Niky Kamran Discusses Orthogonal Polynomials
New Hot Paper Commentary, November 2010
Article: An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
Authors: Gomez-Ullate, D;Kamran, N;Milson,
Niky Kamran talks with ScienceWatch.com and answers a few questions about this month's New Hot Papers paper in the field of Mathematics.
Why do you think your paper is highly cited?
The conclusions of our research came as kind of a surprise, because it was thought that the classical orthogonal polynomials, which are captured by Bochner's celebrated 1928 theorem, covered all complete orthogonal systems defined by Sturm-Liouville problems. It turns out that this is not the case and that these classical polynomials admit an unexpected and fruitful generalization, which we present in our paper.
Does it describe a new discovery, methodology, or synthesis of knowledge?
It is a synthesis of several bodies of knowledge and techniques, building on our earlier work and on the works of many researchers, who are cited in the paper. We believe that the applications of Darboux transformation and polynomial flag analysis to orthogonal polynomials hold great promise for new insights and applications.
Would you summarize the significance of your paper in layman's terms?
"Collaboration with excellent colleagues and a continuous effort are key."
Orthogonal polynomials and their properties appear in many important questions in mathematics and in physics. They are like "building blocks" and "shadows" for deeper properties of many structures. The exceptional polynomials we have discovered have the potential of broadening the realm of investigation of these ideas.
How did you become involved in this research, and how would you describe the particular challenges, setbacks, and successes that you've encountered along the way?
There are many wonderful researchers who have been involved in the general questions that have led us to this paper, with whom we have either collaborated, or discussed these questions at length. I would like to particularly mention Peter Olver, Artemio Gonzalez-Lopez, Sasha Turbiner, and Norrie Everitt, who have all been important sources of inspiration and encouragement for us. The specific collaboration that led to this paper is part of an exciting journey that my collaborators David Gomez-Ullate, Rob Milson, and I have been involved in. Collaboration with excellent colleagues and a continuous effort are key.
Where do you see your research leading in the future?
There are a number of exciting questions and generalizations to pursue: applications to quantum mechanics, integrable systems, and exact solutions of other equations of mathematical physics. We are happy to see that many of these questions are being investigated by researchers from different groups.
Department of Mathematics and Statistics
Montreal, Quebec, Canada
KEYWORDS: DIFFERENTIAL-EQUATION; BOCHNER; THEOREM.