More Research, and the Next LinAlg Activity

Originally Posted 01-24-2012

I had a pleasant and still day in the office. I had no formal teaching duties, but I did have two research oriented meetings, and a nice informal chat with a former student.

Meeting Number One: Undergraduate Project

I am working on a combinatorics/number theory project with an undergraduate student. Our meeting today had a bit of a “wow” moment, too. We have been studying some families of polynomials that “arise naturally” from the problem. The student has found a ton of identities involving these polynomials, all experimentally. We have not succeeded in finding a proof. So, as an alternative to feeling stuck, I thought we might change gears a bit and instead of looking at the coefficients (which is what brought us to this point), we could look at the roots. We finally got some Sage/Python code up and running to compute and plot these roots in a meaningful way, and, ‘pop’, they appeared to lie on a logistic S-shaped curve.
I can’t explain it, but it probably means something deeper is going on.

Meeting Number Two: Professional Research

My colleague Bill Wood and I had an exploratory meeting this afternoon. We attended a workshop on Discrete and Computational Geometry at the Joint Mathematics Meetings earlier this month, and we wanted to find a problem to work on together. After two hours of just talking, I think we had three or four problems that we agree are interesting. Most of them seem a bit out of our reach, but one just might be a good way to get going. So, it looks like I’ll be trying out something new this term. I am excited.

Linear Algebra: The Indoctrination Educating Continues

Feeling good about the last two days of linear algebra, I have designed some more for them to handle tomorrow. The activity is meant as a way to introduce a lot of important language quickly, including the concepts of span, linearly (in)dependent sets, dimension, basis, and determinant. It seems impossible that this will work so well, but I have to try. The focus is again on solving simple sounding problems presented in several equivalent ways, and noting what it all really means.
Also, I have designed a Sage based homework assignment. If either of these goes at all well, I’ll share them later.

The Summer Research Program for Students

I spent some time organizing our summer research program for undergraduate students today, too. This was mostly simple grunt work to prepare to advertise the opportunity. The program has run for three consecutive summers, so much of the organization is on auto-pilot now. Right now, we only have money to support two students–which is the other project I am supposed to be thinking about. Oh, look at the time…