Mathematical Sciences: RUI: Geometric Measure Theory and the Topology of 3-Manifolds

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One investigator will study the behavior and singular structure of area-minimizing surfaces using calibrations. On Riemannian manifolds calibrations provide distinguished representatives of homology classes. A wide range of open problems arising from the study of isoperimetric inequalities, materials science, and soap films compel the consideration of integrands more general than area and of new classes of surfaces. A second investigator will continue his studies of volumes of hyperbolic 3-manifolds and 3-orbifolds. These include the study of horoball packings which has shed new light on the geometry of hyperbolic 3-manifolds. An example of a calibration of the torus is the set of longitudinal circles. As in this case, elements of a calibration are homology class generators. These have minimal length. One investigator will use the emerging theory of calibrations to understand assorted integrals on area-minimizing surfaces. The second investigator will study the geometry of three-dimensional surfaces, or manifolds.