What is the complexity of matrix multiplication?

Naive matrix multiplication is O(n^3); Strassen reduced this to O(n^2.81) in 1969. The current best is around O(n^2.371). The true exponent ω satisfies 2 ≤ ω < 2.372, but its exact value is unknown.

Many believe ω = 2 (i.e. essentially linear in the input size), but recent advances proceed via tiny improvements with vastly complex constructions.