This year marks the twentieth anniversary of one of the most successful programs for undergraduate mathematics research. Since 1977 Joseph Gallian of the University of Minnesota, Duluth has been inviting students to spend the summer with him to work on mathematical research problems. Over the years, his work with seventy-five students has resulted in approximately seventy published research papers in professional mathematics journals.
When Gallian started there were only a handful of undergraduate research programs around the nation. Now there are twenty Research Experiences for Undergraduates (REU) in mathematics sponsored by the National Science Foundation, one of which is Gallian's.
Not long ago, most mathematicians believed that undergraduates simply were not able to perform professional quality research in mathematics. However, the REU programs have helped to change the minds of many, and the Duluth program has been particularly instrumental in this regard. Anant Godbole, who runs an REU at Michigan Technological University in Houghton, Michigan, says, ``Joe's program was an inspiration to many of us that high-quality undergraduate research is possible.''
In the Duluth program students work on problems from graph theory, combinatorics, or finite group theory. Gallian says, ``Discrete mathematics is a field that is accessible to people without much background. Solutions to problems rely more on creativity, insight and raw talent and less on using existing results and mastering known techniques than is the case in most other disciplines.'' Thus these subjects are especially suitable for undergraduate research.
At the beginning of the summer, Gallian gives each student a recently written research paper. Typically the paper has several conjectures or unanswered questions. Gallian searches year-round through the mathematics literature for suitable papers. He also has developed a network of people who provide him problems. Finding appropriate problems is by far the hardest part of his job as the director of the program. Very few papers will yield a suitable research project for an undergraduate. Gallian looks for problems that have recently appeared in a well-regarded journal and that discuss new areas not already thoroughly investigated. It is all the better if the article is authored by well-known mathematicians.
Even with many years of experience in assigning problems, it is often the case that the problems Gallian gives students lead to no results. Sometimes after a few weeks of hard work, a student makes no significant progress. In these situations he gives a new problem to the student. In 1993 for example, David Dorrough spent weeks of fruitless effort on two problems before finally getting some interesting results on a third. Dorrough says, ``It was amazing to me that during the entire ten-week program, the only substantial result I got was essentially born out of a half hour's worth of ideas. I'm glad I had the opportunity to try my hand at a few different problems in discrete mathematics, and to experience both success and failure in my research efforts.'' Dorrough's paper has now been published in Discrete Mathematics.
Gallian explains, ``Matching students and problems is a tricky process that requires a lot of educated guessing tempered by experience.'' Aaron Abrams, a participant in 1992, says, ``Joe has a knack for matching people with problems. Although my problem continued to confound me, I found it fascinating, and I was extremely eager to solve it.''
In a few cases, a student will write more than one paper in a summer. For these reasons, Gallian starts the summer armed with many more problems than students.
Gallian absolutely insists that students write their work in a format suitable for publication in a research journal. He says, ``If you don't write up your results, it's just as if you didn't even get them at all. Writing for publication is an important aspect of being a professional mathematician." He helps the students through the sometimes lengthy and arduous publication process, even when on occasion it lasts for several years. But the extra effort has paid off. Work done by undergraduates in the program has resulted in the publication of papers in such journals as Crelle's Journal, Journal of Algebra, Journal of Combinatorial Theory, Discrete Mathematics, Applied Discrete Mathematics, Annals of Discrete Mathematics, and Journal of Graph Theory. This first experience with the publishing side of mathematics is one of the most valuable things the students gain from their time spent in the program. Gallian explains, ``The publication process can be a very discouraging environment, especially for an inexperienced mathematician. Without help, most students would not even realize that their work is of professional caliber.''
Although it is rare for a student to reach the end of the summer without any results worthy of publication, Gallian emphasizes that publication is not the only measure of success. Gallian likes to tell a story about a phone conversation he once had with the mother of one student a few months after the program had ended. Even though the student did not obtain results substantial enough for publication, his mother said he had come home very enthusiastic about his experience in Duluth. The student had benefited in other ways through networking with other mathematicians and exposure to the nature of professional research mathematics; he went on to a successful career in mathematics graduate school at the University of Chicago.
The Duluth program entails a lot more than just hard work on mathematics problems. For example, Gallian provides networking opportunities for the students by inviting many visitors throughout the summer. Most visitors are former participants who are in graduate school or are professors, but other mathematicians such as well-known combinatorists Ron Graham and Fan Chung have visited. Gallian explains, ``I'd like Duluth to be the Tanglewood [a famous summer music camp] of mathematics, a place where young mathematicians work and live with experienced mathematicians. It takes years to develop this kind of environment.'' Weekly field trips to state parks or other local attractions are an essential part of the program as well. Gallian believes that it is important for the students to have fun while doing research.
Where does a mathematician with such a successful program go from here? Gallian says, ``I always set goals for myself. Some of my current goals are to have 100 students in the program and have 100 research articles published as a result of work done in the program. In the distant future, I hope to pass my program on to one of the former participants!''