Define \(\omega = \inf\{p :\ n\times n\) matrix multiplication over a field runs in \(O(n^p)\) arithmetic operations\(\}\). Trivially \(\omega \le 3\); Strassen (1969) shocked everyone with \(\omega < 2.81\) — seven multiplications for \(2\times 2\). Conjecture: \(\omega = 2\).
A single line of attack dominated for decades: Schönhage's asymptotic sum inequality, Bini's approximate algorithms, the laser method of Strassen (1987), and the Coppersmith–Winograd record ω < 2.376 (1990) — whose descendants (Stothers 2010, Vassilevska Williams 2012, Le Gall 2014, Alman–Vassilevska Williams 2021) inched to ω < 2.3728596 amid growing structural-barrier results. In October 2022 DeepMind's AlphaTensor used reinforcement learning to discover concrete faster algorithms (e.g. 47 multiplications for 4×4 modulo 2), the first improvement over Strassen's 49-year-old 2×2 composition in specific settings. Whether ω truly equals 2 — and whether tensors can get us there — is one of the great open questions at the algebra/algorithms interface.
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