Abstract for DMS - 0905909 Singularity 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.
" A central geometric theme for much of twentieth century mathematics was the notion of a Riemannian manifold. This is roughly a space which at sufficiently small scales, seems, in the sense of distances and angles, to be very close to ordinary flat Euclidean space of some dimension n . (e.g., a point, curve, or surface for n = 0,1, or 2) Thus a single soap bubble is a 2 dimensional Riemannian manifold. Calculations can be done both in manifolds or with manifolds. For example, finding a shortest path in a bumpy surface produces a special curve in the surface. But some such geometric variational problems may produce spaces with singularities, which are nonmanifold points. For example, shortest networks connecting multiple points have crossings and compound soap bubbles have edges and corners. Or the ambient space may already be singular, like a jagged surface. A four dimensional example from general relativity would be a curved space-time manifold with singularities such as blackholes. Problems with both singularities in the minimizing objects and singularities in the ambient space have been addressed in this project. For the possibly very singular context of a general metric space, 50 year old theories based on manifolds, have been generalized to allow for the existence and at least partial regularity (nonsingularity) of least mass variational problems in all dimensions. The ambient singular spaces include examples ranging from exotic fractals to sets, called semi-algebraic, that are defined by polynomial equations and inequalities. An example of the latter is the robotics example of finding an optimal path in a configuration space for a ladder being carried through complicated hallways or stairwells. Semi-algebraic sets and their properties were treated in other parts of the project where new results on geometric and topological properties were given based on the behavior of special semi-algebraic subsets. Finally a third phase of the project addressed some new positive numbers for a fixed Riemannian manifold which were discovered using various energies and integrals of maps from the boundary of a ball (of some special dimension) into the Riemannian manifold. The results of the project lead to new research directions connecting several areas of pure and applied mathematics.