The Isoperimetric Problem with Density
Abstract: The classical isoperimetric theorem says that the round sphere is the least-perimeter way to enclose given volume in R^n. Since their appearance in Perelman’s proof of the Poincaré conjecture, there has been great interest in enhancing space with a positive density that weights both perimeter and volume. Even if the density is radial, spheres about the origin usually no longer minimize weighted perimeter for given weighted volume. We discuss some open problems and recent results using geometric measure theory, some by undergraduates.
Biographical Information: Frank Morgan works in minimal surfaces and studies the behavior and structure of minimizers in various dimensions and settings. His proof with colleagues and students of the Double Bubble Conjecture is featured at the NSF Discoveries site. He has six books: Geometric Measure Theory: a Beginner’s Guide (5th ed. 2016); Calculus Lite 2001, republished as Calculus 2012; Riemannian Geometry: a
Morgan went to MIT and Princeton, where his thesis advisor, Fred Almgren, introduced him to minimal surfaces. He then taught for ten years at MIT, where he served for three years as Undergraduate Mathematics Chairman, received the Everett Moore Baker Award for excellence in undergraduate teaching, and held the Cecil and Ida Green Career Development Chair. He spent leave years at Rice, Stanford, the Institute for Advanced Study, Princeton, and Berkshire Community College. He served on the NSF Math Advisory Committee from 1994-97, and as chair of the Hudson River Undergraduate Mathematics Conference in 1997. In January, 1993, he received an inaugural MAA national award for distinguished teaching. In 1995 he represented mathematics research at the exhibition for Congress by the Coalition for National Science Funding. He received the Allen High School Distinguished Alumni Award and an honorary doctorate from Cedar Crest College. For 1997-98 he held the first Visiting Professorship for Distinguished Teaching at Princeton University. From 2000-2002 he served as Second Vice-President of the Mathematical Association of America, from 2009-2012 as Vice-President of the American Mathematical Society, when he launched the AMS Graduate Student Blog, by and for mathematics graduate students.
Morgan served at Williams as Mathematics Department Chair and founding director of an NSF undergraduate research project. He is currently Webster Atwell ’21 Professor of Mathematics, Emeritus, and Editor-in-Chief of Notices of the American Mathematical Society.
See also Wikipedia and “A Math Chat with Frank Morgan” by Donald J. Albers, Math Horizons, The Mathematical Association of America, September 1997, 14-17.