Mathematical Sciences: Singularity Behavior in Some Geometric Variational Problems

主要研究人员将研究调和映射、极小曲面和某些力学问题中出现的奇点的行为。要考虑的问题包括规定奇点的位置及其行为的可能性，以及关于具有奇点的调和映射的空间结构的问题。具体的研究将集中在液晶、铁磁性、p-调和映射和各种测地线上。在我们的物理环境模型中出现的许多函数、映射和空间都涉及奇点。在两个最常见的环境中，奇点研究扮演着重要角色，它们是液晶和极小表面。液晶通常用于在数字手表和笔记本电脑屏幕上显示字符。极小曲面是横跨一条闭合曲线且面积最小的曲面。一个实际的例子是横跨导线回路的肥皂泡。通过在循环中引入扭曲，人们可以很容易地产生带有奇点的肥皂泡。首席研究员将使用几何测度论中的技术来研究这些奇点。这项工作的一个主要方面是开发通用技术，然后这些技术可以应用于广泛的问题。

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.