google-deepmind / google-deepmind/formal-conjectures

Formalize the matrix multiplication exponent conjecture (ω = 2)

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Description

The matrix multiplication exponent conjecture states that ω = 2: for every ε > 0, two n × n matrices over ℂ can be multiplied using O(n^(2+ε)) arithmetic operations.

A tensor-rank formulation asks that, for every ε > 0, there exists C > 0 such that the rank of the matrix multiplication tensor is at most C n^(2+ε) for every positive n. The tensor is the coefficient tensor of (A, B, C) ↦ tr(ABC), with three modes of size n².

References:
- Cohn, Kleinberg, Szegedy, and Umans, [Group-theoretic Algorithms for Matrix Multiplication](https://arxiv.org/abs/math/0511460), introduction and §1.1.
- Chiantini, Hauenstein, Ikenmeyer, Landsberg, and Ottaviani, [Polynomials and the exponent of matrix multiplication](https://arxiv.org/abs/1706.05074), §1.

I am preparing a draft formalization using Mathlib's `Holor.cprank`. The existing Green 36 statements concern related sufficient combinatorial conjectures.

Contributor guide

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Research direction

Start by reading Mathlib's `Holor.cprank` and the cited introductions and §1.1, then compare the existing Green 36 statements in this repository. The work is done when the matrix multiplication exponent conjecture is stated formally in Lean using the tensor-rank formulation described here.

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Assessment

Domain
devtools
Issue type
Feature
Difficulty
5/5
Estimated time
Over a week
Activity status
Active
Clarity
Needs clarification
Newbie friendliness
30/100

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