``Regularity questions in the geometric calculus of variations and in geometric flow problems'' (Leon Simon and Brian White)
Leon Simon plans to pursue various questions related to the structure of the singular sets of minimal submanifolds and energy minimizing maps. More specifically Simon aims to establish first order regularity (i.e. that the singular set lies in a locally finite union of continuously differentiable submanifolds) for various multiplicity 1 classes near points where there is a tangent cylinder with a cross-section which admits a calibration. Such a result would in particular apply to mod-2 minimizers near ``top-dimensional'' singular points. Brian White plans to study regularity properties of mean curvature flow, including non-uniqueness properties and ``fattening.'' In addition he will continue his work on the singular structure of minimizing cones with coefficients in a metric group, and his work on 2 dimensional minimal surfaces with particular emphasis on the study of branch points.
An understanding of singularities, and how singularities are formed, is a fundamental element in our overall understanding of many physical and geometric phenomena. For example, in cosmology singularities of space-time (e.g. ``black holes'') play a fundamental role, and the understanding of singularity formation in geometric flow problems is a key ingredient in the approach of Hamilton, Perelman and others to Thurston's Geometrization Conjecture. Likewise in the study of the ``canonical'' objects which arise naturally in topology and geometry, singularities arise in a very natural and unavoidable manner, and the understanding of these singularities is an absolutely fundamental problem. As with most non-linear phenomena, there is not a single general theory which applies in a wide range of different contexts. Rather, each different context has its own collection of effective techniques, and it is the development and application of such techniques in the context of the geometric calculus of variations which is the focus of the present research proposal. Specifically, Simon and White propose to continue their efforts toward a more complete understanding of singularities, and how they are formed, in the context of area minimizing submanifolds and energy minimizing maps, and in the context of various geometric flow problems.
