Math 197S: Symmetry, Geometry, and Optimization
What is this Document?
This page contains information about the Fall 1998 version of
Math 197S, entitled "Symmetry, Geometry, and Optimization" being
taught by
Robert L. Bryant.
Course Synopsis
We will look at some classical problems involving soap bubbles and films,
curves of shortest descent (Brachistochrone problems), shortest paths on
curved surfaces, and the motion and shape of elastic rods and strings.
All of these will be used as motivation for introducing the ideas
of the calculus of variations and studying how they interact with geometric
notions, such as symmetry, both in problems and solutions. If time permits,
we may study some higher dimensional problems, such as Poincaré's famous
analysis of the three-body problem in celestial mechanics.
More description
If you are interested in learning something about the field
of optimization, particularly as it applies to problems in geometry
and physics, I'd like to encourage you to consider taking a new
course, Math 197S, that is being taught this fall by Professor Bryant.
If you've ever wondered what mathematics can be applied to such
diverse problems as:
- finding the shortest graph connecting a specified set
of vertices, in the plane or in space,
- finding the shortest path between two points that lies
on a given surface containing the points,
- determining the shape of soap films and soap bubbles,
- figuring out why and how rivers meander and what this
has to do with the shape an elastic wire takes when
you clamp the ends in any given position,
- how we can most effectively use symmetry in a given
problem involving differential equations to help us
solve it,
- what physicists mean when they say space is `curved'
and how do we observe and predict these effects.
All this and more will be treated in Math 197S this fall. If
you enjoyed Frank Morgan's DUMU talk this spring, and are looking
for some way to follow up on the sort of issues that he raised,
this would be a good place ot start.
The background that Professor Bryant will be assuming is a
facility with vector calculus and some familiarity with the basics
of linear algebra and differential equations. (Don't worry, you
won't be required to know all sorts of tricks for solving differential
equations. In fact, one of the subjects of the course will be
just where these tricks come from, so Professor Bryant will be
going over this material anyway when it comes up in the course.)
There's no textbook to buy, instead Professor Bryant will
be handing out lecture notes every week. What you should bring to
the course is plenty of curiosity and a willingness to share in
the work.
What, when, and where
- Lectures: 2:15--3:30, Tuesday and Thursday, in Physics 218
- Text: None. Lecture notes will be provided by the professor.
Grading Policies
You'll be assigned problems on a regular basis and will
be expected to present your solutions in class. You'll also
have to write-up an extended project (about 15 to 20 pages) by the end of the
term, explaining and giving your solution to a problem selected by
you in consultation with Professor Bryant.
Robert L. Bryant <bryant@math.duke.edu>
email address: bryant@math.duke.edu
office 'phone: (919)-660-2805
office number: 128A Physics
Updated: 19 August 1998