Proves $\omega \le 9/4$ over $\mathbb{C}$. A companion manuscript proves $\omega < 2.258$ and rectangular bounds. Released October 6, 2026 in OpenAI's math repository (family 107): An Upper Bound of 9/4 for the Matrix Multiplication Exponent; Complex Matrix Multiplication Below…
Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii, Abbas Mehrabian, Francisco J. R. Ruiz, Abigail See, Renfei Zhou, Josh Alman, Virginia Vassilevska Williams, Matej Balog
\omega=2.371177
Proves $\omega < 2.371177$, improving the laser-method analysis with numerical optimization and AlphaEvolve. Paper: arXiv:2608.16884 (August 2026).
Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou
\omega=2.371339
Proves $\omega < 2.371339$. J. Alman, R. Duan, V. Vassilevska Williams, Y. Xu, Z. Xu, R. Zhou, "More asymmetry yields faster matrix multiplication", SODA 2025, arXiv:2404.16349.
Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, Renfei Zhou
\omega=2.371552
Proves $\omega < 2.371552$. V. Vassilevska Williams, Y. Xu, Z. Xu, R. Zhou, "New bounds for matrix multiplication: from alpha to omega", SODA 2024, arXiv:2307.07970.
Proves $\omega < 2.3728596$. J. Alman, V. Vassilevska Williams, "A refined laser method and faster matrix multiplication", SODA 2021; TheoretiCS (2024), arXiv:2010.05846.
Proves $\omega < 2.3728639$. F. Le Gall, "Powers of tensors and fast matrix multiplication", ISSAC 2014, 296–303, doi:10.1145/2608628.2608664, arXiv:1401.7714.
Proves $\omega < 2.3737$ by analysing the fourth tensor power of the Coppersmith–Winograd tensor; the published version gives $\omega < 2.373689703$. A. Stothers, Complexity of matrix multiplication, PhD thesis, University of Edinburgh (2010), era.ed.ac.uk/handle/1842/4734. A. M…
Proves $\omega < 2.375477$ with the laser method applied to the Coppersmith–Winograd tensor. The bound stood until 2010. D. Coppersmith, S. Winograd, "Matrix multiplication via arithmetic progressions", STOC 1987; Journal of Symbolic Computation 9 (1990), 251–280…
Introduces the laser method and proves $\omega < 2.479$. V. Strassen, "The asymptotic spectrum of tensors and the exponent of matrix multiplication", FOCS 1986, 49–54, doi:10.1109/SFCS.1986.52.
Proves $\omega < 2.517$. F. Romani, "Some properties of disjoint sums of tensors related to matrix multiplication", SIAM J. Comput. 11 (1982), 263–267, doi:10.1137/0211020.
Proves $\omega < 2.496$. D. Coppersmith, S. Winograd, "On the asymptotic complexity of matrix multiplication", FOCS 1981; SIAM Journal on Computing 11 (1982), 472–492, doi:10.1137/0211038.
Proves $\omega < 2.522$ via the asymptotic sum inequality ($\tau$-theorem) for disjoint sums of matrix multiplication tensors. A. Schönhage, "Partial and total matrix multiplication", SIAM Journal on Computing 10 (1981), 434–455, doi:10.1137/0210032.
Dario Bini, Milvio Capovani, Francesco Romani, Grazia Lotti
\omega=3\log_{12} 10
Proves $\omega \le 3\log_{12} 10 \approx 2.7799$ using approximate bilinear algorithms (border rank). D. Bini, M. Capovani, F. Romani, G. Lotti, "$O(n^{2.7799})$ complexity for $n\times n$ approximate matrix multiplication", Information Processing Letters 8 (1979), 234–235…
Proves $\omega \le \log_{70} 143640 \approx 2.7951$ by trilinear aggregation, the first improvement on Strassen's exponent. V. Y. Pan, "Strassen's algorithm is not optimal", FOCS 1978, 166–176, doi:10.1109/SFCS.1978.34.