# 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