Singularity Behavior in Some Geometric Variational Problems

William Marsh Rice University, Houston TX

Investigators

Abstract

Abstract Award: DMS-0072486 Principal Investigator: Robert M. Hardt This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, or geometric structures subject to geometric or analytic constraints. Specific investigations focus on the relation between energy and topological obstruction in mappings between manifolds, ferromagnetic and liquid crystal materials, improper slicing of polynomial varieties, compactness of currents in Carnot groups, and the regularity of relaxed energy minimizers. We will investigate the energy concentration along area-minimizing sets for limits of singularities of p-energy minimizing maps as p approaches a critical power. In various higher dimensional cases, energy concentration of limits of smooth mappings may occur along sets of infinite measure and is related to homotopically nontrivial mappings of spheres. Singularities in ferromagnetic and liquid crystal materials will be also studied in both stationary and dynamic contexts. The theories of improper intersections of polynomial zero sets from algebraic geometry will be investigated to understand related behavior in analysis and partial differential equations. A theory of currents in Carnot groups will be studied with an eye on applications to variational problems. Underlying many physical phenomena is a least-energy principle whereby certain configurations or fields or geometric shapes are distinguished by their property of having less energy or area than competing objects. The external constraints often lead to singularities, which are special points characterized by rapid changes of structure occurring in very small spatial regions. For example, one observes dislocation faults in solids under stress, domain walls in magnetized materials, vortices in superconducting materials, liquid edges and corners in soap films, and point, curve, and surface defects in various liquid crystal materials. We deal with new mathematical structures and theories necessary to explain and predict such phenomena. In these problems, the theoretical studies of pure mathematics, the numerical computational studies of applied mathematics, and the phenomenological studies from physics all benefit each other and all have a crucial scientific role.