in each column of the matrix, the n vectors in that column form a basis of V. the n vectors in row i are the members of the ith basis B i (in some order), andĢ. Rota’s basis conjecture states that if B 1, B 2, …, B n are n bases of an n-dimensional vector space V (not necessarily distinct or disjoint), then there exists an n × n grid of vectors ( v ij) such thatġ. This means that you should comment freely in the comments section below.
I have discussed the possibility with the polymath blog admins and since I do not have a blog of my own, they have generously agreed to host the project here on the polymath blog itself. Recently, I emailed several likely participants, and a number of them responded enthusiastically, enough in my opinion to warrant an attempt to start a Polymath project. If you have not read my proposal, I strongly recommend that you read it now, because in it, I sketched some reasons why I thought this would make a good Polymath project, as well as some known partial results and potential avenues for progress. I proposed Rota’s basis conjecture as a possible Polymath project on MathOverflow last year. This post tentatively kicks off Polymath 12 on Rota’s basis conjecture.
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