Technology has the potential to provide incredible benefits in the modern world. This is especially true in mathematics, where calculations for analysis and problem solving can be tedious and time-consuming. Having powerful and reliable tools allows users to feel confident in their results. However, it is not enough to simply have these tools; they must be used responsibly, with their capabilities and limitations in mind. Rob Corless, a professor at Western University Canada, is a believer in technology tools and uses Maple to support his mathematics research and teaching. He is passionate about reliability and views Maple as a trustworthy tool that enhances his work.

Corless is one of the first users of Maple, having used its first iteration while a student at the University of Waterloo in the early 1980s. He then used it in support of his Master’s thesis and has been using it ever since for teaching and research. He has also written two books, based on his work with Maple. The first, Essential Maple, was published in 1995, and the more recent one is on numerical methods. “The programming elements of Maple have held up well over the years,” Corless said. “It’s also very quick at doing a large amount of calculations. The biggest thing for me, though, is the Linear Algebra package. Maple’s eigenvalues routines are as fast as anybody’s and they’re completely reliable.”

Eigenvalues have been a significant factor in Corless’ work, including his substantial research into Bounded Height Matrices of Integers, which are known by the mnemonic term he coined, Bohemian Matrices. These are defined as a family of matrices whose entries are all chosen from the same finite set.  

His work on Bohemian Matrices began at the Centre for Experimental and Constructive Mathematics in Vancouver, in cooperation with renowned mathematician Dr. Peter Borwein from Simon Frasier University. While working on problems with polynomials, Borwein drew a picture of the roots associated with one of the problems, “and it was gorgeous,” Corless said. “It got me thinking, and when I later came up with a new algorithm for finding roots of polynomials that were expressed in Lagrange bases, I used something similar. I had found a companion matrix, that is, a matrix whose eigenvalues were the desired roots, and I began doing tests on polynomials whose values were + or – 1 on roots of unity. It was basically a Bohemian Matrix before it had the name,” he said. “I was able to create some nice pictures for an academic meeting in 2004, using Maple, which inspired me to draw some sets of eigenvalues. I came up with the term Bohemian Matrices and it snowballed from there.”

Plotting Bohemian Matrices creates visually stunning images. Though it began as a form of artwork, interest has expanded due to its many mathematical applications. There are serious mathematicians interested in these applications, wanting to discover their relevance, Corless said. “At first, we developed some test problems. We generated matrices with basic elements. Then we noticed some interesting math in there, interesting connections to graph theory, number theory, and then we thought maybe there’s more to this, so we dug deeper,” he said. “Many people have used these kinds of problems, some stated and solved, with connections to other, unexpected mathematics. This leads to new questions and discoveries.”

With one particular family of Bohemian Matrices, researchers count the number of matrices, and then count the number of characteristic polynomials and eigenvalues. “They’re different, yet they share some values. Now you test algorithms with the worst possible examples. You try to find new lower bounds on growth. Software like Maple processes this data in a way humans can’t,” Corless said.

“Bohemian Matrices have a broad range of appeal,” he continued. “There is enormous potential for mathematical implications, but there is also an artistic aspect. You get these beautiful images that allow people to connect with the math in a fun and creative way. You definitely need a sense of play in this field.”

Rob Corless and student Steven Thornton prepare for Steven’s Ph.D. defence:
Algorithms for Bohemian Matrices.

A big proponent of computer algebra systems like Maple, Corless believes they have the potential to greatly enhance mathematical research. “Using a computer algebra system can amplify intelligence if you use it responsibly,” he said. “It puts calculation power in the hands of anyone who can type. In principle, the combination of human and machine makes it much better. Knowing computer algebra makes you a stronger thinker.”

Lately, Corless has been working on what he refers to as computational epistemology, studying the reliability of computing. The name is derived from the philosophical study of knowledge and truth. Corless’ objective is to discover how users can know when their computer is telling the truth. “A lot of modern science is being done via computers,” he said. “You have no other access to the results a specific computer is giving you, except more computation. How do you assess validity? Reliability is an important factor when dealing with computers.”

This is one of the reasons Corless regularly teaches his students computer algebra and tools like Maple. He stresses the reliability of software, coupled with intelligent and responsible use of technology. “When you look at some applications for using this technology, like designing car engines for example, accuracy can literally be a matter of life or death,” he said. “Tools like Maple have great potential when you understand how they work and use it to your advantage.”

Eunice Chan, a Ph.D. student and research assistant to Rob Corless, poses with posters she created from Maple fractals, using graphs created by students.

The use of computer algebra must be integrated into teaching and assessment, said Eunice Chan, a Ph.D. student working with Corless on his research. “Maple’s ease of use makes it an ideal tool for introducing students to computer algebra and mathematics software tools,” she said. “New Maple users don’t necessarily need to know mathematical command inputs; there are buttons for integration and a toolbar that makes it easier. Today’s students are already proficient in that type of interface. New users can learn Maple without being overwhelmed.”

Corless and his team of students have a number of research projects in the works for the near future, including working on a rule-based integration project. The project, called RUBI, was developed by Applied Logician Albert Rich. Corless’ team is helping to translate Rich’s program into Maple. He also has more work planned with Bohemian Matrices in Maple. “We want to use the technology to generate images faster,” he said. “The faster they can be generated in Maple, the more people will want to use it.”

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