| MATHÉMATIQUES What is the Role of Algebra in Applied Mathematics? (2008) | |||||||||||||
Abstract | |||||||||||||
| When I first encountered abstract algebra in the late 1960s, I was drawn to its power and beauty. One of the liberating ideas of the subject is that in dealing with algebraic structures, you don’t need to worry about what the objects are; rather, it is how they behave that is important. Algebra is a wonderful language for describing the behavior of mathematical objects. My fascination with algebra led me to algebraic geometry, which was then among the most abstract areas of pure mathematics. At the time, I would never have predicted that 25 years later I would be writing papers with computer scientists, where we use algebraic geometry and commutative algebra to solve problems in geometric modeling. The algebra that I learned as pure and abstract has come to have significant applications. What do these and other applications imply about the relation between algebra and applied mathematics? The purpose of this essay is explore some aspects of this relation in the hope of provoking useful discussions between pure and applied mathematics. Here, “applied mathematics ” includes not only what students learn in mathematics and applied mathematics departments, but also the mathematics learned in computer science, engineeering, and operations research departments. I begin with examples from geometric modeling, economics, and splines to illustrate the possible applications of algebra. I then discuss computer algebra and conclude with remarks on the role of algebra in the applied mathematics curriculum. Geometric Modeling, Cramer’s Rule, and Modules My first example concerns an unexpected application of Cramer’s Rule and the Hilbert-Burch Theorem on the structure of certain free resolutions. While Cramer’s Rule should be familiar, free resolutions (whatever they are) may sound rather abstract. As we will see, there are questions in geometric modeling where these topics arise naturally. My part of the story begins with the research of Tom Sederberg and Falai Chen on parametric curves in the plane. If we are given relatively prime polynomials a(t), b(t), c(t) of degree n, then the parametric equations (1) x = a(t) b(t), y = c(t) c(t) | |||||||||||||
Publication details | |||||||||||||
| |||||||||||||