Ciro Santilli @cirosantilli 40

This project initiated by Terence Tao aims to find the relations between various statements in abstract algebra by using a combination of automated theorem proving and human effort. As mentioned by Terence himself, this is a bit similar to the idea of the Busy Beaver Challenge:

A "commutative matrix multiplication algorithm" is a matrix multiplication algorithm that requires the ring to be commutative. Such algorithms are inferior because you cannot use them to create more efficient algorithms for general matrix matrix multiplication by decomposing the bigger matrix into smaller ones.

For example, the Strassen algorithm is based on reduction to non-commutative 2x2 matrix multiplication optimized to be done in 7 multiplications rather than 8 as in the native algorithm.

For 3x3 matrix multiplication, the best algorithms as of 2025 are:and beating the Strassen algorithm using 3x3 matrices would require a non-commutative algorithm with 21 multiplications.

Bibliography:

Blog post: deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/

Whitepaper: storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf

Basically they require users to hand-code a metric and provide a program skeleton with some parts of the code marked to be replaced, and then the system focuses on modifying the code regions in question to optimize the metric.

All the novel results they announced were in constraint satisfaction problems or optimization problem. Their results are still awesome, but it's not very different from AlphaGo style things.

DeepMind likes coming up with new improved algorithms for these more specific cases, e.g. it was announced in 2025 that AlphaEvolve found a novel 4x4 complex valued algorithm that uses 48 multiplications.

Bibliography:

Related: Antihydra in Magic: The Gathering.