Abstract for NSF Proposal DMS-0306294, "Singularity Behavior in Some Geometric Variational Problems", Robert Hardt, P.I.
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 analytical constraints. Specified investigations focus on the relation between energy and topological obstruction in mappings between manifolds, liquid crystal materials, improper slicing of polynomial varieties, transport problems, and the regularity of relaxed energy minimizers. 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 spaces. This concentration behavior corresponding to any rational homotopy invariant of the target mapping can now be described. Singularities in liquid crystal materials will be also studied in several 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. Combined transport systems such as occurs in postal routes, the circulatory system, and plant root systems can be described variationally with recently discovered geometric structures.
Underlying many physical phenomena is a least energy principle whereby certain configurations or fields or geometric shape are distinguished by their property of having less energy or area than competing objects. The external constraints often lead to singularities, which are 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, 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 observation and experiment all benefit each other and all have a crucial scientific role. The present proposed research has many problems motivated by and applicable to science and engineering. The new geometric variational tools are not presently well-known in the broader scientific community, and communication of these mathematical topics and techniques will be very useful.
