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Greg Coxson

IEEE; currently MAA Governor at-Large, Outside Academia

*From: The BIG Notebook, A Newsletter of the MAA Special Interest Group for Mathematics in Business, Industry & Government, Volume 4, Number 1, August 2008 ; with permission from the author.*

It might seem too good to be true to have a team of bright mathematics graduate students working on one of your problems for a week, for free no less. But this is essentially the situation I found myself in this past June.

It all started with an email from Professor Margaret Cheney of Rensselaer Polytech. She asked if I would be interested in participating in the 2008 Mathematical Problems in Industry workshop at Worcester Polytech. If so, I should draw up a list of problems from radar engineering, my company’s particular area of work, and to submit them for consideration.

Before I go on, I want to say that anyone who has read this newsletter for the past three years knows that summer research experience workshops, and the REU at WPI (this is separate from the MPI workshop, and is for undergraduate rather than graduate students) in particular, are close to the my heart and that of BIG SIGMAA. But I had never participated in one. So there was little question of going if I could. The decision was made even clearer when word came that the National Science Foundation had waived the problem submission fee, said to have been about $2000 in recent years.

It is actually an interesting challenging finding problems for such a workshop. I was able to list about ten problems quite easily. However, in talking with colleagues at my company, I determined that most were ill-advised. For example, several were problems being worked by teams at the company and would have caused political problems, and one of them was said to be too valuable to the company to risk having it solved in such an open academic forum. I ended up whittling my list down to one interesting, theoretical and quite well-known problem. At the outset, I though the problem was likely to be impossible to solve in a week, even by a team of eager and well-trained graduate students.

The title of the problem I chose was “Do the Barker Codes End?”. This problem goes back at least to the 1950s and boils down to the existence of binary codes of even lengths longer than 4 having the Barker property of autocorrelation sidelobe magnitudes no greater than 1. It is combinatorial in nature and closely linked to another difficult conjecture by Ryser on the existence of circulant Hadamard matrices of order greater than 4.

I voiced my concerns to Dr. Cheney and to the workshop director before the workshop. I was assured that it is often the seemingly impossible problems that make the best problems and, “the students may surprise you”.

I came to learn that the workshop rarely gets combinatorial questions like mine. They often get problems involving mathematical modeling of physical problems arising in manufacturing. In fact, the other two problems at the workshop were of this latter type; one concerned glass formation and the other, the design of lithium-ion batteries. One professor at the workshop came up to me and told me “we do not get problems like yours”. I was not clear whether he meant that was a good or bad thing, but by week’s end, I concluded it was an approving comment.

The week started with problem presentations by myself and the other ``problem proposers’’. Then the teams were chosen and went right to work. The teams had four days to work on the problem. The final day, Friday, was for presentations. It was a a real whirlwind.

I felt a bit rusty, I must admit (the students picked apart my presentation and quickly found three errors). They were all wonderful, sharp, and easy to like. There were 12 students on the team, among them Mike from Arizona, who quickly went to work on search algorithms, Yicong a chinese kickboxing Statistics student from Clemson who worked on curve fitting of data sets, Rajaa, a Fulbright scholar from Syria, who concentrated on assessing an existing approach by Golay, and Jeremy from Indiana University, who ended up working on the problem in another room with a team of professors who became interested. Jeremy’s team ended up developing an approach from first principles in an intense and, one might say, ``muscular,’’ form of mathematical chalkwork.

One aspect of this process that can be a bit challenging is to bring students up to speed on fundamental concepts. For someone like me, who has been in a discipline for a while, I found it was necessary to answer questions on ``basic’’ concepts, which are not basic to someone coming to the area for the first time. Some time has to be reserved at the start of the week for the team to ask questions, and to acquire these basic concepts.

Another interesting aspect are the team advisors. My vision of this in the beginning was that I would be proposing an interesting problem arising in my discipline and a team of students would work on it for a week, basically deciding the direction to take. Then I learned that the project team would need advisors. Furthermore, the director was finding it difficult to find experts to serve as advisors for this particular problem. As a result, I was asked to suggest a list of potential advisors. Remarkably, two wonderful folks whose work I admired accepted the invitation, and the NSF grant covered their travel expenses from their posts in Canada. The opportunity to meet and work with them, by itself, made the workshop worthwhile and exciting for me.

One day we had a special treat and were given presentations by the undergraduates participating in the summer REU at WPI. Unlike the graduate students who were there for a week, the undergraduates are in their program for two months. They were remarkably prepared, poised and knowledgeable.

Mathematical Problem in Industry workshops are a popular and widespread phenomenon overseas. We had a few workshop veterans from overseas, all of them very fun, engaging and experienced professors, most from Britain or Canada. The MPI workshop I participated in is an American version that moves year-to-year between several colleges on the east coast of the US. These include RPI, the University of Delaware, WPI and Franklin Olin College.

Everyone from the Math office staff to the directors and professors were welcoming and helpful. The organizers took care of all expenses, making it easier to concentrate on the workshop. Sometime later this summer, a final report will be produced, prepared by one of the team advisors (the rules state that a faculty member has to write the report; luckily one of the advisors on the Barker problem team fit that description). In addition, the event rekindled my interest in the Barker problem (as proof, my satchel now holds several books I had held off buying before!).

Did the team solve the Barker problem? Actually, no. However, I have come to the conclusion that the problem was not a bad choice. It can be approached with number theory, or matrix theory, with lots of algebra and analysis mixed in.

I recommend this workshop to any of my BIG colleagues. What problem would you and your organization like to put to a team of eager graduate students to solve in a week? Think about participating as a problem proposer at the 2009 MPI workshop at the University of Delaware.

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Greg Coxson

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