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Matrix multiplication exponent equals 2

Determine whether the exponent ω of square matrix multiplication over the reals is equal to 2.

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Background

Recent nearly-optimal linear programming algorithms have arithmetic complexity that depends on the matrix multiplication exponent ω. If ω were 2, these LP algorithms would achieve near-quadratic time (up to polylogarithmic factors).

The long-standing conjecture that ω = 2 remains open, and resolving it would immediately sharpen the best-known LP complexity bounds described in the paper.

References

Were the exponent of matrix multiplication ultimately shown to be 2 (as is conjectured, although progress in reducing it has slowed considerably), the complexity of the algorithms in would become \tilde{O}(n{2+1/6}) while for the algorithm of it would become O(n{2+1/18}).

— Optimization in Theory and Practice (2510.15734 - Wright, 17 Oct 2025) in Section 4, Linear Programming, Subsection "Recent Developments in Complexity Analysis of Interior-Point Methods"