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The 12th Polymath project has started: resolve Rota’s basis conjecture

By Christian Lawson-Perfect. Posted February 26, 2017

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

What’s that? It’s this:

… if $B_1$, $B_2$, $\dots$, $B_n$ 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!