
Mathematician Joel Spruck will spend the next year in Paris with a Guggenheim, tracking the elusive soap bubble. An expert in partial differential equations, Spruck will apply his analytical skills to modern geometric problems related to what is generally known as the Isoperimetric Problem.
"A simple spherical soap bubble," Spruck explained, "which possesses an exquisite perfection of form, arises as the equilibrium configuration of a liquid surface of least area enclosing a prescribed volume of trapped air." Unlike some mathematicians, who prefer to labor silently in the vineyard of their discipline, Spruck is delighted to talk about his work. "There is a rich area in mathematics that has developed in understanding this problem. Mathematically, a soap bubble is a closed constant mean curvature surface. The fact that a child's soap bubble is spherical is physical evidence for the conjecture that the only mathematical soap bubble is the sphere." The famous problem was finally completely solved around 1984, when it was demonstrated that there are unstable soap bubbles that are topologically different from the sphere. "I've been fascinated by this area of research over the course of my career, and now I will have the time and freedom to pursue it again." The Guggenheim Fellowship is the first in recent times to be given to a member of the Mathematics faculty at the Homewood campus. In the past, when Spruck took a sabbatical to do his work, he found himself teaching a course, and "hanging around" the Math Department talking with students or faculty. The time off was never perfectly satisfactory. "I wanted to take a year off with total freedom to work" said Spruck, who has been a member of the faculty here since 1992. "I wanted to allow time to devote myself full time to the project." The historical aspects of the Isoperimetric Problem also hold a peculiar fascination for the professor because the problem is rooted in myth. The fundamental questions raised were first attributed to Queen Dido of Tyre, according to ancient mythology. "The Queen was offered property that bordered the sea, and she was told she could have as much property as she could enclose with a certain length of rawhide," Spruck explained. "The question then was, What shape encloses the most area?" The answer, of course, is a circle. Obvious as it seems, Spruck said, it took "a couple of thousand years" to prove rigorously. In the 20th century, this has in turn led to questions about surface area and volume in higher dimensional Euclidean space, and more generally in Riemannian manifolds. This is precisely where the mathematician finds himself in hot pursuit of the elusive soap bubble. Spruck, who is presently the chairman of the Mathematics Department, said he, his wife and his 15yearold daughter are excited about the prospect of living in Paris for the next year. He will do some of his research at the University of Paris. Professor Steve Zelditch will take over as chairman of the department in July.
