Timothy Chow of MIT has proposed a new Polymath project: resolve Rota’s basis conjecture.

What’s that? It’s this:

… if $B_1$, $B_2$, $\ldots$, $B_n$ are $n$ bases of an $n$-dimensional vector space $V$ (not necessarily distinct or disjoint), then there exists an $n \times n$ grid of vectors ($v_{ij}$) such that

1. the $n$ vectors in row $i$ are the members of the $i$th basis $B_i$ (in some order), and

2. in each column of the matrix, the $n$ vectors in that column form a basis of $V$.

Easy to state, but apparently hard to prove!

If the Polymath project is new to you, it’s the name for a series of collaborative research endeavours to solve long-standing problems by sharing lots of contributions, of any size, from people all over the world. Previous Polymath projects have successfully led to proofs of the density version of the Hales-Jewett theorem, the Erdős discrepancy problem, and famously reduced the bound on the smallest gaps between primes to within shouting distance of the ultimate goal.

On the Polymath blog, Chow suggests three ideas for ways to make progress, based on previous work on the conjecture.

Even if you can’t make an intelligent contribution, Polymaths are an excellent opportunity to see the messy business of how maths is made – something that you don’t get by reading only finished papers and textbooks.

### More information

Rota’s Basis Conjecture: Polymath 12? on the Polymath blog

Timothy Chow’s original proposal on MathOverflow

Papers published under the collective pseudonym “D.H.J Polymath” on the arXiv