IAS/Park City Mathematics Institute
Co-sponsored by Chautauqua Programs
For the faculty members whose main focus is teaching undergraduate students, PCMI offers the opportunity to renew excitement about mathematics, talk with peers about new teaching approaches, address some challenging research questions, and interact with the broader mathematical community.
Each year the theme of the UFP bridges the research and education themes of the Summer Institute. In 2002, Daniel Schaal, South Dakota State University, will lead the UFP program Directing Undergraduate Research Programs in Mathematics with an emphasis on combinatorics.
Directing Undergraduate Research Programs in Mathematics
In this program we will address some common problems faced by anyone directing an undergraduate research program in mathematics. We will consider how to select and recruit students and how to find appropriate problems for students to investigate. We will discuss how to quickly teach students the background material they will need and how to get students started on research. We will also discuss possible sources of funding and the application procedures of some funding agencies. We will consider large and small programs, summer and academic year programs, and programs with or without students from outside the director's home institution.
To illustrate the ideas of problem selection, student preparation and problem investigation, we will conclude the program with a short investigation in Ramsey theory. Ramsey theory is an area of graph theory and combinatorics that deals with colorings of the edges of a graph and colorings of the natural numbers. A coloring is simply a function that partitions the domain of the function into equivalence classes. Among other things, Ramsey theory addresses the question of how big the domain of the coloring must be to guarantee that at least one equivalence class contains a "special" subset. Ramsey theory is an especially good area for undergraduate research due to the small amount of background material needed and the abundance of open problems. We will briefly cover the basics of Ramsey theory and some proof techniques that are often used in this area. We will then investigate some open problems and may even solve some.
It is not the goal of this program to prepare the participants to direct undergraduate research in Ramsey theory. The brief investigation of Ramsey theory is intended as an example of how an undergraduate research program can be structured. The techniques of problem selection, student preparation and problem investigation that will be illustrated in this investigation are applicable to undergraduate research in all areas of mathematics. One of the programs for undergraduate students at the PCMI will investigate expander graphs, another area of graph theory and combinatorics. Participants in this program may also choose to attend some or all of the sessions on expander graphs. The program on expander graphs can serve as a second example of how an undergraduate research program can be structured.
All mathematics faculty members interested in undergraduate research are invited to apply to this program. No knowledge of combinatorics or prior experience directing undergraduate research is expected or required. However, faculty members with experience directing undergraduate research are encouraged to apply and will be able to enrich the program by sharing their experiences with the other participants.
The UFP explores one central course of topic in the undergraduate curriculum from the dual perspectives of the mathematics itself and its teaching. Also, the UFP is one of the fundamental sources of meaningful interaction between PCMIs constituent groups and programs. Some UFP participants attend courses of the Graduate Summer School each year. A large number are attracted to the Undergraduate Program, both for the interesting mathematics in the courses and for the kind of research experiences for undergraduates that the mathematics of the courses typically generates. Finally some engage teachers of the High School Teachers program in the examination of transitional issues between high school and early undergraduate mathematics instruction.
College faculty with a strong interest in undergraduate education are encouraged to apply to PCMI's Undergraduate Faculty Program.
Prerequisites: two years of undergraduate mathematical teaching experience. This program is generally not for graduate students or new PhD's.
The Coordinator of PCMI's Undergraduate Faculty Program, Daniel Goroff, is Professor of the Practice of Mathematics at Harvard University and Associate Director of the Derek Bok Center for Teaching and Learning.
questions or concerns should be directed to C. Giesbrecht