Timothy Chow starts Polymath12

This is a quick post to draw attention to the fact that a new and very interesting looking polymath project has just started, led by Timothy Chow. He is running it over at the Polymath blog.

The problem it will tackle is Rota’s basis conjecture, which is the following statement.

Conjecture. For each $i$ let $B_i=\{e_{i1},\dots,e_{in}\}$ be a basis of an $n$ -dimensional vector space $V$ . Then there are $n$ disjoint bases of $V$ , each containing one element from each $B_i$ .

Equivalently, if you have an $n\times n$ matrix where each row is a basis, then you can permute the entries of the rows so that each column is also a basis.

This is one of those annoying problems that comes into the how-can-that-not-be-known category. Timothy Chow has a lot of interesting thoughts to get the project going, as well as explanations of why he thinks the time might be ripe for a solution.

3 Responses to “Timothy Chow starts Polymath12”

help. Says:
March 22, 2017 at 3:38 am | Reply
a basis for $\Bbb R^n$ is a set of $n$ independent vectors. If rows are independent then columns are as well. Why permute? do not understand. could you please explain?
- woutervandoorn Says:
  March 23, 2017 at 8:29 am
  Each row and each column have to be bases, not basiselements
danicao Says:
March 26, 2017 at 10:08 pm | Reply
Maybe it is some kind of stupid idea and I have not thought too much about it, but would it not be possible to construct something like the determinant of your matrix (with vectorial entries) with sums (with sign) of all possible combinations of vectorial products? Anycase, not sure if this would help because the transformations that preserve determinant do not break rows or columns (and you needed to mix the basis).

Anyway, I find it very curious and interesting.

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