Statement of the Problem

My father is a mathematician pretty well known in his field of linear algebra. (I'll tell you what linear algebra is in the next blog post.) He is a professor at the College of William and Mary in Williamsburg, Virginia. He spends most of his time doing research (another blog topic) but he also really likes teaching, as long as it is either at a fairly high level or his students are pretty sharp.

He especially enjoys running and teaching at an Research Experience for Undergraduates (REU) program in the summer. The program is made up of college math students from around the country, and the professors present a bunch of small problems that they find interesting but don't know the answer to. Each student picks a problem and a professor and tries to get results by the end of the summer.

I asked my dad if he could give me a math problem to work on, since he is very good at collecting problems that are narrow in scope and good for beginners. I am a career writer, currently working as a technical writer for an online advertising startup, where I write about our products and software. My first love is creative nonfiction, especially essays. I took linear algebra in college in 1998, and did perfectly well but not brilliantly. I am better and more comfortable at general math than the average American, but have never showed any particular talent at the college level. I am interested in math as a concept, but have never been drawn to problems the way my dad is.

I am pretty terrified of this problem, because I am insecure about my high-level mathematics ability, and truthfully, I would really like to impress my dad. So I have been shying away from it. I thought maybe chronicling my progress on it through a blog, where I will then be responsible to my millions of clamoring fans, might help.

So. Here is the problem. Feel free to work on it yourself, but if you solve it, don't tell me the answer. In the next post I will define all the terms and add a little more information about it from my dad, which he imparted while perched precariously on a stool at a paella restaurant near my apartment in New York City.

Dear Em. Let me try to give a rough description of the problem. A matrix is called totally positive TP) if ALL its minors (determinants of square submatrices) are positive. A weaker condition (TP2) is that only the entries and 2-by-2 minors are required to be positive.(However Shaun and I have shown the remarkable result that some entry-wise power of a TP2 matrix is TP.). Now, it is a theorem that a matrix is TP2 if and only if it is entry-wise positive and all its contiguous 2-by-2 minors (just the ones with consecutive row indices and consecutive column indices) are positive. There are many fewer of these. Now, consider, in place of a conventional (rectangular) matrix an arbitrary array that may take on any rectilinear shape (perhaps a fat "L" or a fat "U", among many). The question is, for which connected shapes does contiguous 2-by-2 minors positive (in addition to positive entries) imply that all 2-by-2 minors are positive? Note that I get a 2-by-2 minor ANY time I have a square 2-by-2 submatrix, even if there is space in between.