Mathematical Sciences: Singularity Behavior in Some Geometric Variational Problems

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The principal investigator will study the behavior of singularities arising in harmonic maps, minimal surfaces, and certain problems in mechanics. The questions to be considered include the possibility of prescribing the location of singularities and their behavior as well as questions on the structure of spaces of harmonic maps with singularities. Specific investigations will focus on liquid crystals, ferromagnetics, p-harmonic maps, and geodesics in varieties. Many functions, mappings, and spaces arising in models of our physical surroundings involve singularities. Two of the most familiar settings in which the study of singularities plays an important role are liquid crystal and minimal surfaces. Liquid crystals are commonly used to display characters in digital watches and laptop computer screens. Minimal surfaces are those surfaces which span a closed curve and have smallest area. A physical example is a soap bubble which spans a wire loop. By introducing twists in the loop, one may easily produce a soap bubble with singularities. The principal investigator will study these singularities using techniques from geometric measure theory. A major aspect of the work is to develop general techniques which can then be applied to a wide range of problems.