Math 428, Optimal Geometry, Spring 1999

Meetings:
in 341 Atgeld Hall, MWF at 10am (sect C1).
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, Mon,Thu 2-3pm, or by appointment.
Prerequisites:
Basic knowledge of the geometry of curves and surfaces in space, as from Math 323 or Math 423.
Recommended Texts:
Oprea, Differential Geometry and Its Application, Prentice Hall
Morgan, Riemannian Geometry: A Beginner's Guide, 2nd Ed, A K Peters
Morgan, Geometric Measure Theory: A Beginner's Guide, 2nd Ed, Academic Press
Outline:
This course will cover variational problems in geometry, primarily the geometry of surfaces in (euclidean or spherical) space. Specifically, we will consider problems of minimizing area (leading to minimal surfaces, or constant-mean-curvature surfaces if there is a volume constraint) and of minimizing elastic bending energy (leading to Willmore surfaces), among others. Such surfaces arise physically in soap films, foams, cell membranes and elsewhere. For some of these problems, there is not yet enough theory to get explicit solutions, except numerically. Students will learn to use the Surface Evolver to find numerical solutions for these geometric optimization problems.