AF: Small: Symbolic Computation and Difference and Differential Equations

North Carolina State University, Raleigh NC

Investigators

Abstract

Many of the functions of interest in mathematics and physics are defined by difference or differential equations. Understanding their algebraic properties is key to using these functions to describe physical and mathematical phenomena. The investigator will develop algorithms that will reveal these properties. In particular, he will develop algorithms to determine the algebraic and differential relations that occur among solutions of a given set of linear difference equations. In addition, he will develop theory and give algorithms to measure the algebraic behavior of solutions of parameterized linear differential equations as one varies the parameter. Finally, he proposes to attack the problem of factoring underdetermined systems of partial differential equations. Although disparate in appearance, these problems will be attacked using techniques based on studying the underlying symmetries of the defining equations. In the past, the investigator has contributed to the development of theory and algorithms to solve differential and difference equations, in particular to the Galois theories of these equations and algorithms to solve them in closed form. The present project in part refines and extends this work but goes beyond to attack the broader problems mentioned above. This research addresses foundational and computational issues concerning the algebraic behavior of systems of linear difference and differential equations. It allows researchers in many scientific fields to understand aspects of the qualitative behavior of solutions of these equations. The researcher will develop algorithms that that will be useful to number theorists, combinatorists and analysts. In addition, the investigator anticipates that these algorithms will form the foundation on which Maple and Mathematica code are based and so have an impact on the education and day-to-day work of engineers and other scientists. Key components of this project are the development of human resources, the fostering of interactions with other scientific fields and the advancement of international collaborations. The investigator will continue to not only train his Ph.D. students but to further develop with his colleagues a program at NC State University to train students in a broad range of topics in Symbolic Computation. He will sponsor a postdoctoral scholar, involving this scholar in the research proposed here as well as develop the scholar's teaching skills and integrate the scholar into the scientific community. He will continue and expand his work on revising the undergraduate Abstract Algebra curriculum to include Symbolic Computation as a core topic. In addition he will continue to organize workshops aimed at students and colleagues in diverse fields to disseminate to a broad scientific community the ideas of Symbolic Computation in general and Symbolic Analysis in particular. He will continue his recent collaborations with researchers in Germany, France and China and will involve graduate students from NC State University in these projects, allowing them to integrate themselves in the international research community.