# What will be the exponent of the fastest known polynomial-time matrix multiplication algorithm in 2029?

The computational complexity class of an algorithm is a measure of how the runtime increases as the input becomes larger. Often, these are written in big-O notation, where an algorithm running in $O(f(n))$ time means that there is some constant $k$ for which the runtime will never exceed $kf(n)$ for an input of length $n$.

In the case of matrix multiplication, the best-known algorithm runs in polynomial time; multiplication of two square n×n matrices runs in $O(n^\omega)$ time for some $ω$. Over time, the smallest known ω has been decreasing - faster algorithms have been discovered.

Naive matrix multiplication, from directly evaluating the sum of the definition, has complexity in $O(n^3)$ time. In 1969, Strassen discovered Strassen's algorithm, which has complexity in $O(n^{2.807})$. By 1990, the Coppersmith-Winograd algorithm was discovered, which has complexity in $O(n^{2.376})$; this has been improved slightly since, with the current best-known algorithm being Le Gall's, which has complexity in $O(n^{2.3728639})$ and was discovered in 2014.

The best known lower bound on matrix multiplication is $O(n^2 \log(n))$; it is known that there is no algorithm faster than this. So further improvement on Le Gall's algorithm has not yet been ruled out.

In 2029, what will be the smallest $ω$ for which there is known to exist an algorithm to multiply two square n×n matrices which has complexity in $O(n^ω)$?

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