Calculus of Variations:

A Conference for Undergraduate Mathematics Majors

Rice University, November 8-10, 1996

The general goal of this conference is to expose undergraduates from many universities to current research in mathematics___the particular focus of this meeting being the calculus of variations and its applications. Several research mathematicians in that subject will discuss classical and current results and open questions at a level accessible to junior and senior mathematics majors. Interested graduate students and faculty are also welcome to attend.


Steven Cox (Rice University)
Robin Forman(Rice University)
Frank Jones (Rice University)
Barbara Keyfitz (University of Houston)
Frank Morgan (Williams College)
Michael Wolf (Rice University)

Conference Schedule

Friday, November 8:

6:30 - 7:30 pm: Complimentary Buffet behind Herman Brown Hall

8:00 pm (BL131): Frank Jones (Rice University)

Abstract: An explanation of the basic definitions and beginnings of this subject, showing the natural extension from the basic ideas of single-variable calculus, through (finite dimensional) multi-variable calculus, to (infinite dimensional) "variational" calculus. The famous Euler-Lagrange equation!

Saturday, November 9:

9:00 am (HB438) Coffee, donuts, fruit

9:30 am (HB227): Robin Forman (Rice University)
Title: How Many Equlibria are There? An Introduction to Morse Theory

Abstract: When watching a dynamical system, the states which are easiest to observe, at least for more than a fleeting moment, are those which do not change over time. These states are said to be "stable" or "in equilibrium ." A fundamental problem in the study of dynamical systems is to identify these equilibrium states.

Dynamical systems which occur in nature can often be described by saying "the system is trying to minimize energy." We will describe a very powerful method to study such dynamical systems, developed by Marston Morse in the 1940's and 50's, which converts the problem into an essentially equivalent problem in topology. We will also discuss the difficulties one encounters when trying to apply these methods to the important special cases that some of the other lecturers will be speaking about.

11:00 am (HB227): Steven Cox (Rice University)
Title: Aye, Now There's the Rub

Abstract: So why does a plucked string return to rest after only 5 or 6 seconds? Some believe the string rubs against the air, others that it rubs against itself, and still others that it rubs against the bridge. We shall present experimental and mathematical evidence in our indictment of the bridge.

12:00 noon: Brunch at Baker College with Rice undergraduates

1:30 pm (HB227): Frank Morgan (Williams College)
Title: The Double Soap Bubble Breakthrough and Undergraduate Research

Abstract: We know that a single round soap bubble provides the least-area way to enclose a given volume of air, but it remains an open question today whether the familiar double soap bubble provides the least-area way to enclose and separate two given volumes. The computer breakthrough heralded in the press this past year can be traced back to undergraduate research.

3:00 pm: Three concurrent "lab" activities: Steven Cox, Robin Forman, Frank Morgan

I. Steve Cox (HB423): A Laboratory for Linear & Nonlinear Waves

We shall view linear & nonlinear waves in circuits, on springs, cords, strings, and on plates. With respect to the latter we shall reproduce the arabesques described in,

To the small amount of physical apparatus which Adrian's father had at his command belonged a round glass plate, resting only on a peg in the centre and revolving freely. On the this glass plate the miracle took place. It was strewn with fine sand, and Jonathan, by means of an old cello bow which he drew up and down the edge from top to bottom made it vibrate, and according to its motion the excited sand grouped and arranged itself in astonishingly precise and varied figures and arabesques. This visible acoustic, wherein the simple and the mysterious, law and miracle, so charmingly mingled, pleased us lads exceedingly; we often asked to see it, and not least to give the experimenter pleasure.

___Thomas Mann, Doctor Faustus

II. Robin Forman (HB453): What's Hot and What's Not: Open Problems in Mathematics

We will discuss the current state of some of the famous (and not so famous) open problems in mathematics, along with a general discussion of what mathematicians are doing these days. Attendees are encouraged to bring their favorite problems.

III. Frank Morgan (HB427): Soap Bubble Geometry

Why do you get those funny shapes?

6:00 pm: Pizza party at Nancy's house (5119 Caroline at Southmore) with Rice graduate students

Sunday, November 10:

9:00 am (HB438): Coffee, donuts, fruit

9:30 am (HB227): Michael Wolf (Rice University)
Title: Minimal Surfaces of Least Total Curvature and a Problem in Plane Geometry

Abstract: Minimal Surfaces in Euclidean space can be found both as a limit of surfaces which tend to a critical point of area, or in terms of some complex analytic objects on a surface, viewed as a complex domain. We adopt the latter view and then consider these objects as describing several flat geometries on the surface. It then turns out that the existence of some interesting minimal surfaces is then equivalent to finding a polygonal line in the plane, so that the two flat domains that it bounds are equivalent as complex polygons.

11:00 am (HB227): Barbara Keyfitz (University of Houston)
Title: Hold That Light! Conservation Law Models for Traffic Flow

Abstract: A simple continuum model for heavy traffic on a one-way street leads to an equation which models traffic jams very convincingly, displaying waves (stop-and-go patches) and shocks (slam-on-the-brakes time). Through variational methods, a formula is found which shows that there will always be waves and shocks.

12:00 noon: Lunch (in Village?)

Afternoon: games (frisbie, soccer, chess)


All activities will be held in the Herman Brown Building on the Rice University campus. The Herman Brown Building is located near entrance 14, on Rice Boulevard. Near this entrance there should be adequate visitor parking available. If not, proceed west to the stadium lot.

For a map and further information see "Travel Information" in the Rice Mathematics Department homepage (


Please submit--by October 30th--the following information by e-mail ( or fax (713-285-5231) or snail-mail (Math. Dept., Rice University, PO Box1892, Houston, TX 77251).

Phone Number:
Class (junior, senior, etc.):
Arrival date and time:
Departure date and time:
Do you plan to attend the complimentary buffet on Friday?
Do you plan to attend the Saturday evening party?