RUI: Manifolds with Density and Isoperimetric Problems

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Frank Morgan and his students will study manifolds with density, a generalization of Riemannian manifolds, long prominent in probability and of rapidly growing interest in geometry and applications. The density function weights volume and area equally, unlike a conformal change of metric. Manifolds with density are the smooth case of Gromov's mm spaces, although we also consider singularities. The grand goal is to generalize appropriate parts of Riemannian geometry to manifolds with density. Advances should improve our understanding of Riemannian geometry; for example, Ricci curvature has many different generalizations to manifolds with density, which just happen to coincide for Riemannian manifolds. Isoperimetric problems provide an excellent entry point. Isoperimetric theorems on Gauss space, the premier example of a manifold with density, have had applications in probability theory, in isoperimetric problems in Riemannian geometry, and specifically in Perelman's work on the Poincaré Conjecture. Methods will include standard and innovative applications of geometric measure theory, Riemannian geometry, and second variation. Spaces with singularities are central to theory and applications, and isoperimetric problems once again provide an excellent entry point. The density we consider on a ann-dimensional manifold (such as a 2-dimensional surface or a 3-dimensional universe, perhaps with singularities), is the same kind of density one considers in freshman physics, a weighting that varies from point to point. Such densities arise naturally throughout mathematics; recent applications include Brownian motion in physics, stock option pricing, and Perelman's work on the Poincaré Conjecture. Studying this natural generalization provides new insights into classical geometry. As one important example, for the classical geometry of unit density, the _isoperimetric problem_, central to geometry research since the time of the Ancient Greeks, seeks a region of given volume of least perimeter; in Euclidean space, the solution is a round ball. On a manifold with variable density, the isoperimetric problem seeks the region of given weighted volume of least weighted perimeter. Work in the more general context provides new results in the classical context. Morgan's undergraduate research students have solved interesting sample cases and are continuing the work. Morgan lectures widely at venues ranging from popular forums, high schools, and summer schools for students and young faculty to university colloquia and research seminars.