Singularity Behavior in Some Geometric Variational Problems

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Abstract for DMS - 0905909Singularity Behavior in Some Geometric Variational Problems Robert Hardt (Rice University)This project lies in the area of geometric calculus of variations, which treats the formation and behavior of singularities and concentration structures for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. The first specific class of projects involves continuing work with T. Riviere, on relations between the pth power energy of a map between Riemannian manifolds and its homotopy class. In various higher dimensional cases, energy concentration of limits of smooth mappings to a manifold may produce new geometric topologically nontrivial objects and is related to the nonvanishing homotopy of the manifold. This concentration behavior corresponding to any nontorsion homotopy invariant can now be described, and bubbling related to torsion invariants is being investigated for variational problems. We also are attacking higher order Sobolev spaces which seem more natural for certain homotopy classes, but for which basic approximation results and constructions have not been previously studied. Work with Thierry De Pauw involves the study of chains, cochains, charges, and the higher dimensional calculus of variations in general metric spaces with general coefficient groups. We consider a variety of mass-type functionals and the notion of a flat chain which generalizes the finite mass metric-space currents of Ambrosio-Kirchheim and the rectifiable and flat Euclidean-space G- chains of B.White. Semi-algebraic maps, chains, forms, and various structures generalized from geometric measure theory continue proving useful in work with Pascal Lambrechts on the topology of algebraic varieties, including the real homotopy theory. Also metric properties of varieties are to be approached using special classes of metric chains and cochains. Other studies include microstructure computation, combined transport-shape problems with applications to imaging, and the existence and regularity of optimal trusses.Solutions to many variational problems in both pure and applied mathematics often are forced to have singularities, that is, to involve regions where large oscillations occur. For example a nematic liquid crystal material in a spherical container whose optical axis is forced to point outward on the container necessarily will have singularities inside (observable through cross-polarizers or x-ray diffraction). In this example the optical axis has an energy density, which measures its local rate of change and whose integral tends to have a minimum value among all possible configurations. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of these problems. We have derived new notions which allow the treatment and precise geometric and analytic description of a wide variety of problems from soap films (which locally minimize area) and their higher dimensional generalizations to optimal transport paths in various complex media. The goal in these applications of geometric calculus of variations is to develop sufficient mathematical tools to model, compute, and predict physical behavior. Geometric constraints which occur naturally in many physical problems have led to new mathematical and computational issues. In particular, three that we are now studying involve flow problems in image processing, microstructure formation in certain crystalline materials, and geometric analysis of large data sets.