Lecture Publication Series PCMI Math Forum Archive 2003 Program About the Program


Institute for Advanced Study
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IAS/Park City Mathematics Institute
Undergraduate Faculty Program (UFP) 2003

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, Professor Andrew J. Bernoff (Harvey Mudd College) will lead the UFP program "Harmonic Analysis and Partial Differential Equations in the Undergraduate Curriculum." The program will concentrate on two facets of this material; effective methods for incorporating technology (in particular MAPLE) into the classroom and strategies for introducing undergraduates to research in these subjects.

Harmonic Analysis and Partial Differential Equations in the Undergraduate Curriculum;
Leader: Professor Andrew J. Bernoff, Harvey Mudd College

15 2-hour sessions (split between lectures and small group work)

Harmonic analysis and partial differential equations arise naturally in the application of mathematics to physical problems such as the oscillation of a drumhead, the conduction of heat in a metal bar, or the shape of a fluid droplet. Consequently the subject is often taught both as a service course to other disciplines (notably Engineering and Physics) and as a first course applying analysis to physically inspired problems for our own majors. One of the major obstacles in the teaching of this course is relating the tedious algebraic manipulation of infinite sums of eigenfunctions to the visualization of the underlying phenomena. However, the course provides an ideal opportunity for kindling students curiosity about areas of active research in mathematics and its applications.

In this program we will address these issues in two separate threads:

1) Effective methods for incorporating technology into the undergraduate curriculum, and

2) Strategies for involving students in an undergraduate research program.

We will also use these threads for addressing the complementary roles of the classical lecture style and the modern trend toward small group work as effective methods of teaching.

In the first thread we will address strategies for incorporating technology into a course (in particular MAPLE), with the goal of helping students obtain a more physical and visual grasp of the material. This portion of the course will be closer to a traditional lecture format (with some computer lab components). We will cover the derivation of some basic PDEs (Heat equation, Wave equation, Laplace's equation),the method of separation of variables, eigenfunction expansions, Sturm-Liouville problems and special functions (in particular Bessel functions).

The second thread will be devoted to strategies for promoting undergraduate research in partial differential equations and harmonic analysis. This portion of the course will be run as a seminar, and will concentrate on working in small groups and developing presentation skills. We will discuss setting up a problem solving seminar aimed at freshmen and sophomores as a natural forum for interesting students in research and helping them develop problem solving skills. The participants will also be encouraged to take on research projects in small groups, concentrating on problems such as modeling mixing in fluids via Monte Carlo methods (corresponding to solving an advection-diffusion equation) and determining static configuration of fluid droplets via energy minimization (corresponding to a minimal surface problem).

All mathematics faculty members interested in harmonic analysis and partial differential equations in the undergraduate curriculum are invited to apply to this program. Only a modest acquaintance with the material will be assumed. However, faculty members with experience teaching partial differential equations in the undergraduate curriculum and/or supervising undergraduate research in the area 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 or 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 PCMI's 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.