Math 428, Minimal Surfaces, Spring 2002

Meetings:
in 343 Atgeld Hall, MWF at 1pm (sect E1).
Web address:
Course information is available online at http://www.math.uiuc.edu/~jms/m428
Professor:
John M. Sullivan, jms@uiuc.edu, 326 Illini Hall,
244-5930 (with answering machine); mailbox in 250 Altgeld.
Office hours:
Tentatively, Wed 11am, Thu 10am or by appointment.
Prerequisites:
Basic knowledge of the geometry of curves and surfaces in space, as from Math 323 or Math 423.
Recommended Texts:
Almgren, Plateau's Problem: An Invitation to Varifold Geometry, Revised Ed, AMS (Stud.Math.Lib 13)
Morgan, Geometric Measure Theory: A Beginner's Guide, 3nd Ed, Academic Press
Morgan, Riemannian Geometry: A Beginner's Guide, 2nd Ed, A K Peters
Oprea, Differential Geometry and Its Application, Prentice Hall
Osserman, Geometry V, Springer (Enc.Math.Sci 90)
Outline:
This course will cover variational problems in geometry, primarily the geometry of surfaces in (euclidean or spherical) space. Specifically, it will concentrate on problems of minimizing area, which lead to minimal surfaces, or constant-mean-curvature surfaces if there is a volume constraint.

This course will give an overview of the different methods (from geometric measure theory, partial differential equations, and complex analysis) which have been used to study minimal surfaces.

Minimal surfaces arise physically in soap films and foams. We will also consider their varied mathematical applications, which include the study of three-manifolds and the positive-mass conjecture in relativity.

As time permits, we will look at other geometric optimization problems, like surfaces minimizing Willmore's elastic bending energy. The course will include an introduction to Brakke's Evolver, a piece of mathematical software for numerical simulation of solutions to such problems.