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!
