8:00 pm (BL131): Frank Jones
(Rice University)
Title: CALCULUS OF VARIATIONS: What Does "VARIATIONS"
Mean?
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!
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.
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
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)
For a map and further information see "Travel Information" in the Rice Mathematics Department homepage (http://math.rice.edu).