 2.371339 via the authors’ method. A trivial lower bound ω ≥ 2 follows from output size, and despite decades of work no larger lower bound is known. This context has led to the prevailing belief in the field that ω might be exactly 2.

The authors explicitly note this widely held conjecture while discussing known limitations of existing approaches, motivating further methodological innovations such as their more asymmetric laser-method analysis.