
IAS/Park City Mathematics Institute Cosponsored 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; 15
2hour 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, SturmLiouville
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 advectiondiffusion 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. 