A few years ago, my colleague Lazaros Kikas gave a short seminar on his doctoral dissertation, in which he found results for the "k-disjoint path problem" in the alternating group graph. He had proven that disjoint paths existed in these graphs, but had left as an unsolved problem the question of whether an alogorithm could be found to efficiently generate the paths. I found the problem interesting, and took it home to think about on a Friday afternoon.
The next thing I knew, it was Monday morning. I had hardly eaten and not slept a wink, lost in thought about the problem (this happens to me on occasion). But I held in my hands what ended up being about 90% of a solution to the problem. I showed Lazaros the next day, and within weeks we had begun a graph theory research group on the UDM campus, and I had done my first research in pure mathematics. Click here for the paper that resulted, which was published in Congressus Numerantium.
We just submitted two more papers to Congressus. This paper discusses the use of algebraic factorization to create a path-generation algorithm for Cayley groups in general -- this could be an important breakthrough in parallel processing. Meanwhile, I continue to research the "k-disjoint path problem," and have invented the "Nova Graph," which provides a guaranteed three disjoint paths in an symmetry group-styled interconnection network, using the minimum number of edges and vertices to do so. Click here for the Nova Graph paper.
On occasion, I also have the pleasure of applying mathematics to other fields. Mark Benvenuto, my incredibly prolific colleague from chemistry, asked me to help out with some statistical analysis regarding the chemical compositions of ancient coins. Click here for the first and here for the second of the papers we've co-authored with undergrduate chemistry students in the field of archaeological chemistry.
My research interests include inventing and improving upon mathematical pedagogy. For my doctoral dissertation, I wrote a textbook for a basic course linear algebra which uses MATLAB software in self-tutorial exercises. Math education, particularly involving the proper use of instructional technology, was my first academic passion, but these days I've expanded to research in science education in general, and interdisciplinary pedagogy.
A few years ago, I was invited to Montreal to be a session speaker at the 1999 Winter meeting of the Canadian Mathematical Society (Societe Mathematique du Canada). Soon after, I was invited to deliver a one-hour lecture to the general audience of the Ninth International Colloquium on Numerical Analysis and Computer Science with Applications, held last August in Plovdiv, Bulgaria. My related paper was published in its proceedings. Click here for a copy of the paper, in PDF format, which offers an overview of instructional technology usage in teaching linear algebra and numerical analysis, with an admittedly North American bias.
I later took an additional interest in K-12 mathematics education, probably because I now act as Director of UDM's Master of Arts in the Teaching of Mathematics (MATM) program. Nancy Dwyer, Kathy Zhong, Betty Causey-Lee, and I worked on a grant from the Michigan Mathematics Forum (MMF) to revamp UDM's MTH 477 course. Subsequently, Nancy and I have coauthored a paper for the journal "Teaching Mathematics in the Middle School," along with Sharon Laing (an MATM graduate student) and Mark Fratella (a local middle school teacher), a copy of which can be found here in PDF format. I apologize for the poor quality of the scan.
A few years later, with Stokes Baker (UDM Biology) and Deb Hydorn (Mary Washington College, Statistics), I was asked to co-chair the Math/CS/Biology branch of a Project Kaleidoscope initiative, funded jointly my the Mathematical Association of America (MAA) and the National Science Foundation (NSF). It involved a nationwide study to identify the common trends and characteristcs of productive interdisciplinary collaboration between mathematics and biology at two-year colleges and four-year liberal arts and comprehensive universities. Click here for a .PDF file of the 19-page paper that came out of it, published in the MAA tradebook Math and BIO 2010: Linking Undergraduate Disciplines.
Click here for a paper (pending review) regarding the social experiences and difficulties of "blue-collar mathematics students." I also have some experiments that I'd like to try dealing with the use of game play as a form of differentiated instruction at the middle school level. I'll get around to it one of these days....