Patrick Van Fleet’s initial mathematical research as a graduate student focused on spline functions. Before computers, architects used a flexible
rubber tool called a spline to help them render irregular curves. Thus spline functions are the mathematical analog of an architect’s spline – they
can be used to mathematically describe complex curves. In two dimensions, splines are used in cartography or font design. In three dimensions, splines are enormously popular in applications such as auto body design and computer generated imagery (CGI). If you’ve watched Pixar movies such as "Toy Story," then you have seen splines at work.
Van Fleet started graduate school at Southern Illinois University in 1985. Deciding to study the large-scale evaluation of spline functions, he remembers a faculty member telling him he’d chosen a popular research area, and that in six years he would have his pick of jobs. This prediction did not ring true. In 1991, the year he graduated, a glut of new Ph.D.s meant that tenure-track jobs were difficult to secure and in Van Fleet’s case, not in the cards. He was fortunate, however, to be offered a one-year postdoctoral position at Vanderbilt University where he had the opportunity to work with one of the world’s experts on spline functions. Provided with this ideal research opportunity, he naturally did the unthinkable and switched research areas.
Van Fleet’s postdoctoral adviser traveled much of the year and, as a result, he became engaged in conversations with two young Vanderbilt colleagues
who were doing research in the merging area of wavelet theory. The opportunity to delve into a research area in its infancy was rare and the promise wavelets showed in digital applications made it too good an opportunity to pass up. Wavelet functions seemed to finally provide a way to address the time/frequency conundrum that has plagued mathematicians who work on data/image processing applications.1
Wavelets are used today in applications such as the JPEG2000 image compression standard and is the method of choice of the FBI for compressing and storing digital images of fingerprints. His career path changed again in 1996, when an undergraduate student walked into his office wanting to know how wavelets could be used to compress digital images. Van Fleet was embarrassed to say that he did not know how wavelets were used in this application. “I had formulated and proved theorems about wavelets and understood the abstract theory behind them, but I could not use them in a
‘meaningful’ way!” he recalled.
The student’s question ignited a spark. He knew immediately that he wanted to continue doing this type of work with undergraduates, and so began his first foray into undergraduate research. In 1998 he was hired by St. Thomas as the director of the Center for Applied Mathematics (CAM). He continued to publish papers on wavelet theory but augmented this research with grant work that, in part, allowed him to develop a course on wavelets and to write a couple of books on the topic. He noted, “I really enjoy teaching students about wavelets. The subject lends itself quite well to inquiry-based learning, and students truly enjoy learning how to use mathematics to perform tasks such as data compression, image feature detection or audio de-noising.”
Undergraduate research is truly part of the culture at the St. Thomas and, indeed, it is a prominent component of the CAM. Shortly after his arrival, the CAM developed a summer research program in which small groups of students team up with faculty advisers to mathematically investigate a wide variety of problems. The program has grown steadily over the years. Last summer, the CAM supported 25 students and 12 faculty working on 14 research projects, which have included modeling portfolio risk, characterizing tumors, modeling cell movement and analyzing tornadoes and supercell storms. The students present their work throughout the summer, culminating in a talk by one of the groups as part of the spring
semester CAM Lecture Series.
“I have been amazed at the high quality of research performed by our students and the student lecture in the spring is always a highlight of the academic year for me. Students and alumni provide positive feedback on their research experiences and the CAM provides research opportunities and support for students that are not readily available at other institutions,” Van Fleet said.
According to Van Fleet, wavelet theory has been fully developed and its scope in applications is well understood. He plans to teach the wavelets course again this fall. Students will use a final draft of the second edition of his first book on wavelets. The fall group of students will benefit from great feedback he received from his former students and from faculty around the world who have used the original book in their classes.
Where does he go from here? Back to where he started, of course! “After all these years, I am collaborating with one of my former Vanderbilt colleagues and looking again at open problems involving spline functions. I am smarter this time – we have published some results but I am already thinking about how to make the theory and applications available to my students. Undoubtedly, there are some good student research projects waiting to be discovered.”
1 To understand this problem, think about having sheet music with incomplete tempo or pitch information!