Principal Investigator: Frank Morgan
The principal investigator will study various isoperimetric problems in Euclidean space and more general Riemannian manifolds. He will work on Aubin's conjecture, that the Euclidean isoperimetric inequality continues to hold in a simply connected space of nonpositive curvature, at least for small volumes. More general isoperimetric problems in Euclidean space involve clusters of several given volumes, crystalline energies dependent on direction, and immiscible fluids, for which interface cost depends on the fluids separated. He will work on the Double Bubble Conjecture for general volumes and in higher dimensions. A distinguishing feature of many of these problems is the appearance of singularities.
Isoperimetric problems consider the cost of enclosing volume: soap bubbles enclosing volumes of air, crystalline arrangements of molecules in materials, the interplay between area and volume in the curvature of the universe. Singularities often play a critical role: soap films meeting in threes, structural defects in materials, black holes in the universe. Fundamental questions remain open. The Double Bubble Conjecture, for example, says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air. The recent computer proof for the case of equal volumes by Hass and Schlafly can be traced back to work by Morgan's NSF undergraduate research Geometry Group, which will continue to work on the general problem. Morgan gives some forty talks a year. To popular audiences he likes to explain that the way to understand the geometry of the universe is first to understand the geometry of soap bubbles. He has a Math Chat TV show and column, both available at the Mathematical Association of America website at http://www.maa.org, and is currently making final preparations for the publication of a popular Math Chat Book.
