RESEARCH INTERESTS
My principal work has been in valuation theory and ordered fields, an area of abstract algebra in which we study analogues of the absolute value function. That allows the introduction of some topological language and techniques from analysis.
I have recently begun work with a team of researchers on areas involving an overlap of graph theory, ring theory, group representations, Galois theory and other areas.
I also enjoy working on “recreational mathematics” where we look at problems whose statements are easily understood at an elementary level, but whose solutions may sometimes involve non-elementary techniques. I was first motivated to work on Ducci Sequences, for example, after seeing an arithmetic exercise for third graders!
I am currently doing work on some aspects of noncommutative ring theory, group representations, and graph theory with Jàn Minàč (University of Western Ontario), Lyle Muller (University of Western Ontario), Sunil Chebolu (Illinois State University), Tung T. Nguyen (University of Western Ontario), Federico Pasini (Huron University College), and Nguyễn Duy Tân (Hanoi University of Science and Technology), Papers with this group:
On the Join of Group Rings, J. Pure Appl. Algebra 227 (2023), Issue 9 (link is to preprint version posted on arXiv before publication)
Spectral perturbation by rank m matrices, Operators and Matrices 17 (2023), Issue 3, 867-874 (link is to preprint posted on arXiv before publication)
Recent publications on valuation theory, joint work with Ron Brown, Professor of Mathematics. University of Hawaii:
The space of R-places on a rational function field. J. Algebra 565 (2021), 489-512
The space of real places on R(x,y). Ann. Math. Sil. 32 (2018), no. 1, 99-131
The main invariant of a defectless polynomial. J. Algebra Appl. 12 (2013), no. 1, 16 pp.
Invariants of defectless irreducible polynomials. J. Algebra Appl. 9 (2010), no. 4, 603-631
Work on Ducci sequences with Professor Ron Brown, University of Hawaii:
(What the heck is a Ducci sequence? Click here.)
- The number of Ducci sequences with given period. Fibonacci Quart. 45 (2007), no.2, 115-121
- The length of Ducci’s four-number game. Rocky Mountain J. Math. 37 (2007), no. 1, 45-65
- Limiting behavior in Ducci sequences. Period. Math. Hungar. 47 (2003), no. 1-2, 45-50
Topology paper with Professor Inga Johnson, Willamette University:
A class of left ideals of the Steenrod algebra. Homology Homotopy Appl. 9 (2007), no. 1, 185-191.
This topology paper involves a “postage stamp” problem. Check here.