Report on


a VIGRE-REU program in the
Mathematics Department at the
University of Illinois, Urbana.

5 August 2001

George Francis

Part I. Description, Summary and Overview.
Part II. The Projects and their Significance.
Part III. Evaluation and Advice for Next Year.

My REU program is referred to here as "Experience" since we never could decide collectively what to call it. It was not a "summer school" since tables were turned: the students were paid and the instructors worked for free. It was not a "summer math camp" since it was not set up for the exclusive benefit of the students. It was, as it's acronym suggests, an opportunity for undergraduate students to participate in real mathematical research conducted by the mathematicians and computere scientists associated with illiMath2001. Of course, we subscribe to the generic notion of research as used in the computational and physical sciences, rather than the narrow sense used in mathematics. The research experienced in illiMath2001 was the development of new techniques and technology for the acquisition and exposition of new knowledge and ideas. That in the course of one project ( Bishop Coaster) we discovered some (possibly) new theorems was an added bonus.

The Experience was structured in the manner of a VIGRE-ALP in that I placed great emphasis on vertical integration. Each of the nine undergraduate was the principal student investigator (PSI) of a project, and collaborated in other projects. Each project had one or more associated mentor (AM), who was not only the expert consultant but also expressed an active interest. The AMs were recruited from the Departmental faculty, the NCSA, and Beckman Institute. A number of the AMs were extramural and participated in various ways: a campus visit, e-mail and telephone consultations, and attending the student presentations at the PME-MAA MathFest in Madison. Some AMs worked more consistently and closely with several SPIs, including two graduate students and a postdoc who volunteered their time this summer.

The staff of illiMath2001 consisted of its director, a post-doc and two graduate students. VIGRE-postdoc, Karen Shuman, helped select the students initially, and subsequently became the special mentor for one ofthem. Karen also informally monitored the student happiness with their Experience and organized lunches for the women mathematicians in the group. Like Dr. Shuman, graduate students Michael Pelsmajer and Elizabeth Denne not only worked intensely with particular students, but participated actively in the daily seminars and workshops. Mike has worked on many CAVE projects since he took my Geometrical Graphics course. He was the RA for the grafiXlab last summer. Elizabeth is working on her dissertation with John Sullivan.

The nine students came in groups of three, ordered by their lenghth of association with me. The first group had been in the NCSA sponsored "Audible Sketchpad for the CAVE" REU-project. Matt Woodruff for two summers, Doug Nachand and Ben Shanbaum for one summer. They also worked in the CAVE and the grafiXlab during the year. For example, Doug upgraded the grafiXlab network from 10bT to 100bT and helped with our all volunteer systems-administration of the grafiXlab.

In the second group, Mark Flider, Ben Farmer, Ben Bernard, are also students from the elite UIUC Campus Honors Program (CHP) who took my Freshman Honors Seminar, Math198, in earlier years. Their qualifications and dedication were familiar to me. The final three students, Alison Ortony, Lorna Salaman, Robert Shuttleworth, were new to the activities of the grafiXlab. Lorna and Robert were extramural (Puerto Rico and Youngstown, Ohio). These three students were math majors and gave PME talks on their projects at MathFest.

The typical illiMath2001 weekday began with a workshop in REL (Beckman) on basic graphics programming in C/OpenGL/GLUT for those students who had not taken my course. This was concurrent with a CAVE workshop for those projects already under CAVE development. After lunch, illiMath2001 moved to Altgeld where we had daily project seminars and workshops. The former were attended by all REUs and often had outside visitors. The workshops were directed to specific technical and mathematical problems shared by only some students and visitors. Tuesday evenings we had a supper for the students, the mentors, outside speakers, visitors etc. Thursday evenings were spent in the CAVE.

The pro-seminars and workshops were conducted equally by students and associated mentors. The mentors spoke more frequently at first, recruiting students to their projects, and the students spoke more later, reporting their results and achievements.

The senior, associated, and corresponding mentors were Richard Bishop (Emeritus, Math, UIUC ), Stuart Levy (NCSA, UIUC), Guy Garnett (School of Music, UIUC), Paul McCreary (Xavier University), Ben Schaeffer (ISL, Beckman Institute), John M. Sullivan (Math, UIUC), Jeff Weeks (MacArthur Fellow, Canton NY). Their indispensible roles are reported in connection with the projects, reported on in the second part of this report.

In the aggregate, the students achieved all the goals they and their mentors had set themselves, even if in particular cases some plans remained unrealized. Students learned a great deal of mathematics related to their projects, and programmed real-time interactive computer animations (RTICA) of mathematical concepts. They learned TeX and how to write expository papers, build web-pages and prepare power-point presentations. They taught each other skills, and shared hardware and software expertise. Their RTICAs helped their mentors to discover new geometrical theorems. Their eagerness to learn helped their mentors prepare and give polished expository talks. Their codes, documentation and summaries will be invaluable for future research on the projects.